1. PHYS451: MPhys Project Final Report
Non-tensorial Properties of Higher Order Vectors & Their
Combination with the Connection
Alexander C Booth
Project Supervisor - Dr. Jonathan Gratus
April 24, 2015
Abstract
The concept of combining the connection with higher order vectors on a manifold is intro-
duced, demonstrating two different ways in which this can be done. Definitions in both index
and coordinate free representations are suggested, then written in terms of useful geometric
quantities such as the torsion and curvature. Large emphasis is given to the methods which
have been developed to deal with the problem of taking the definitions from coordinate
to coordinate free. Some possible applications are described, most notably rewriting an
equation from general relativity and viewing higher order vectors as a new source of matter.
1
3. 1 Introduction
The connection is a highly useful geometric object which appears in many areas of physics
and mathematics. It is a idea that will be a familiar to many through its applications in gen-
eral relativity and fluid physics, featuring in both the geodesic deviation and Navier-Stokes
equations[10][12]. In these equations it acts as the covariant or directional derivative, providing
a way of differentiating one vector field along another vector field on a manifold. A seeming
unrelated concept at first is that of a higher order vector, introduced in a paper by Duval which
studied differential operators on manifolds[5]. Their application to systems of ordinary differen-
tial equations was then investigated by Aghasi et al in 2006, yet they remain a rather abstract
concept[1]. Although higher order vectors do not lend themselves to an intuitive introduction,
a natural relationship exists between them and connection. This relationship becomes evident
when each of their transformation laws are calculated. It will be shown that both the connection
and higher order vectors are non-tensorial, that is to say in general they are dependent on the
choice of coordinate basis. With this shared property in mind, it can be asked whether the non-
tensorial nature of the two objects can be exploited in such a way, that they can be combined to
form an overall tensor. Tensors of course do not depend on the choice of basis, a property which
makes them far more useful for constructing physical theories. This project began with nothing
more than the assumption that such tensorial objects should exist, at least when working with
the ‘lowest order,’ higher order vectors. The method then being to take products of the various
non-tensorial objects in such a way that if searching for a vectorial component for example, only
one free contravariant index is left. The transformation properties of this newly constructed
object are then worked out by direct computation, confirming whether or not a true vector has
been built. As far as we are aware, the combination of higher order vectors and the connection
in this way has not been seen before. Up until this point, research has been centred around
second and third order vectors. It is believed however that an inductive definition, describing
how the connection and a vector of arbitrary order can be combined, does exist. This possibility
will be explored in more detail in later sections.
Throughout the project, classical tensor calculus is the primary technique which is used. This is
the manipulation of tensorial and tensor-like objects using index notation. It is a very common
algebraic method which features heavily at undergraduate level, in topics such as general rela-
tivity. One of the main problems with this classical approach is that it requires reference to a
coordinate system, which in turn means the introduction of a metric. From the project’s outset,
the research has been focussed on defining in a coordinate free way, how higher order vectors
can be combined with the connection. At least at low orders, our research has found that from
this viewpoint, the concepts of torsion and curvature are naturally introduced. These are two
physical quantities which play central roles in modern theories of nature. Curvature has long
been considered in general relativity as the ‘source’ of gravity, whereas the possible significance
of torsion was only more recently recognised[12]. Potential areas of application which could ex-
ploit this natural appearance of torsion, are discussed more closely toward the end of the report.
The description of objects and physical laws without reference to a basis is not a new idea. It is
the foundation of a field known as differential geometry, an extremely powerful tool in theoretical
physics. In this language for example, all four of Maxwell’s equations can be reduced to just two,
describing fully relativistically, the electro-magnetic fields in any spacetime[12]. Furthermore,
a classical vector is no longer defined by its transformation properties, but by a set of basic
algebraic rules. It is believed that a coordinate free approach to higher order vectors has not yet
been attempted. As well as the final definitions themselves, the report puts much emphasis on
the process by which the definitions evolve from coordinate, to coordinate free. During research,
a number of tools were developed to do this effectively.
3
4. Many of the coordinate free manipulations and definitions which appear in the project involve
concepts which should be familiar. Basic knowledge of multivariable calculus along with covari-
ant differentiation and tensors in index notation is assumed. However, to aid the reader who
is unfamiliar with these ideas from the perspective of differential geometry, section 2 has been
included. This is an in depth discussion which covers all of the necessary background mathe-
matics, restated in coordinate free language. Also, appendix A has been written to support the
use of exterior calculus seen briefly in section 6. With these two parts included, it is hoped that
this document is completely self contained. That is to say, no reading beyond what is written
herein should be required. Furthermore, there is wide use of both standard and non-standard
notation. Any notation which is not explained in the main body of the report can be found in
section 8, a comprehensive glossary of all notation. Finally, the paper’s key results have been
highlighted by borders for quick reference.
2 Preliminary Mathematics
2.1 Formal Treatment of Coordinate Systems & Taylor’s Theorem
Calculating transformation laws constitutes a large part of many sections in this report, it is
important to discus therefore exactly what is meant by a coordinate system. A coordinate
xa is simply a scalar function which takes a point p = (p1, · · · , pm) on an m-dimensional
manifold M and maps it to a subset of R. It is assumed throughout that the manifold
M is m-dimensional. A coordinate system therefore is just a set of m of these functions,
(x1, · · · , xm). An alternative coordinate system is given by a different set of scalar functions
y1(x1, · · · , xm), · · · , ym(x1, · · · , xm) , which are all functions of the old coordinate functions.
The chain rule can therefore be used to relate an object O = O(x1, · · · , xm) in one coordinate
system, to that same object ˆO = ˆO(y1, · · · , ym), in another coordinate system. This convention
of ‘hatted’ and ‘un-hatted’ frames will be used throughout. For further clarity, when working in
a hatted frame, Greek indices will be used. When working in an un-hatted frame, Latin indices
will be used.
Two incredibly useful relations that will be required when investigating transformation prop-
erties will now be derived. Firstly, an expression relating the second order derivatives of frame
(x1, · · · , xm) with respect to yα and second order derivatives of frame (y1, · · · , ym) with respect
to xa.
Lemma 1. Given two coordinate frames (x1, · · · , xm) and (y1, · · · , ym), the following relation
holds true.
∂2yα
∂xa∂xb
∂xc
∂yα
= −
∂yα
∂xa
∂yβ
∂xb
∂2xc
∂yα∂yβ
(1)
Proof.
0 =
∂
∂xa
δc
b =
∂
∂xa
∂yα
∂xb
∂xc
∂yα
=
∂2yα
∂xa∂xb
∂xc
∂yα
+
∂yβ
∂xb
∂
∂xa
∂xc
∂yβ
=
∂2yα
∂xa∂xb
∂xc
∂yα
+
∂yβ
∂xb
∂yα
∂xa
∂2xc
∂yα∂yβ
Rearranging the final line gives exactly (1).
Now a slightly more complicated expression is considered, relating the third order coordinate
partial derivatives.
4
5. 2.1 Formal Treatment of Coordinate Systems & Taylor’s Theorem
Lemma 2. Given two coordinate frames (x1, · · · , xm) and (y1, · · · , ym), the following relation
holds true.
∂3y
∂xa∂xb∂xc
= −
∂yγ
∂xc
∂y
∂xd
∂2yα
∂xa∂xb
∂2xd
∂yα∂yγ
+
∂y
∂xd
∂yβ
∂xa
∂2yα
∂xb∂xc
∂2xd
∂yα∂yβ
(2)
+
∂y
∂xd
∂yα
∂xb
∂2yβ
∂xa∂xc
∂2xd
∂yα∂yβ
+
∂yγ
∂xc
∂y
∂xd
∂yα
∂xb
∂yβ
∂xa
∂3xd
∂yα∂yβ∂yγ
Proof. The result follows from partially differentiating each side of equation (1). Beginning
with the left hand side.
∂
∂yγ
∂2y
∂xa∂xb
∂xc
∂y
=
∂xd
∂yγ
∂xc
∂y
∂3y
∂xa∂xd∂xb
+
∂2y
∂xa∂xb
∂2xc
∂yγ∂y
Now the right hand side.
∂
∂yγ
−
∂yα
∂xb
∂yβ
∂xa
∂2xc
∂yα∂yβ
= −
∂xd
∂yγ
∂2yα
∂xb∂xd
∂yβ
∂xa
∂2xc
∂yα∂yβ
+
∂xd
∂yγ
∂yα
∂xb
∂2yβ
∂xa∂xd
∂2xc
∂yα∂yβ
+
∂yα
∂xb
∂yβ
∂xa
∂3xc
∂yα∂yβ∂yγ
Rearranging and multiplying each side by ∂yγ
∂xf
∂y
∂xg gives
∂yγ
∂xf
∂y
∂xg
∂xd
∂yγ
∂xc
∂y
∂3y
∂xa∂xd∂xb
= −
∂yγ
∂xf
∂y
∂xg
∂2yα
∂xa∂xb
∂2xc
∂yγ∂yα
+
∂yγ
∂xf
∂y
∂xg
∂xd
∂yγ
∂2yα
∂xb∂xd
∂yβ
∂xa
∂2xc
∂yα∂yβ
+
∂yγ
∂xf
∂y
∂xg
∂xd
∂yγ
∂yα
∂xb
∂2yβ
∂xa∂xd
∂2xc
∂yα∂yβ
+
∂yγ
∂xf
∂y
∂xg
∂yα
∂xb
∂yβ
∂xa
∂3xc
∂yα∂yβ∂yγ
=⇒ δd
f δc
g
∂3y
∂xa∂xd∂xb
= −
∂yγ
∂xf
∂y
∂xg
∂2yα
∂xa∂xb
∂2xc
∂yγ∂yα
+ δd
f
∂y
∂xg
∂2yα
∂xb∂xd
∂yβ
∂xa
∂2xc
∂yα∂yβ
+δd
f
∂y
∂xg
∂yα
∂xb
∂2yβ
∂xa∂xd
∂2xc
∂yα∂yβ
+
∂yγ
∂xf
∂y
∂xg
∂yα
∂xb
∂yβ
∂xa
∂3xc
∂yα∂yβ∂yγ
=⇒
∂3y
∂xa∂xf ∂xb
= −
∂yγ
∂xf
∂y
∂xg
∂2yα
∂xa∂xb
∂2xg
∂yγ∂yα
+
∂y
∂xg
∂2yα
∂xb∂xf
∂yβ
∂xa
∂2xg
∂yα∂yβ
+
∂y
∂xg
∂yα
∂xb
∂2yβ
∂xa∂xf
∂2xg
∂yα∂yβ
+
∂yγ
∂xf
∂y
∂xg
∂yα
∂xb
∂yβ
∂xa
∂3xg
∂yα∂yβ∂yγ
=⇒
∂3y
∂xa∂xb∂xc
= −
∂yγ
∂xc
∂y
∂xd
∂2yα
∂xa∂xb
∂2xd
∂yα∂yγ
+
∂y
∂xd
∂2yα
∂xb∂xc
∂yβ
∂xa
∂2xd
∂yα∂yβ
+
∂y
∂xd
∂yα
∂xb
∂2yβ
∂xa∂xc
∂2xd
∂yα∂yβ
+
∂yγ
∂xc
∂y
∂xd
∂yα
∂xb
∂yβ
∂xa
∂3xd
∂yα∂yβ∂yγ
This is exactly equation (2).
Since only the transformation properties of vectors up to and including third order are dealt
with in this report, there is no need for any higher order relationships.
In section 3, the most general basis of a third order vector is stated and proved. Central to this
proof is the following version of Taylor’s theorem[9].
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6. 2.2 First Order Vectors & 1-Forms
Theorem 3. Given any function f ∈ ΓΛ0M that is differentiable at least q-times and described
by coordinates (x1, · · · , xm), it can be expressed about the point p = (0, · · · , 0) as
f(x1
, · · · , xm
) =
|I|≤q
DIf
I! p
xI
+
|I|=q
EI(x1
, · · · , xm
)xI
(3)
Where E(x1, · · · , xm) is a finite error term with the property that it is continuous and
lim
xa→0
EI(x1
, · · · , xm
) = 0 (4)
A full explanation of multi-index notation can be found in the glossary of notation, section 8.
Equipped with this formal treatment of coordinate systems, vector fields are considered next.
2.2 First Order Vectors & 1-Forms
Before talking about higher order vectors, it is useful to introduce the coordinate free definition
of a ‘regular’ vector. Regular vectors refer to the type of vector usually dealt with in classic
physics, such as those in mechanics. That is to say, in index notation they are defined as all
objects u = ua ∂
∂xa , whose components ua obey the following transformation law[12].
ˆuα
=
∂yα
∂xa
ua
(5)
For the remainder of the document, these vectors will be known as first order vectors. The
claim that all first order vectors can be written in the form u = ua ∂
∂xa will be covered by a more
general theorem in section 3.
In the language of differential geometry, a vector field v is defined as a function which takes a
scalar field f and gives v f , a new scalar field[11]. Here angular brackets are used for clarity,
avoiding any confusion between this type of action and simply listing a function and its variables.
For example, g(x, y) is a scalar field in x and y. In order for this to be a full and completely
equivalent definition of a vector field, the function must satisfy two properties[11].
Definition 4. Given f, g ∈ ΓΛ0M, a vector field v ∈ ΓTM is a function v : ΓΛ0M → ΓΛ0M,
with v : f → v f such that it satisfies
v f + g = v f + v g (6)
v fg = fv g + gv f (7)
Equation (6) ensures that a vector acting upon a sum of scalar fields, gives a sum of the vector
acting on each scalar. This is known as plus linearity. Equation (7) says that a vector acting
upon a product of scalars obeys the Leibniz rule.
Useful to keep in mind, yet far less important for the purposes of this project are 1-form fields.
They are defined in a similar way to vector fields but instead of following a Leibniz rule, they
are ‘f-linear’[11].
Definition 5. Given f ∈ ΓΛ0M and v, w ∈ ΓTM, a 1-form field µ ∈ ΓΛ1M is a function
µ : ΓTM → ΓΛ0M, with (v) → µ : v such that it satisfies
µ : (v + w) = µ : v + µ : w (8)
µ : (fv) = fµ : v (9)
6
7. 2.3 The Connection
Intuitively if a particular operation is f-linear, it means that a scalar field can be ‘pulled out’ of
the operation. This is what is shown in equation (9). Exactly what is meant by f-linearity will
become clear as the report moves forward. An alternative definition of a tensor for example is
to view them as objects that are both plus and f-linear. Note the use of the colon, also seen
later in the project to represent a higher order vector combining with the connection.
In complete analogy with a first order vector field, given an m-dimensional manifold M with
coordinates (x1, · · · , xm), dxa for a = 1, · · · , m denotes a 1-form basis on this manifold[12]. It
is possible to construct differential forms of arbitrary degree, the process by which this is done
is explained in section A. These higher order differential forms are a far more well established
tool in mathematics and physics than higher order vectors.
2.3 The Connection
Of central importance to this project is the connection, appearing greatly in sections 5
onwards. As previously explained, when dealing with vectors it is sometimes called the covariant
derivative and represents differentiation of one vector field along another. This research only
considers the combination of the connection with first and higher order vectors, although its
action is defined on any tensor. Before considering the connection in a coordinate free way, it
is useful to look at it using index notation. To do this, the following objects must be defined.
Definition 6. Given a general connection on M,
Γc
ab = ∂a ∂b xc
(10)
Are the Christoffel Symbols of the second kind[11].
It can be shown that given a metric compatible and torsion free connection, the Christoffel
symbols are objects which can be written as a product of partial derivatives of the metric and
the inverse metric[11]. Metric compatibility describes the condition that the covariant derivative
of the metric is zero. In this project, the explicit form of these symbols is never required. With
definition 6 in mind and using classical tensor analysis, the covariant derivative of a vector
v ∈ ΓTM in the direction of a vector u ∈ ΓTM can be calculated.
( uv)c
= ua
∂a vb
∂b
c
= ua
∂a vb
c
∂b + ua
vb
∂a ∂b
c
(11)
The covariant derivative of vb, ∂a vb is just the partial derivative of vb with respect to xa and
using equation (10) it follows that
( uv)c
= ua ∂vc
∂xa
+ ua
vb
Γc
ab = u vc
+ ua
vb
Γc
ab (12)
As with a first order vector, defining the connection in a coordinate free way involves viewing
it as a function[11][12].
Definition 7. Given first order vector fields u, v, w ∈ ΓTM and f ∈ ΓΛ0M, a general con-
nection on M is a function : ΓTM × ΓTM → ΓTM, with (u, v) → uv such that it
satisfies
u (v + w) = uv + uw u (fv) = u f v + f uv (13)
(u+w)v = uv + wv (fu)v = f uv (14)
The equations in (13) ensure that the connection is plus linear and Leibniz in the vector being
differentiated. The equations in (14) on the other hand ensure that it is plus linear in the
direction being differentiated in, but instead of being Leibniz in this argument it is f-linear.
One further piece of notation featuring later in the text is 0, which is used to denote a torsion
free connection.
7
8. 2.4 Torsion & Curvature
2.4 Torsion & Curvature
Torsion and curvature are both tensorial quantities which appear in differential geometry, pro-
viding a way to quantify the warped nature of a particular manifold. Although Einstein’s
theory of gravity assumes a Levi-Civita connection, that is to say a connection which is metric
compatible and torsion free, curvature plays a central role. The Riemann curvature tensor fea-
tures explicitly not only in Einstein’s equation but also the geodesic deviation equation. This
relation quantifies the tidal forces between particles on neighbouring geodesics, a second or-
der effect[12]. It would be reasonable to assume therefore that somewhere in their definitions,
second derivatives and products of derivatives are involved.
Definition 8. Given first order vector fields u, v, w ∈ ΓTM, the curvature R of a connection
on M, is a function R : ΓTM × ΓTM × ΓTM → ΓTM, with (u, v, w) → R(u, v)w such
that
R(u, v)w = u vw − v uw − [u,v]w (15)
It is plus and f-linear in all of its arguments.
This object is sometimes known as the curvature vector[11]. The equivalent coordinate expres-
sion, viewing the curvature as a classical (1, 3) tensor is given by the following equation[12].
Re
bac = Γd
abΓe
cd + ∂aΓe
cb − Γd
cbΓe
ad − ∂cΓe
ab (16)
This report mostly deals with the coordinate free result.
Despite Einstein’s gravity only talking about the Levi-Civita connection, where possible in the
report, new objects are kept completely general. There are many alternative theories of gravity
such as Einstein-Cartan theory, which do involve torsion[4]. As such, the torsion tensor is now
defined[11][12].
Definition 9. Given first order vector fields u, v ∈ ΓTM, the torsion T of a connection on
M is a function T : ΓTM × ΓTM → ΓTM, with (u, v) → T (u, v) such that
T (u, v) = uv − vu − [u, v] (17)
It is plus and f-linear in all of its arguments.
As a classical tensor[12].
T c
ab = Γc
ab − Γc
ba (18)
Both the torsion and the curvature will be seen again in section 5, where an equation relating
the two will be required. An expression which does just this is Bianchi’s First Identity[8].
Ω R(u, v)w = Ω ( uT )(v, w) + T (T (u, v), w) (19)
Here Ω denotes the cyclic sum over u, v and w. Most notably when working in the torsion free
regime, this immediately reduces rather nicely to the following[12].
R(u, v)w + R(w, u)v + R(v, w)u = 0 (20)
All of the tools which form the foundation of the report’s proofs and definitions have now been
introduced. Next it is shown how higher order vectors are defined mathematically.
8
9. 3 Introducing Higher Order Vectors
The main focus of this thesis is higher order vectors. There is a complete theory surrounding
differential forms of arbitrary order, yet work on arbitrary order vectors rarely features in the
literature. As has been mentioned, the notion of a higher order operator was introduced by
Duval in 1997 and their application to ordinary differential equations was recognised shortly
after[1][5]. All of what are believed to be new results established in this project, involve second
and third order vector fields. In the first part of this section therefore, particular attention is
paid to these. The second part of this section, section 3.2, introduces how higher order vectors
can be defined in general. Such a definition would be necessary if our research were to be
extended to arbitrary orders.
3.1 Second & Third Order Vectors
Beginning with the most simple extension to regular vector fields, second order vector fields.
The space of all second order vector fields is denoted ΓT2M. As with first order vectors, it
is possible to define them in a coordinate free way by means of a plus linearity condition and
Leibniz rule.
Definition 10. Given f, g ∈ ΓΛ0M, a second order vector field U ∈ ΓT2M is a function
U : ΓΛ0M → ΓΛ0M, with U : f → U f such that it satisfies
U f + g = U f + U g (21)
U fg = fU g + gU f + U(1,1) f, g (22)
Where
U(1,1) −, − : ΓΛ0
M × ΓΛ0
M → ΓΛ0
M , U(1,1) ∈ Γ (TM ⊗ TM) (23)
It is clear that this definition is similar to that of a first order vector field, equation (22) however
says that second order vector fields do not obey the standard Leibniz rule. When acting upon a
product of scalar fields there is the usual Leibniz part fU g +gU f as would be expected, but
then an extra term U(1,1) f, g . This object belongs to the set Γ (TM ⊗ TM) and is defined as
a function U(1,1) −, − : ΓΛ0M × ΓΛ0M → ΓΛ0M. These two properties mean that it is itself
Leibniz in both arguments. That is to say given h ∈ ΓΛ0M also
U(1,1) fg, h = fU(1,1) g, h + gU(1,1) f, h , U(1,1) f, gh = gU(1,1) f, h + hU(1,1) f, g (24)
Many will have written down a second order vector without realising. The Lie bracket of vectors
for example u, v , itself a vector, when written in a coordinate free way expands as
u, v f = u v f − v u f = u ◦ v f − v ◦ u f = (u ◦ v − v ◦ u) f (25)
Here the new notation u ◦ v is introduced, meaning ‘u operate v’. It is straightforward to
show that the object u ◦ v is a second order vector (see section 5.1). This simple example also
highlights the fact that it is possible to write a first order vector as a linear combination of
second order vectors. Not only does this rule extend to higher order vectors but implies that
ΓTM ⊂ ΓT2M. When looking for a general basis for this new space, it should include terms
similar to those bases of a first order vector. It will be proven at third order, but for now simply
stated in lemma 11, the most general form a second order vector field can take.
9
10. 3.1 Second & Third Order Vectors
Lemma 11. Any second order vector field U ∈ ΓT2M can be expressed
U = Ua ∂
∂xa
+
Uab
2
∂2
∂xa∂xb
(26)
Where
Ua
= U xa
, Uab
= U(1,1) xa
, xb
(27)
Proof. This result will follow immediately from lemma 13, since ΓT2M ⊂ ΓT3M. That is to
say, a second order vector is effectively a special case of a third order vector.
It is useful to note the symmetry Uab = Uba due to the equality of mixed partial derivatives.
This observation will be a of great importance in later sections. Such a basis makes sense
if second order vectors are viewed in analogy with differential forms. An example of a basis
element for a general 2-form is dxa ∧ dxb (see appendix A), ∂2
ab can be written ∂a ◦ ∂b. The
transformation and symmetry properties of second order vector fields can be exploited in such a
way, that they can be combined with the connection to give a new first order vector field. This
will be demonstrated in section 5. A third order vector field will now be defined, the extension
is not as obvious as perhaps would be expected.
Definition 12. Given f, g ∈ ΓΛ0M, a third order vector field V ∈ ΓT3M is a function
V : ΓΛ0M → ΓΛ0M, with V : f → V f such that it satisfies
V f + g = V f + V g (28)
V fg = fV g + gV f + V(1,2) f, g + V(2,1) f, g (29)
Where
V(1,2) −, − : ΓΛ0
M × ΓΛ0
M → ΓΛ0
M , V(1,2) ∈ Γ TM ⊗ T2
M (30)
V(2,1) −, − : ΓΛ0
M × ΓΛ0
M → ΓΛ0
M , V(2,1) ∈ Γ T2
M ⊗ TM (31)
Defining V(1,2) and V(2,1) as belonging to sets Γ TM ⊗ T2M and Γ T2M ⊗ TM respectively
means that V(1,2) is Leibniz in its first argument but not in its second and V(2,1) is Leibniz in
its second but not in its first. This is best interpreted by introducing the following quantity.
V(1,1,1) −, −, − : ΓΛ0
M×ΓΛ0
M×ΓΛ0
M → ΓΛ0
M , V(1,1,1) ∈ Γ (TM ⊗ TM ⊗ TM) (32)
This object is Leibniz in all of its arguments and is analogous to the additional term in equation
(22). With this in mind and taking f, g, h ∈ ΓΛ0M, the Leibniz properties of V(1,2) can be
written down less abstractly.
V(1,2) fg, h = fV(1,2) g, h + gV(1,2) f, h (33)
V(1,2) f, gh = gV(1,2) f, h + hV(1,2) f, g + V(1,1,1) f, g, h (34)
Similar equations apply for V(2,1). To see why such a definition may be reasonable, consider the
specific case that V ∈ ΓT3M is such that for u ∈ ΓTM and U ∈ ΓT2M, V = u◦U. It is shown
later in lemma 26 that u ◦ U is indeed a third order vector field. Simply using the definition
V fg = fV g + gV f + V(1,2) f, g + V(2,1) f, g (35)
10
11. 3.1 Second & Third Order Vectors
= f(u ◦ U) g + g(u ◦ U) f +
1
2
(u ⊗ U) f, g + (U ⊗ u) f, g
Writing it in this way and comparing the two lines, it is clear to see why V(1,2) −, − would be
Leibniz in the first argument and V(2,1) −, − Leibniz in the second argument. Definition 12
can be used to show that third order vectors have the following basis in general.
Lemma 13. Any third order vector field V ∈ ΓT3M can be expressed
V = V a ∂
∂xa
+
V ab
2
∂2
∂xa∂xb
+
V abc
6
∂3
∂xa∂xb∂xc
(36)
Where
V a
= V xa
, V ab
= V(1,2) xa
, xb
+ V(2,1) xa
, xb
, V abc
= V(1,1,1) xa
, xb
, xc
(37)
Proof. To prove this result requires Taylor theorem as stated in theorem 3, to express f ∈ ΓΛ0M
about point p ∈ M. The action of a general third order vector on this scalar field will then be
considered. It will be shown that the lemma holds for a third order vector, V ∈ T3
p M at point
p = (0, · · · , 0). This is sufficient since there is always the freedom to chose the origin of the
coordinate system used. Furthermore the third order vector basis only involves derivatives up
to third order, this means the error term can be introduced at this order. Hence,
f(x1
, · · · , xm
) = f
p
+
∂f
∂xa
p
xa
+
1
2
∂2f
∂xa∂xb
p
xa
xb
+
1
6
∂3f
∂xa∂xb∂xc
p
xa
xb
xc
+ Eabcxa
xb
xc
(38)
Therefore
V f = V f p
+ xa
∂af p
+
1
2
xa
xb
∂2
abf p
+
1
6
xa
xb
xc
∂3
abcf p
+ Eabcxa
xb
xc
= V f p
+ V xa
∂af p
+ V
1
2
xa
xb
∂2
abf p
+ V
1
6
xa
xb
xc
∂3
abcf p
+ V Eabcxa
xb
xc
= 0 + xa
p
V ∂af p
+ ∂af p
V xa
+ V(1,2) xa
, ∂af p
+ V(2,1) xa
, ∂af p
+
1
2
(xa
xb
) p
V ∂2
abf p
+ ∂2
abf p
V xa
xb
+ V(1,2) xa
xb
, ∂af p
+ V(2,1) xa
xb
, ∂af p
+
1
6
(xa
xb
xc
) p
V ∂3
abcf p
+ ∂3
abcf p
V xa
xb
xc
+ V(1,2) xa
xb
xc
, ∂af p
+ V(2,1) xa
xb
xc
, ∂af p
+ Eabc p
V xa
xb
xc
+ (xa
xb
xc
) p
V Eabc + V(1,2) Eabc, xa
xb
xc
+ V(2,1) Eabc, xa
xb
xc
When a vector acts upon a constant, the result is 0. In addition recall that point p is in fact
the origin, therefore all coordinate functions evaluated at p are zero. Finally, by definition of
the error function in Taylor’s theorem (see theorem 3), it is zero in the limit that (x1, · · · , xm)
tends to (0, · · · , 0). Applying all of these observations implies that
V f = ∂af p
V xa
+
1
2
∂2
abf p
V xa
xb
+
1
6
∂3
abcf p
V xa
xb
xc
+ V(1,2) Eabc, xa
xb
xc
+ V(2,1) Eabc, xa
xb
xc
11
12. 3.2 Vectors of Arbitrary Order
= ∂af p
V xa
+
1
2
∂2
abf p
xa
p
V xb
+ xb
p
V xa
+ V(1,2) xa
, xb
+ V(2,1) xa
, xb
+
1
6
∂3
abcf p
(xb
xc
) p
V xa
+ xa
p
V xb
xc
+ V(1,2) xa
, xb
xc
+ V(2,1) xa
, xb
xc
+ xa
p
V(1,2) Eabc, xb
xc
+ (xb
xc
) p
V(1,2) Eabc, xa
+ V(1,1,1) Eabc, xa
, xb
xc
+ xa
p
V(2,1) Eabc, xb
xc
+ (xb
xc
) p
V(2,1) Eabc, xa
= ∂af p
V xa
+
1
2
∂2
abf p
V(1,2) xa
, xb
+ V(2,1) xa
, xb
+
1
6
∂3
abcf p
V(1,2) xa
, xb
xc
+ V(2,1) xa
, xb
xc
+ V(1,1,1) Eabc, xa
, xb
xc
= ∂af p
V xa
+
1
2
∂2
abf p
V(1,2) xa
, xb
+ V(2,1) xa
, xb
+
1
6
∂3
abcf p
xb
p
V(1,2) xa
, xc
+ xc
p
V(1,2) xa
, xb
+ V(1,1,1) xa
, xb
, xc
+ xb
p
V(2,1) xa
, xc
+ xc
p
V(2,1) xa
, xb
+ xb
p
V(1,1,1) Eabc, xa
, xc
+ xc
p
V(1,1,1) Eabc, xa
, xb
= ∂af p
V xa
+
1
2
∂2
abf p
V(1,2) xa
, xb
+ V(2,1) xa
, xb
+
1
6
∂3
abcf p
V(1,1,1) xa
, xb
, xc
= V xa
∂a +
1
2
V(1,2) xa
, xb
+ V(2,1) xa
, xb
∂2
ab +
1
6
V(1,1,1) xa
, xb
, xc
∂3
abc f p
= V a
p
∂a +
1
2
V ab
p
∂2
ab +
1
6
V abc
p
∂3
abc f p
Since this is true for all f
V = V a
p
∂a +
1
2
V ab
p
∂2
ab +
1
6
V abc
p
∂3
abc (39)
This expression is equation (36) evaluated at point p. A vector field is simply a collection of
vectors at points, therefore lemma 13 holds.
As with the basis of second order vector fields, this basis makes sense by analogy with three
form fields, whose basis is of the form dxa ∧ dxb ∧ dxc. Third order vector fields become very
important in section 5. Like second order vector fields, their specific transformation properties
and natural symmetry of coefficients can be exploited. Note once again that V ab = V ba and
V abc = V cba = V cab = · · · . They can be combined with the connection to construct both
vectorial and non-trivial scalar quantities. It will next be shown how a vector of nth order can
be defined.
3.2 Vectors of Arbitrary Order
Comparing the different bases of first, second and third order vector fields which have already
been seen, there is a clear pattern emerging. Although this report will not explicitly use vectors
of fourth order and above, such a definition would be useful if research in this area were to
be taken any further. The definition of an nth order vector field will now be given in a form
introduced by Gratus, Banachek, Ross and Rose but is as yet unpublished. Definitions 10 and
12 are of course specific cases of this more general definition.
12
13. 3.3 Jet Spaces
Definition 14. Given f, g ∈ ΓΛ0M, an nth order vector field W ∈ ΓTnM is a function
W : ΓΛ0M → ΓΛ0M, with W : f → W f such that it satisfies
W f + g = W f + W g (40)
W fg = fW g + gW f +
a+b=n
W(a,b) f, g (41)
Where
W(a,b) −, − : ΓΛ0
M × ΓΛ0
M → ΓΛ0
M , W(a,b) ∈ Γ Ta
M ⊗ Tb
M (42)
In the case of a first order vector field n = 1, W(i,j) = 0 for all i and j since ΓT0M is not
explicitly defined. The summation runs over all possible combinations of a and b such that
a + b = n. It is clear to see that at large orders, things quickly become complicated. Take for
example W ∈ ΓT5M, equation (41) will include a term of the form W(3,2) ∈ Γ T3M ⊗ T2M .
In order to do any meaningful calculations, W(3,2) must be broken down into terms which are
Leibniz in most or all of their arguments, using a similar approach to that seen in the third
order case.
The most general basis of an nth order vector is as one would expect by extension of lemmas
11 and 13. The proof of the exact expression is however beyond the level of the report and is
largely irrelevant since our research involves vectors of order no higher than three.
3.3 Jet Spaces
It has been repeatedly highlighted that it is the specific transformation properties of higher order
vector components and the connection, which allows them to be combined in a meaningful way.
The foundation of this ‘natural relationship’ is in prolongation and jet spaces. Here a brief
overview of these ideas is presented. Consider first of all a scalar function f : M → R such that
f = f(x1, · · · , xm). The rth order jet space of f is denoted Jr(M → R) and is best understood
by considering the first few values of r. The zero jet of f, J0(M → R) is simply the set of all
functions {f : M → R} and is the bundle R × M over M[15]. It can be described therefore
by coordinates (x1, · · · , xm, f), meaning that the dimension of this jet space is m + 1. Higher
order jets can then be defined in a similar way, the table below shows the next three orders
of jets of f along with their corresponding coordinate system and dimension. The fractions
which appear in the expressions for the dimension of each space, are there to account for the
symmetries fab = fba, fabc = fbca = fcab = · · · and so on.
Jets of f. Bundle. Coordinate System. a, b, c ∈ [1, · · · , m] Dimension.
J0(M → R) R × M (xa, f) m + 1
J1(M → R) R × T∗M (xa, f, fa) 2m + 1
J2(M → R) - (xa, f, fa, fab) 1
2m2 + 2m + 1
J3(M → R) - (xa, f, fa, fab, fabc) 1
6m3 + 1
2m2 + 2m + 1
Table 1: Jets of f.
Here T∗M refers to the dual space of TM. Now take for example the third order jet of f, J3f
and consider the most general form of the third order vector V ∈ ΓT3M shown in equation
(36). Given an element of this jet space 3ϕ = (xa, ϕ, ϕa, ϕab, ϕabc) and the higher order vector
components V a, V ab and V abc, they can be combined in the following way.
V : 3
ϕ = V •
f(3
ϕ) + V a
fa(3
ϕ) +
1
2
V ab
fab(3
ϕ) +
1
6
V abc
fabc(3
ϕ) (43)
13
14. 3.3 Jet Spaces
Where V • is known as the secular component and is included in some definitions of higher order
vectors. Duval’s work on differential operators for example does include this term[5]. In this
report however it was decided that the term be quotiented out of the higher order vector space.
This is equivalent to taking V • = 0. An element of the third order jet space is said to be the
third prolongation of f, if all of the Latin subscripts correspond to partial differentiation. That
is to say, fa(3ϕ) = ∂aϕ, fab(3ϕ) = ∂2
abϕ and so on. If it is assumed that in equation (43),
V • = 0 and it is the prolongation being dealt with then
V : 3
ϕ = V a ∂ϕ
∂xa
+
1
2
V ab ∂2ϕ
∂xa∂xb
+
1
6
V abc ∂3ϕ
∂xa∂xb∂xc
= V ϕ
That is to say, combining a third order vector with the third prolongation of f (secular term
quotiented out), corresponds to our definition of a higher order vector acting upon a scalar field.
The third order vector components therefore belong to the dual of jet J3f, denoted (J3f)∗.
It will later be shown in section 5 that taking combinations of higher order vector compo-
nents and the connection, leads to the cancellation of non-tensorial terms. This is because the
Christoffel symbols which ultimately define the connection, belong to the first order jet space
on M. That is to say given a connection on M, Γ : M → J1M where J1M is the set of all
first order jets on M[14].
14
15. 4 Investigation of Transformation Properties
Equipped with the coordinate free definitions of second and third order vectors and having
shown what their most natural coordinate bases look like, their transformation properties can
be calculated. These transformation laws are the main motivation for this work, highlighting
an intimate relationship between the Christoffel symbols and certain higher order vector com-
ponents. This section requires the proper treatment of coordinates as in section 2.1, yet most
of the results are achieved simply by repeated application of the chain rule.
4.1 The Christoffel Symbols
It is well known that the Christoffel symbols are not tensorial, yet the derivation of this result
is usually done using the Levi-Civita expression for the symbols. That is to say, assuming that
the connection is metric compatible and torsion free. Here the relationship is shown using just
the definition of Γc
ab in terms of the connection.
Lemma 15. Consider the Christoffel symbols of the second kind, Γc
ab on an m-dimensional
manifold M in coordinate frame (x1, · · · , xm). In another coordinate frame (y1, · · · , ym), the
Christoffel symbols are denoted ˆΓγ
αβ. The symbols in each frame are related in the following
way.
ˆΓγ
αβ =
∂xa
∂yα
∂xb
∂yβ
∂yγ
∂xc
Γc
ab +
∂yγ
∂xd
∂2xd
∂yα∂yβ
(44)
Proof.
ˆΓγ
αβ = ˆ∂α
ˆ∂β yγ
= ∂
∂yα
∂
∂yβ
yγ
= ∂
∂yα
∂xb
∂yβ
∂
∂xb
yγ
=
∂2xb
∂yα∂yβ
∂
∂xb
+
∂xb
∂yβ ∂xa
∂yα
∂
∂xa
∂
∂xb
yγ
=
∂2xb
∂yα∂yβ
∂
∂xb
yγ
+
∂xa
∂yα
∂xb
∂yβ ∂
∂xa
∂
∂xb
yγ
=
∂2xb
∂yα∂yβ
∂yγ
∂xb
+
∂xa
∂yα
∂xb
∂yβ ∂
∂xa
∂
∂xb
∂yγ
∂xc
xc
=
∂yγ
∂xd
∂2xd
∂yα∂yβ
+
∂xa
∂yα
∂xb
∂yβ
∂yγ
∂xc
Γc
ab
The Christoffel symbols therefore transform into two terms. There is a tensorial term as would
be expected from a (1, 2) tensor and one extra term dependent on a second order partial deriva-
tive. If tensorial expressions are to be formed from the Christoffel symbols and other objects,
then these other objects must transform in a way such that this additional term is ‘cancelled
out.’ It turns out that the higher order vector components are what is required.
It is straightforward to show that the Christoffel symbols of the first kind, defined in terms of
the second kind and metric tensor as Γcab = gcdΓd
ab transform in a similar fashion.
ˆΓγαβ =
∂xa
∂yα
∂xb
∂yβ
∂xc
∂yγ
Γcab + gab
∂2xa
∂yα∂yβ
∂xb
∂yγ
(45)
This relation will be useful when investigating combining a higher order vectors with the con-
nection to obtain a scalar quantity.
15
16. 4.2 Second & Third Order Vectors
4.2 Second & Third Order Vectors
It has been demonstrated that the Christoffel symbols transform as a (1, 2) tensor with the
addition of an extra term dependent on the second derivative. It will now be shown that the
Ua component of a second order vector shares a similar property. To do this, the invariance
of scalar fields under a change of coordinate system is used. By definition, for f ∈ ΓΛ0M
and U ∈ ΓTnM, U f ∈ ΓΛ0M also. That is to say for two coordinate frames, hatted and
un-hatted
U f = ˆU ˆf (46)
With this in mind, the following lemma is proposed.
Lemma 16. Consider a second order vector field U ∈ ΓT2M with components Ua and Uab
in coordinate frame (x1, · · · , xm). In another coordinate frame (y1, · · · , ym), its components
are ˆUα and ˆUαβ. The components in each frame are related in the following way.
ˆUα
= Ua ∂yα
∂xa
+ Uab 1
2
∂2yα
∂xa∂xb
, ˆUαβ
= Uab ∂yα
∂xa
∂yβ
∂xb
(47)
Proof. For f ∈ ΓΛ0M
ˆU ˆf = ˆUα ∂ ˆf
∂yα
+ ˆUαβ 1
2
∂2 ˆf
∂yα∂yβ
= ˆUα ∂xa
∂yα
∂f
∂xa
+ ˆUαβ 1
2
∂
∂yα
∂xb
∂yβ
∂f
∂xb
= ˆUα ∂xa
∂yα
∂f
∂xa
+ ˆUαβ 1
2
∂2xb
∂yα∂yβ
∂f
∂xb
+
∂xa
∂yα
∂xb
∂yβ
∂2f
∂xa∂xb
= ˆUα ∂xa
∂yα
+ ˆUαβ 1
2
∂2xa
∂yα∂yβ
∂f
∂xa
+ ˆUαβ 1
2
∂xa
∂yα
∂xb
∂yβ
∂2f
∂xa∂xb
The right hand side must be equal to U f by (46), therefore
ˆUα ∂xa
∂yα
+ ˆUαβ 1
2
∂2xa
∂yα∂yβ
∂f
∂xa
+ ˆUαβ 1
2
∂xa
∂yα
∂xb
∂yβ
∂2f
∂xa∂xb
= Ua ∂f
∂xa
+ Uab 1
2
∂2f
∂xa∂xb
(48)
Since the expression is true for all f, by comparing the coefficients of ∂af and ∂2
abf, then using
the freedom to relabel and interchange the hatted and un-hatted frame yields exactly (47).
Notice that the expression for ˆUα involves a tensorial term and a term dependent on a second
order derivative of yα. The non-tensorial term yielded by the Christoffel symbols is a second
derivative of xa, however it was shown in section 2.1 that there is a simple expression, equation
(1) relating the two.
The transformation laws for the components of a third order vector, V ∈ ΓT3M are now
considered. Although more complex, a similar sort of pattern is followed.
16
17. 4.2 Second & Third Order Vectors
Lemma 17. Consider a third order vector field V ∈ ΓT3M with components V a, V ab
and V abc in coordinate frame (x1, · · · , xm). In another coordinate frame (y1, · · · , ym), its
components are ˆV α, ˆV αβ and ˆV αβγ. The components in each frame are related in the
following way.
ˆV α
= V a ∂yα
∂xa
+ V ab 1
2
∂2yα
∂xa∂xb
+ V abc 1
6
∂3yα
∂xa∂xb∂xc
ˆV αβγ
= V abc ∂yα
∂xa
∂yβ
∂xb
∂yγ
∂xc
(49)
ˆV αβ
= V ab ∂yα
∂xa
∂yβ
∂xb
+ V abc 1
3
∂yα
∂xa
∂2yβ
∂xb∂xc
+
∂yα
∂xb
∂2yβ
∂xa∂xc
+
∂yβ
∂xc
∂2yα
∂xa∂xb
(50)
Proof.
ˆV ˆf = ˆV α ∂ ˆf
∂yα
+ ˆV αβ 1
2
∂2 ˆf
∂yα∂yβ
+ ˆV αβγ 1
6
∂3 ˆf
∂yα∂yβ∂yγ
= ˆV α ∂xa
∂yα
∂f
∂xa
+ ˆV αβ 1
2
∂
∂yα
∂xb
∂yβ
∂f
∂xb
+ ˆV αβγ 1
6
∂
∂yα
∂
∂yβ
∂xc
∂yγ
∂f
∂xc
= ˆV α ∂xa
∂yα
∂f
∂xa
+ ˆV αβ 1
2
∂2xb
∂yα∂yβ
∂f
∂xb
+
∂xa
∂yα
∂xb
∂yβ
∂2f
∂xa∂xb
+ ˆV αβγ 1
6
∂
∂yα
∂2xc
∂yβ∂yγ
∂f
∂xc
+
∂xb
∂yβ
∂xc
∂yγ
∂2f
∂xb∂xc
= ˆV α ∂xa
∂yα
∂f
∂xa
+ ˆV αβ 1
2
∂2xb
∂yα∂yβ
∂f
∂xb
+
∂xa
∂yα
∂xb
∂yβ
∂2f
∂xa∂xb
+ ˆV αβγ 1
6
∂3xc
∂yα∂yβ∂yγ
∂f
∂xc
+
∂2xc
∂yβ∂yγ
∂xa
∂yα
∂2f
∂xa∂xc
+
∂xa
∂yα
∂xb
∂yβ
∂xc
∂yγ
∂3f
∂xa∂xb∂xc
+
∂xb
∂yβ
∂2xc
∂yα∂yγ
∂2f
∂xb∂xc
+
∂2xb
∂yα∂yβ
∂xc
∂yγ
∂2f
∂xb∂xc
= ˆV α ∂xa
∂yα
+ ˆV αβ 1
2
∂2xa
∂yα∂yβ
+ ˆV αβγ 1
6
∂3xa
∂yα∂yβ∂yγ
∂f
∂xa
+ ˆV αβ ∂xa
∂yα
∂xb
∂yβ
+ ˆV αβγ 1
3
∂2xa
∂yα∂yβ
∂xb
∂yγ
+
∂xa
∂yβ
∂2xb
∂yα∂yγ
+
∂2xb
∂yβ∂yγ
∂xa
∂yα
1
2
∂2f
∂xa∂xb
+ ˆV αβγ ∂xa
∂yα
∂xb
∂yβ
∂xc
∂yγ
1
6
∂3f
∂xa∂xb∂xc
As with the second order vector transformation laws, by (46) the right hand side must be equal
to V f for all f. The freedom to relabel and interchange the frames can be used again yielding
exactly equations (49) and (50) by comparing coefficients.
Taking U and V to be the higher order vectors used in lemmas 16 and 17, one can see that the
‘largest order coefficients’ Uab and V abc both transform tensorially. Each of the other coefficients
have a tensorial piece and extra terms involving partial derivatives, whose maximum degree
corresponds to the order of the vector. The transformation of V a for example yields third
order partial derivatives of ya. This means that to form a vector quantity from V a and a
linear combination of other objects, one of these other objects must involve a third order partial
derivative of xa when transformed. It will later be seen that the derivative of a Christoffel
symbol provides such a term.
17
18. 5 Combining Higher Order Vectors with the Connection
In the last section, it was shown that some of the components of second and third order vectors
transform in a similar way to the Christoffel symbols for a general connection. With these
transformation properties in mind, one can ask the following question. Is it possible to build
a tensorial object from a sum of terms, composed of these vector components and Christoffel
symbols? In this section it is shown that many of these combinations do indeed exist. Since
it is transformation laws that are being dealt with, it is far more intuitive to start working in
index notation. Once a new object has been established, it is then a case of working backwards
to extract a sensible coordinate free definition. This is the method of approach used through-
out this section of research. It is believed that all of the material presented in this section is
completely new and absent from the literature.
It is sensible to demonstrate first of all, how a first order vector combines with the connection.
This result is included here as it can almost be trivially defined. The combination is largely
uninteresting, however allows the introduction of the colon notation used throughout.
Definition 18. Given a first order vector field u ∈ ΓTM and a general connection on
M,
u : = u (51)
This definition alone does not hold any new mathematics, but will be required later when it is
extended to higher orders, becoming something more meaningful.
5.1 Second Order Vectors & the Connection
The most basic object with a non-tensorial transformation property is the Ua component of a
second order vector field U ∈ ΓT2M, as shown in section 4.2. With this in mind the following
object is defined.
Definition 19. Given a second order vector field U ∈ ΓT2M such that U = Ua∂a + Uab
2 ∂2
ab
and a general connection on M,
(U : )c
=
Uab
2
Γc
ab + Uc
(52)
The choice of notation, that is to say the use of , will become clear when this object is defined
in a coordinate free manner. It will now be shown that (U : )c transforms as a bona fide
vector.
Lemma 20. Given a second order vector U ∈ ΓT2M, the object (U : )c is a vector
quantity. That is to say
U :
γ
=
∂yγ
∂xc
(U : )c
(53)
18
19. 5.1 Second Order Vectors & the Connection
Proof.
U :
γ
=
ˆUαβ
2
ˆΓγ
αβ + ˆUγ
=
Uab
2
∂yα
∂xa
∂yβ
∂xb
∂yγ
∂xc
∂xd
∂yα
∂xe
∂yβ
Γc
de +
∂yγ
∂xd
∂2xd
∂yα∂yβ
+ Ua ∂yγ
∂xa
+ Uab 1
2
∂2yγ
∂xa∂xb
=
Uab
2
δd
aδe
b
∂yγ
∂xc
Γc
de +
∂yα
∂xa
∂yβ
∂xb
∂yγ
∂xd
∂2xd
∂yα∂yβ
+
∂2yγ
∂xa∂xb
+ Ua ∂yγ
∂xa
=
Uab
2
∂yγ
∂xc
Γc
ab −
∂2yγ
∂xa∂xb
+
∂2yγ
∂xa∂xb
+ Ua ∂yγ
∂xa
=
∂yγ
∂xc
Uab
2
Γc
ab + Uc
=
∂yγ
∂xc
(U : )c
The penultimate line is reached using lemma 1. In equation (25) it was reasoned that the Lie
bracket, although itself a first order vector, is a sum of second order vectors. These second
order vectors were of the form ‘first order vector operate first order vector.’ Such a second order
vector is useful here, the following coordinate free object based on equation (52) is defined, for
the specific case that U ∈ ΓT2M is such that U = v ◦ w.
Definition 21. Given v, w ∈ ΓTM and a general connection on M, (v ◦w) : ∈ ΓTM
is such that
(v ◦ w) : = vw −
1
2
T (v, w) (54)
The motivation for this definition becomes clear when the next lemma is considered. For the
proof, the definitions of the connection and torsion tensor introduced in sections 2.3 and 2.4
respectively are required.
Lemma 22. Let a second order vector field U ∈ ΓT2M have components given by
Ua
= vd ∂wa
∂xd
, Uab
= va
wb
+ vb
wa
(55)
Then
Uab
2
Γc
ab + Uc ∂
∂xc
= vw −
1
2
T (v, w) (56)
Proof.
Uab
2
Γc
ab + Uc ∂
∂xc
=
1
2
va
wb
+ vb
wa
Γc
ab + vd ∂wc
∂xd
∂
∂xc
= va
wb
Γc
ab +
1
2
vb
wa
− va
wb
Γc
ab + vd ∂wc
∂xd
∂
∂xc
= va
wb
Γc
ab + vd ∂wc
∂xd
+
1
2
vb
wa
Γc
ab −
1
2
vb
wa
Γc
ba
∂
∂xc
= va
wb
Γc
ab + vd ∂wc
∂xd
+
1
2
vb
wa
(Γc
ab − Γc
ba)
∂
∂xc
19
20. 5.1 Second Order Vectors & the Connection
= ( vw)c
+
1
2
vb
wa
T c
ab
∂
∂xc
= ( vw)c
−
1
2
vb
wa
T c
ba
∂
∂xc
= ( vw)c
−
1
2
T (v, w)c ∂
∂xc
= vw −
1
2
T (v, w)
To begin analysing this result, the choice of the second order vector components Ua and Uab
must be justified. As discussed in section 3.1, it is a straight forward exercise to prove that for
v, w ∈ ΓTM, v ◦ w is a second order vector. This simple result will now be shown.
Lemma 23. Given v, w ∈ ΓTM, then U ∈ ΓT2M if
U = v ◦ w (57)
Furthermore in index notation this may be written
U = va ∂wb
∂xa
∂
∂xb
+
vbwa + vawb
2
∂2
∂xa∂xb
(58)
Proof. This proof begins using definition 4 of a first order vector field.
U fg = (v ◦ w) fg = v w fg
= v fw g + gw f = v fw g + v gw f
= v f w g + fv w g + gv w f + v g w f
= fU g + gU f + v g w f + v f w g
= fU g + gU f + U(1,1) f, g
Where
U(1,1) f, g = v g w f + v f w g (59)
It is clear that U(1,1) f, g is Leibniz in both of its arguments, therefore v ◦ w ∈ ΓT2M by
definition 10. Next consider a similar calculation using indices and with f ∈ ΓΛ0M.
U f = (v ◦ w) f = v w f
= v wa ∂f
∂xa
= vb ∂
∂xb
wa ∂f
∂xa
= vb ∂wa
∂xb
∂f
∂xa
+ vb
wa ∂2f
∂xb∂xa
= vb ∂wa
∂xb
∂
∂xa
+ vb
wa ∂2
∂xb∂xa
f = vb ∂wa
∂xb
∂
∂xa
+
vbwa + vawb
2
∂2
∂xa∂xb
f
The final step exploits the natural symmetry in the definition of a second order vector, namely
Uab = Uba. This is true for all f therefore after relabelling, the final line is exactly equation
(58).
The notation (v◦w) : used in definition 21 is therefore perfectly logical. The choices of Ua and
Uab made in this definition correspond exactly to the calculated first and second components
of v ◦ w respectively.
There are two other observations which further justify the suitability of this definition. First of
all, it is clear to see that any first order vectors v, w ∈ ΓTM must satisfy by definition of the
Lie bracket
v ◦ w − w ◦ v − [v, w] = 0 (60)
Immediately then, the following is also true.
(v ◦ w − w ◦ v − [v, w]) : = (v ◦ w) : − (w ◦ v) : − [v, w] : = 0 (61)
Definition 21 must be consistent with this equation.
20
21. 5.2 Third Order Vectors & the Connection
Lemma 24. Given a general connection on M, (u ◦ v) : ∈ ΓTM satisfies
(v ◦ w) : − (w ◦ v) : − [v, w] : = 0 (62)
Proof. Since the Lie bracket of two vector fields is itself a vector field, definition 18 of a first
order vector field combining with the connection will be required.
(v ◦ w) : − (w ◦ v) : − [v, w] : = vw −
1
2
T (v, w) − wv −
1
2
T (w, v) − [v, w]
= ( vw − wv − [v, w]) −
1
2
T (v, w) +
1
2
T (w, v)
= T (v, w) − T (v, w)
= 0
The second observation is that the coordinate expression for (U : )e is very nearly the exact
component expansion of vw. It would therefore be expected that any additional terms in the
definition of (v◦w) : would involve first order covariant derivatives only. The torsion between
v and w, as stated in definition 9 is T (v, w) = vw − wv − [v, w], namely a sum of first order
covariant derivatives. Furthermore, the definition involves specifically two vectors, v and w.
A regular vector component must have only one free contravariant (upstairs) index. Simply
by considering the number of upstairs and downstairs indices in the coordinate expression, the
product vawb can only be multiplied by a tensor of the form Qc
ab. Torsion is the only candidate.
5.2 Third Order Vectors & the Connection
During initial research, the motivation for definition 19 originally came from considering the
coordinate expansion of vw. If the idea of combining a second order vector and the connection
is to be extended to third order, a natural consideration would be the coordinate expansion of
u vw.
Lemma 25. Given u, v, w ∈ ΓTM, then
( u vw)e
= uc
va
wb
Γe
abΓd
cd +
∂Γe
ab
∂xc
+ ua
vc
∂cwb
+ uc
wb
∂cva
+ uc
va
∂cwb
Γe
ab (63)
+uc
∂cvb
∂bwe
+ uc
vb
∂2
bcwe
Proof.
u( vw)e
= u va
wb
Γe
ab + va ∂we
∂xa
= uc
va
wb
Γf
ab + va ∂wf
∂xa
Γe
cf + uc ∂
∂xc
va
wb
Γe
ab + va ∂we
∂xa
= uc
va
wb
Γf
abΓe
cf + uc
va
Γe
cf
∂wf
∂xa
+ uc
va
wb ∂Γe
ab
∂xc
+ uc
va
Γe
ab
∂wb
∂xc
+ uc
wb
Γe
ab
∂va
∂xc
+ uc ∂va
∂xc
∂we
∂xa
+ uc
va ∂2we
∂xc∂xa
= uc
va
wb
Γf
abΓe
cf + uc
va
wb ∂Γe
ab
∂xc
+ Γe
ab ua
vc ∂wb
∂xc
+ uc
va ∂wb
∂xc
+ uc
wb ∂va
∂xc
+ uc ∂va
∂xc
∂we
∂xa
+ uc
va ∂2we
∂xc∂xa
At first sight this, equation (63) may not seem too enlightening. It is however straightforward
to show, using an identical method to that used to prove lemma 23, a similar lemma. Here it
is stated without proof.
21
22. 5.2 Third Order Vectors & the Connection
Lemma 26. Given u, v, w ∈ ΓTM, then V ∈ ΓT3M if
V = u ◦ v ◦ w (64)
Furthermore in index notation this may be written
V = ua ∂vb
∂xa
∂wc
∂xb
+ ua
vb ∂2wc
∂xa∂xb
∂
∂xc
+ ub
vc ∂wa
∂xc
+ uc
wa ∂vb
∂xc
+ uc
vb ∂wa
∂xc
∂2
∂xa∂xb
(65)
+ua
vb
wc ∂3
∂xa∂xb∂xc
Proof. Simply apply ua∂a to both sides of equation (58).
It is clear by comparing equations (63) and (65), that the coefficients of the coordinate expansion
of u vw are exactly those of the third order vector u ◦ v ◦ w. As with the second order vector
case, the fully symmetrised coefficients of a third order vector V are now considered. By doing
this, the following third order analogue of definition 19 is constructed.
Definition 27. Given a third order vector field V ∈ ΓT3M such that V = V a∂a + V ab
2 ∂2
ab +
V abc
6 ∂3
abc and a general connection on M,
(V : )e
=
V abc
6
Γd
abΓe
cd + ∂cΓe
ab +
V ab
2
Γe
ab + V e
(66)
It should be recognised that the same notation has been used for this combination of a third
order vector with the connection, (V : )e as with the combination of a second order vector
and the connection (U : )c. The reason for this is as with (U : )c introduced in definition 19,
the right hand side of equation (66) transforms as a vector. This result will now be proven by
means of a rather long calculation. For clarity, colours have been used to distinguish the origins
of each term. This is also useful for tracking terms down the page as the proof continues.
Lemma 28. Given a third order vector V ∈ ΓT3M, the object (V : )e is a vector quantity.
That is to say
V : =
∂y
∂xe
(V : )e
(67)
Proof. This proof requires many of the results already shown in the report. The expansion of
terms at the beginning requires almost all of the results in section 4. To simplify the resulting
expressions toward the end, the identities relating partial derivatives shown in lemmas 1 and 2
will be needed.
V : =
ˆV αβγ
6
ˆΓδ
αβ
ˆΓγδ + ∂γ
ˆΓαβ +
ˆV αβ
2
Γαβ + ˆV
=
V abc
6
∂yα
∂xa
∂yβ
∂xb
∂yγ
∂xc
∂yδ
∂xh
∂xf
∂yα
∂xg
∂yβ
Γh
fg +
∂yδ
∂xf
∂2xf
∂yα∂yβ
∂y
∂xe
∂xi
∂yγ
∂xj
∂yδ
Γe
ij +
∂y
∂xj
∂2xj
∂yγ∂yδ
+
∂
∂yγ
∂y
∂xe
∂xf
∂yα
∂xg
∂yβ
Γe
fg +
∂y
∂xf
∂2xf
∂yα∂yβ
22
24. 5.2 Third Order Vectors & the Connection
+
V abc
6
∂2y
∂xe∂xd
δf
a δg
b δd
c Γe
fg +
V abc
6
∂yγ
∂xc
∂yα
∂xa
∂y
∂xe
δg
b
∂2xf
∂yα∂yγ
Γe
fg +
V abc
6
∂yβ
∂xb
∂yγ
∂xc
∂y
∂xe
δf
a
∂2xg
∂yβ∂yγ
Γe
fg
+
V abc
6
∂y
∂xe
∂Γe
ab
∂xc
+
V abc
6
∂yα
∂xa
∂yβ
∂xb
∂2y
∂xc∂xf
∂2xf
∂yα∂yβ
+
V abc
6
∂y
∂xf
∂yα
∂xa
∂yβ
∂xb
∂yγ
∂xc
∂3xf
∂yα∂yβ∂yγ
+
V ab
2
δf
a δg
b
∂y
∂xe
Γe
fg +
V ab
2
∂y
∂xf
∂yα
∂xa
∂yβ
∂xb
∂2xf
∂yα∂yβ
+
V abc
6
δf
a
∂y
∂xe
∂2yβ
∂xb∂xc
∂xg
∂yβ
Γe
fg
+
V abc
6
δf
b
∂2yβ
∂xa∂xc
∂y
∂xe
∂xg
∂yβ
Γe
fg +
V abc
6
δg
c
∂2yα
∂xa∂xb
∂y
∂xe
∂xf
∂yα
Γe
fg +
V abc
6
∂yα
∂xa
∂2yβ
∂xb∂xc
∂y
∂xf
∂2xf
∂yα∂yβ
+
V abc
6
∂yα
∂xb
∂2yβ
∂xa∂xc
∂y
∂xf
∂2xf
∂yα∂yβ
+
V abc
6
∂yβ
∂xc
∂2yα
∂xa∂xb
∂y
∂xf
∂2xf
∂yα∂yβ
+V a ∂y
∂xa
+
V ab
2
∂2y
∂xa∂xb
+
V abc
6
∂3y
∂xa∂xb∂xc
=
V abc
6
∂y
∂xe
Γd
abΓe
cd +
V abc
6
∂yα
∂xa
∂yβ
∂xb
∂y
∂xe
∂2xd
∂yα∂yβ
Γe
cd
+
V abc
6
∂yγ
∂xc
∂yδ
∂xd
∂y
∂xj
∂2xj
∂yγ∂yδ
Γd
ab +
V abc
6
∂yα
∂xa
∂yβ
∂xb
∂yγ
∂xc
∂yδ
∂xf
∂y
∂xj
∂2xf
∂yα∂yβ
∂2xj
∂yγ∂yδ
+
V abc
6
∂2y
∂xe∂xc
Γe
ab +
V abc
6
∂yα
∂xa
∂yγ
∂xc
∂y
∂xe
∂2xf
∂yα∂yγ
Γe
fb +
V abc
6
∂yβ
∂xb
∂yγ
∂xc
∂y
∂xe
∂2xg
∂yβ∂yγ
Γe
ag
+
V abc
6
∂y
∂xe
∂Γe
ab
∂xc
+
V abc
6
∂yα
∂xa
∂yβ
∂xb
∂2y
∂xc∂xf
∂2xf
∂yα∂yβ
+
V abc
6
∂y
∂xf
∂yα
∂xa
∂yβ
∂xb
∂yγ
∂xc
∂3xf
∂yα∂yβ∂yγ
+
V ab
2
∂y
∂xe
Γe
ab +
V ab
2
∂y
∂xf
∂yα
∂xa
∂yβ
∂xb
∂2xf
∂yα∂yβ
+
V abc
6
∂y
∂xe
∂2yβ
∂xb∂xc
∂xg
∂yβ
Γe
ag
+
V abc
6
∂2yβ
∂xa∂xc
∂y
∂xe
∂xg
∂yβ
Γe
bg +
V abc
6
∂2yα
∂xa∂xb
∂y
∂xe
∂xf
∂yα
Γe
fc +
V abc
6
∂yα
∂xa
∂2yβ
∂xb∂xc
∂y
∂xf
∂2xf
∂yα∂yβ
+
V abc
6
∂yα
∂xb
∂2yβ
∂xa∂xc
∂y
∂xf
∂2xf
∂yα∂yβ
+
V abc
6
∂yβ
∂xc
∂2yα
∂xa∂xb
∂y
∂xf
∂2xf
∂yα∂yβ
+V a ∂y
∂xa
+
V ab
2
∂2y
∂xa∂xb
+
V abc
6
∂3y
∂xa∂xb∂xc
=
∂y
∂xe
V abc
6
Γd
abΓe
cd+
∂Γe
ab
∂xc
+
V ab
2
Γe
ab+V e
+
V ab
2
∂yα
∂xa
∂yβ
∂xb
∂y
∂xf
∂2xf
∂yα∂yβ
+
∂2y
∂xa∂xb
+
V abc
6
∂yα
∂xa
∂yβ
∂xb
∂y
∂xe
∂2xd
∂yα∂yβ
Γe
cd +
∂yγ
∂xc
∂yδ
∂xd
∂y
∂xj
∂2xj
∂yγ∂yδ
Γd
ab
+
∂2y
∂xe∂xc
Γe
ab +
∂yα
∂xa
∂yγ
∂xc
∂y
∂xe
∂2xf
∂yα∂yγ
Γe
fb+
∂yβ
∂xb
∂yγ
∂xc
∂y
∂xe
∂2xg
∂yβ∂yγ
Γe
ag
+
∂y
∂xe
∂2yβ
∂xb∂xc
∂xg
∂yβ
Γe
ag +
∂2yβ
∂xa∂xc
∂y
∂xe
∂xg
∂yβ
Γe
bg +
∂2yα
∂xa∂xb
∂y
∂xe
∂xf
∂yα
Γe
fc
+
V abc
6
∂yα
∂xa
∂yβ
∂xb
∂yγ
∂xc
∂yδ
∂xf
∂y
∂xj
∂2xf
∂yα∂yβ
∂2xj
∂yγ∂yδ
+
∂yα
∂xa
∂yβ
∂xb
∂2y
∂xc∂xf
∂2xf
∂yα∂yβ
+
∂3y
∂xa∂xb∂xc
+
∂y
∂xf
∂yα
∂xa
∂yβ
∂xb
∂yγ
∂xc
∂3xf
∂yα∂yβ∂yγ
+
∂yα
∂xa
∂2yβ
∂xb∂xc
∂y
∂xf
∂2xf
∂yα∂yβ
+
∂yα
∂xb
∂2yβ
∂xa∂xc
∂y
∂xf
∂2xf
∂yα∂yβ
+
∂yβ
∂xc
∂2yα
∂xa∂xb
∂y
∂xf
∂2xf
∂yα∂yβ
=
∂y
∂xe
(V : )e
+
V abc
6
∂yα
∂xa
∂yβ
∂xb
∂y
∂xe
∂2xd
∂yα∂yβ
Γe
cd +
∂yγ
∂xc
∂yδ
∂xd
∂y
∂xj
∂2xj
∂yγ∂yδ
Γd
ab
+
∂2y
∂xe∂xc
Γe
ab +
∂yα
∂xa
∂yγ
∂xc
∂y
∂xe
∂2xf
∂yα∂yγ
Γe
fb+
∂yβ
∂xb
∂yγ
∂xc
∂y
∂xe
∂2xg
∂yβ∂yγ
Γe
ag
24
25. 5.2 Third Order Vectors & the Connection
+
∂y
∂xe
∂2yβ
∂xb∂xc
∂xg
∂yβ
Γe
ag +
∂2yβ
∂xa∂xc
∂y
∂xe
∂xg
∂yβ
Γe
bg +
∂2yα
∂xa∂xb
∂y
∂xe
∂xf
∂yα
Γe
fc
+
V abc
6
∂yα
∂xa
∂yβ
∂xb
∂yγ
∂xc
∂yδ
∂xf
∂y
∂xj
∂2xf
∂yα∂yβ
∂2xj
∂yγ∂yδ
+
∂yα
∂xa
∂yβ
∂xb
∂2y
∂xc∂xf
∂2xf
∂yα∂yβ
+
V abc
6
∂3y
∂xa∂xb∂xc
+
∂y
∂xf
∂yα
∂xa
∂yβ
∂xb
∂yγ
∂xc
∂3xf
∂yα∂yβ∂yγ
+
∂yα
∂xa
∂2yβ
∂xb∂xc
∂y
∂xf
∂2xf
∂yα∂yβ
+
∂yα
∂xb
∂2yβ
∂xa∂xc
∂y
∂xf
∂2xf
∂yα∂yβ
+
∂yβ
∂xc
∂2yα
∂xa∂xb
∂y
∂xf
∂2xf
∂yα∂yβ
+
V ab
2
∂yα
∂xa
∂yβ
∂xb
∂y
∂xf
∂2xf
∂yα∂yβ
+
∂2y
∂xa∂xb
=
∂y
∂xe
(V : )e
+
V abc
6
∂yα
∂xa
∂yβ
∂xb
∂y
∂xe
∂2xd
∂yα∂yβ
Γe
cd +
∂yγ
∂xc
∂yδ
∂xd
∂y
∂xj
∂2xj
∂yγ∂yδ
Γd
ab
+
∂2y
∂xe∂xc
Γe
ab +
∂yα
∂xa
∂yγ
∂xc
∂y
∂xe
∂2xf
∂yα∂yγ
Γe
fb+
∂yβ
∂xb
∂yγ
∂xc
∂y
∂xe
∂2xg
∂yβ∂yγ
Γe
ag
+
∂y
∂xe
∂2yβ
∂xb∂xc
∂xg
∂yβ
Γe
ag +
∂2yβ
∂xa∂xc
∂y
∂xe
∂xg
∂yβ
Γe
bg +
∂2yα
∂xa∂xb
∂y
∂xe
∂xf
∂yα
Γe
fc
+
V abc
6
∂yα
∂xa
∂yβ
∂xb
∂yγ
∂xc
∂yδ
∂xf
∂y
∂xj
∂2xf
∂yα∂yβ
∂2xj
∂yγ∂yδ
−
∂yα
∂xa
∂yβ
∂xb
∂y
∂xj
∂yγ
∂xf
∂yδ
∂xc
∂2xj
∂yγ∂yδ
∂2xf
∂yα∂yβ
=
∂y
∂xe
(V : )e
+
V abc
6
∂yα
∂xa
∂yβ
∂xb
∂y
∂xe
∂2xd
∂yα∂yβ
Γe
cd +
∂2yβ
∂xc∂xa
∂y
∂xe
∂xg
∂yβ
Γe
bg
+
∂yγ
∂xc
∂yδ
∂xd
∂y
∂xj
∂2xj
∂yγ∂yδ
Γd
ab +
∂2y
∂xe∂xc
Γe
ab +
∂yβ
∂xb
∂yγ
∂xc
∂y
∂xe
∂2xg
∂yβ∂yγ
Γe
ag +
∂y
∂xe
∂2yβ
∂xb∂xc
∂xg
∂yβ
Γe
ag
+
∂2yα
∂xa∂xb
∂y
∂xe
∂xf
∂yα
Γe
fc +
∂yα
∂xc
∂yγ
∂xa
∂y
∂xe
∂2xf
∂yα∂yγ
Γe
fb
=
∂y
∂xe
(V : )e
+
V abc
6
∂yα
∂xa
∂yβ
∂xb
∂2xd
∂yα∂yβ
+
∂2yβ
∂xa∂xb
∂xd
∂yβ
∂y
∂xe
Γe
cd +
∂yγ
∂xc
∂yδ
∂xd
∂y
∂xf
∂2xf
∂yγ∂yδ
+
∂2y
∂xd∂xc
Γd
ab
+
∂yβ
∂xb
∂yγ
∂xc
∂2xg
∂yβ∂yγ
+
∂2yβ
∂xb∂xc
∂xg
∂yβ
∂y
∂xe
Γe
ag +
∂yα
∂xa
∂yγ
∂xc
∂2xf
∂yα∂yγ
+
∂2yα
∂xa∂xc
∂xf
∂yα
∂y
∂xe
Γe
fb
=
∂y
∂xe
(V : )e
Where to get to the penultimate line, the symmetry V abc = V cab has been used on two occasions.
Now that it has been shown that equation (66) represents a bona fide vector, it is reasonable
to assume there is a coordinate free interpretation. This was the case with equation (52), the
second order expression, which was found to be linear in torsion. Such a relationship was to be
expected since the object came about by considering vw. This third order expression however
comes about by investigating u vw, which has been shown to involve both derivatives and
products of Christoffel symbols. There is therefore a far wider variety of terms that could appear.
For example, some kind of curvature dependence would be expected, or indeed derivatives or
squares of torsion. Using a similar yet less forceful approach to that of the last section, the case
when V ∈ ΓT3M is such that V = u ◦ v ◦ w is considered. The starting point is with a new
type of method which exploits the f-linearity and Leibniz properties of our object. Two third
order analogues of equation (60) are then used. It is clear that any first order vectors u, v and
25
26. 5.2 Third Order Vectors & the Connection
w must satisfy both of the following identities.
u ◦ v ◦ w − v ◦ u ◦ w − [u, v] ◦ w = 0 (68)
u ◦ v ◦ w − u ◦ w ◦ v − u ◦ [v, w] = 0 (69)
In section 5.2.2, it is shown how these two equations alone can be used to justify a coordinate
free definition of (V : )e given a torsion free connection.
A Brief Aside
Looking back to section 5.1, the coordinate free definition of (v ◦w) : was ‘derived’ by writing
the coordinate expression in terms of vectors which have a specific coordinate free definition.
A significant amount of time was spent attempting to use the same method to get from the
coordinate to coordinate free definition of (V : )e. The main issue was finding the correct
interpretation for the third order vector component V abc. By definition of a higher order vector,
V abc is completely symmetric. That is to say, with the result of lemma 26, it would be expected
that V abc would take the following form for V = u ◦ v ◦ w.
V abc
∝ ua
vb
wc
+ ua
vc
wb
+ uc
va
wb
+ ub
va
wc
+ uc
vb
wa
+ ub
vc
wa
(70)
This way, V abc = V cba = V bca = · · · . On the other hand, in order to show that (V : )e
transforms as a vector (lemma 28), the only symmetry which is used is V abc = V cab. This is in
fact a cyclic permutation of abc and would imply that V abc could look something like
V abc
∝ A ua
vb
wc
+ uc
va
wb
+ ub
vc
wa
+ B ub
va
wc
+ ua
vc
wb
+ uc
vb
wa
(71)
For constants A and B. It is straightforward to show that this form of the component satisfies
V abc = V cab. Due to the length of each calculation that such a method involves, this turned out
to be a highly inefficient way of dealing with the problem and all attempts were unsuccessful.
For this reason a number of new, more indirect methods were developed. These were largely
more successful in directing the research toward a firm definition.
5.2.1 A General Connection
As has already been discussed, it is expected that a coordinate free (V : )e will involve
curvature terms and those which are derivatives of, or are second order in the torsion. One
other possibility are terms of the form T ( −−, −). These arise due to the appearance of
V ab ∝ ucwa∂cvb in the coordinate definition. To see how each of these terms feature, the
full coordinate free definition of (V : )e with a general connection will now be given. A
full justification will follow. As explained, due to lack of time and methods available, the
exact coefficients of each and every term were not calculated, however the overall form of the
expression is clear.
Definition 29. Given u, v, w ∈ ΓTM and a general connection on M,
(u ◦ v ◦ w) : ∈ ΓTM is such that
(u ◦ v ◦ w) : = u vw −
1
3
R(u, v)w −
1
3
R(u, w)v + ¯T3 (72)
Where
¯T3 = −
1
2
T (u, vw) −
1
2
T ( uv, w) −
1
2
T (v, uw) + A( uT )(v, w) + B( vT )(u, w) (73)
+C( wT )(u, v) + DT (T (u, v), w) + ET (T (v, u), w) + FT (T (w, u), v)
A, B, C, D, E and F are constants yet to be determined.
26
27. 5.2 Third Order Vectors & the Connection
It will first be shown by using nothing more than specific f-linear and Leibniz requirements of
(u ◦ v ◦ w) : , that terms of the form T (−, −−) not only must appear, but can also only
have coefficients ±1
2 or 0.
Lemma 30. Given f ∈ ΓΛ0M, u, v, w ∈ ΓTM and a general connection on M. The
f-linearity and Leibniz requirements of (u ◦ v ◦ w) : ∈ ΓTM force it’s coordinate free
expression to be of the form
(u ◦ v ◦ w) : = −
1
2
T (u, vw) −
1
2
T ( uv, w) −
1
2
T (v, uw) + “f-linear terms” (74)
Proof. Investigation of (fu) is trivial and yields nothing new. Consider then (fv).
(u ◦ (fv) ◦ w) : = u f v ◦ w : + f(u ◦ v ◦ w) :
= u f vw −
1
2
T (v, w) + f(u ◦ v ◦ w) :
= u f vw −
1
2
T ( u(fv), w) − fT ( uv, w) + f(u ◦ v ◦ w) :
= u(f vw) − f u vw −
1
2
T ( u(fv), w) − fT ( uv, w) + f(u ◦ v ◦ w) :
= u (fv)w −
1
2
T ( u(fv), w) + f (u ◦ v ◦ w) : − u vw +
1
2
T ( uv, w)
Hence
(u ◦ (fv) ◦ w) : − u (fv)w +
1
2
T ( u(fv), w) = f (u ◦ v ◦ w) : − u vw +
1
2
T ( uv, w)
=⇒ (u ◦ v ◦ w) : = u vw −
1
2
T ( uv, w) + “other terms f-linear in u and v” (75)
This method clearly only highlights terms which must appear in the definition in order to
compensate for the Leibniz structure of the left hand side. There can therefore be any number
of other terms which are f-linear in u and v, hence the additional “+ other terms f-linear in u
and v.” Next consider (fw).
(u ◦ v ◦ (fw)) : = (u ◦ (v f w)) : + (u ◦ (fv ◦ w)) :
= u v f w : + v f u ◦ w : + u f v ◦ w : + f(u ◦ v ◦ w) :
= u v f w − v f uw + v f uw −
1
2
v f T (u, w) + u f vw
−
1
2
u f T (v, w) + f(u ◦ v ◦ w) :
= u v(fw) − f vw −
1
2
T (u, v(fw) − f vw) + u f vw − f u vw
−
1
2
T (v, u(fw) − f uw) + f(u ◦ v ◦ w) :
= u v(fw) − f u vw −
1
2
T (u, v(fw)) +
f
2
T (u, vw)
−
1
2
T (v, u(fw)) +
f
2
T (v, uw) + f(u ◦ v ◦ w) :
27
28. 5.2 Third Order Vectors & the Connection
Hence
(u ◦ v ◦ (fw)) : − u v(fw) +
1
2
T (u, v(fw)) +
1
2
T (v, u(fw)) = f (u ◦ v ◦ w) : (76)
− u vw +
1
2
T (u, vw) +
1
2
T (v, uw)
Combining this result with (75)
=⇒ (u ◦ v ◦ w) : = u vw −
1
2
T ( uv, w) −
1
2
T (u, vw) −
1
2
T (v, uw) (77)
+ “other terms f-linear in u, v and w”
As before, f-linear terms must be accounted for. This is exactly equation (74).
This method uses nothing more than the Leibniz property of first order vectors and our def-
inition of second order vectors combining with the connection. With no other assumptions,
every term which is not f-linear in all of u, v and w has been attained. These terms just hap-
pen to look like a nice extension of the second order result, it may be there is an underlying
pattern. Roughly speaking, going from (v ◦ w) : to (u ◦ v ◦ w) : , vw → u vw and
T (v, w) → T ( uv, w) + T (u, vw) + T (v, uw). Even at this early stage, the pattern points
to a possible generalisation to nth order combination with the connection.
By definition of the connection and torsion tensors, all terms involving derivatives and squares
of torsion are f-linear in all of their arguments. This is the reason definition 29 includes cyclic
sums of both, with the unknown coefficients A through to F. Only these six terms are necessary
due to the antisymmetry of the torsion tensor. Unfortunately, the exact coefficients of these six
terms were never found due to lack of constraints. This problem will be addressed in detail in
sections 6 and 7. However, by considering (u ◦ v ◦ w) : in a torsion free regime, the exact
form of (u ◦ v ◦ w) : 0 can be fully justified. Assuming that lemma 28 holds, it may be that
at higher orders, object such as (V : )e only exist in the absence of torsion. If this is the case
it could point to something more fundamental, which at the current level of understanding is
being overlooked.
5.2.2 A Torsion Free Connection
Considering a torsion free connection makes for a greatly simplified problem, in this case it is
possible to take ¯T3 = 0 in equation (72). It will now be shown that by enforcing equations (68)
and (69), one arrives at the following definition.
Definition 31. Given u, v, w ∈ ΓTM and a torsion free connection 0 on M,
(u ◦ v ◦ w) : 0 ∈ ΓTM is such that
(u ◦ v ◦ w) : 0
= 0
u
0
vw −
1
3
R(u, v)w −
1
3
R(u, w)v (78)
It has been argued by looking at the coordinate form of (V : )e that (u◦v◦w) : 0, in addition
to u vw, will only involve curvature terms. It is not obvious however that these two particular
curvature terms should be in the definition at all, let alone be sure that they are indeed the
only curvature terms that can feature. This said, Bianchi’s first identity, equation (20) requires
28
29. 5.2 Third Order Vectors & the Connection
that the cyclic sum of three curvature tensors be zero. This means no more than two terms
of a cyclic sum can appear. Using Bianchi again, these two cyclic terms can be written as the
negative of the third term in the cyclic sum, hence no two terms which are cyclic permutations
of each other can appear simultaneously. Furthermore there are only two ways to arrange u, v
and w such that their cyclic sums are independent. That is to say, the cyclic sum of (u, w, v) is
only different to (u, v, w), not for example (w, v, u). This is easily verified by writing down all
of the permutations. With these arguments alone, the following hypothesis can be made.
(u ◦ v ◦ w) : 0
= 0
u
0
vw + AR(u, v)w + BR(u, w)v (79)
Where A and B are constants. These constants can then be found using equations (68) and
(69). Instead of doing this calculation explicitly, it will simply be shown that (u ◦ v ◦ w) : 0
given by definition 31, does indeed satisfy both equations.
Lemma 32. Consider a torsion free connection 0 on M and (u ◦ v ◦ w) : 0 ∈ ΓTM such
that (u ◦ v ◦ w) : 0 = 0
u
0
vw − 1
3R(u, v)w − 1
3R(u, w)v. Then
u ◦ v ◦ w : 0
− v ◦ u ◦ w : 0
− [u, v] ◦ w : 0
= 0 (80)
u ◦ v ◦ w : 0
− u ◦ w ◦ v : 0
− u ◦ [v, w] : 0
= 0 (81)
Proof. Beginning with the left hand side of (80).
u ◦ v ◦ w : 0
− v ◦ u ◦ w : 0
− [u, v] ◦ w : 0
= 0
u
0
vw −
1
3
R(u, v)w −
1
3
R(u, w)v
− 0
v
0
uw −
1
3
R(v, u)w −
1
3
R(v, w)u − 0
[u,v]w
= 0
u
0
vw − 0
v
0
uw − 0
[u,v]w −
1
3
R(u, v)w −
1
3
R(u, w)v
+
1
3
R(v, u)w +
1
3
R(v, w)u
= R(u, v)w −
2
3
R(u, v)w +
1
3
R(v, w)u +
1
3
R(w, u)v
=
1
3
R(u, v)w + R(v, w)u + R(w, u)v = 0
By Bianchi’s first identity, equation (20). Now onto the left hand side of (81).
u ◦ v ◦ w : 0
− u ◦ w ◦ v : 0
− u ◦ [v, w] : 0
= 0
u
0
vw −
1
3
R(u, v)w −
1
3
R(u, w)v
− 0
u
0
wv −
1
3
R(u, w)v −
1
3
R(u, v)w − 0
u[v, w]
= 0
u
0
vw − 0
wv − [v, w] −
1
3
R(u, v)w −
1
3
R(u, w)v
+
1
3
R(u, v)w +
1
3
R(u, w)v
= 0
u (T (v, w)) = 0
Since the connection is torsion free.
It would seem therefore that definition 31 correctly reflects how a third order vector combines
with a torsion free connection. For many applications in physics, a torsion free connection is all
that is required for a solid theory. As has already been mentioned, general relativity is based on
the idea of a torsion free connection[12]. There is also the Fundamental Theorem of Riemannian
29
30. 5.3 Third Order Vectors & the Connection, a Scalar
geometry, revisited later in section 6.
The existence of these vectorial objects, constructed using the non-tensorial Christoffel symbols
along with second and third order vector components, is quite astonishing. A natural step
forward would be to look at whether vectors of this form exist when dealing with vectors of nth
order. The work done at these low orders strongly suggests the possibility of such a definition.
This topic will be discussed in greater detail in section 6.
5.3 Third Order Vectors & the Connection, a Scalar
It has now been explicitly shown that it is possible to combine higher order vectors with the
connection and construct a first order vector. During research it was found that it is also
possible to build a scalar quantity from higher order vector components and the Christoffel
symbols. There is no obvious way to do this with a first order vector, but at second order the
result can be written down almost trivially.
Definition 33. Given a second order vector field U ∈ ΓT2M such that U = Ua∂a + Uab
2 ∂2
ab,
a metric g ∈ Γ M and a general connection on M,
U
... =
Uab
2
gab (82)
Here a triple colon is used to distinguish this expression from the object (U : )e, which is of
course a vector. The claim is that U
... transforms as a scalar quantity. It was shown in section
4 that for a second order vector, the component Uab is a symmetric (2, 0) tensor. The metric
is by definition a symmetric (0, 2) tensor, hence when the two tensors are combined the indices
can be contracted and the result is a scalar. The right hand side of equation (82) also leads to
a nice coordinate free definition. Using the same approach as with the vectorial objects, the
specific case of U = v ◦ w is considered.
Definition 34. Given v, w ∈ ΓTM, a metric g ∈ Γ M and a general connection on
M, (v ◦ w)
... ∈ ΓΛ0M is such that
v ◦ w
... = g(v : , w : ) (83)
This definition is easily justified by expanding the right hand side of equation (82) into is fully
symmetric form introduced in lemma 23.
Lemma 35. Let a second order vector field U ∈ ΓT2M have components Uab = vawb+vbwa
and arbitrary Ua. Then
Uab
2
gab = g(v : , w : ) (84)
30
31. 5.3 Third Order Vectors & the Connection, a Scalar
Proof.
Uab
2
gab =
1
2
(va
wb
+ vb
wa
)gab
=
1
2
(gabva
wb
+ gabvb
wa
)
=
1
2
(gabva
wb
+ gabva
wb
)
=
1
2
(2gabva
wb
)
= g(v, w) = g(v : , w : )
It would appear that definition 34 suggests a relationship between first and second order vectors
combining with the connection. Although this does imply the existence of some sort of inductive
definition for the combination of arbitrary order vectors and the connection, a first order vector
combining with the connection is a trivial result. Recall that u : = u. To show that there is
indeed a strong link between subsequent orders, higher orders must be investigated. A possible
definition is now given for a scalar constructed from a third order vector and the connection.
Definition 36. Given a third order vector field W ∈ ΓT3M such that W = Wa∂a +
Wab
2 ∂2
ab + Wabc
6 ∂3
abc and a general connection on M,
W
... = Wabc
Γcab + Wab
gab (85)
Where Γabc = gadΓd
bc are the Christoffel symbols of the first kind.
The existence of such a definition was a great surprise since the right hand side involves only
two of three possible third order components. For this reason, it would be expected that any
symmetry required to show the object’s invariance under coordinate transform, to be broken.
With the following lemma it is clear that this assumption is incorrect.
Lemma 37. Given a third order vector field W ∈ ΓT3M, the object W
... transforms as
a scalar quantity. That is to say
W
... = W
... (86)
Proof. Note the use of equation (45) for the Christoffel symbol of the first kind transformation
law.
W
... = ˆWαβγ ˆΓγαβ + ˆWαβ
ˆgαβ
= Wabc ∂yα
∂xa
∂yβ
∂xb
∂yγ
∂xc
∂xd
∂yα
∂xe
∂yβ
∂xf
∂yγ
Γfde + gde
∂2xd
∂yα∂yβ
∂xe
∂yγ
+ gde
∂xd
∂yα
∂xe
∂yβ
Wab ∂yα
∂xa
∂yβ
∂xb
+ Wabc 1
3
∂yα
∂xa
∂2yβ
∂xb∂xc
+
∂yα
∂xb
∂2yβ
∂xa∂xc
+
∂yβ
∂xc
∂2yα
∂xa∂xb
31
32. 5.3 Third Order Vectors & the Connection, a Scalar
= Wabc
δd
aδe
b δf
c Γfde + Wabc
gdeδe
c
∂yα
∂xa
∂yβ
∂xb
∂2xd
∂yα∂yβ
+ Wab
gdeδd
aδe
b
+ Wabc
gde
1
3
δd
a
∂xe
∂yβ
∂2yβ
∂xb∂xc
+ δd
b
∂xe
∂yβ
∂2yβ
∂xa∂xc
+ δe
c
∂xd
∂yα
∂2yα
∂xa∂xb
= Wabc
Γcab + Wab
gab + Wabc
gdc
∂yα
∂xa
∂yβ
∂xb
∂2xd
∂yα∂yβ
+ Wabc
gae
1
3
∂xe
∂yβ
∂2yβ
∂xb∂xc
+ Wabc
gbe
1
3
∂xe
∂yβ
∂2yβ
∂xa∂xc
+ Wabc
gdc
1
3
∂xd
∂yα
∂2yα
∂xa∂xb
= W
... − Wabc
gdc
∂xd
∂yα
∂2yα
∂xa∂xb
+
1
3
Wabc
gae
∂xe
∂yβ
∂2yβ
∂xb∂xc
+ Wabc
gbe
∂xe
∂yβ
∂2yβ
∂xc∂xa
+ Wabc
gdc
∂xd
∂yα
∂2yα
∂xa∂xb
= W
... − Wabc
gcd
∂xd
∂yα
∂2yα
∂xa∂xb
+
1
3
Wcab
gcd
∂xd
∂yβ
∂2yβ
∂xa∂xb
+ Wbca
gcd
∂xd
∂yβ
∂2yβ
∂xa∂xb
+ Wabc
gcd
∂xd
∂yβ
∂2yβ
∂xa∂xb
= W
... − Wabc
gcd
∂xd
∂yβ
∂2yβ
∂xa∂xb
+
1
3
Wabc
gcd
∂xd
∂yβ
∂2yβ
∂xa∂xb
+
∂xd
∂yβ
∂2yβ
∂xa∂xb
+
∂xd
∂yβ
∂2yβ
∂xa∂xb
= W
... − Wabc
gcd
∂xd
∂yα
∂2yα
∂xa∂xb
+ Wabc
gcd
∂xd
∂yβ
∂2yβ
∂xa∂xb
= W
...
So what would be a coordinate free interpretation of this result, that is to say (u ◦ v ◦ w)
... ?
Unfortunately the same problem is encountered as was had when defining (V : )e. The
coordinate definition includes the fully symmetric component Wabc, an object which has proven
very difficult to interpret. In order for W
... to transform in the correct way, it is demonstrated
in the proof that the only symmetries required are Wabc = Wcab = Wbca. Notice that once
again, these are the cyclic permutations of abc. As before, it is easy to verify that the condition
Wabc = Wcab = Wbca is satisfied if
Wabc
= A ua
vb
wc
+ uc
va
wb
+ ub
vc
wa
+ B ub
va
wc
+ ua
vc
wb
+ uc
vb
wa
(87)
Where A and B are arbitrary constants. The component Wab should also be fully symmetric.
Referring back to the result of lemma 26, a suitable form for this component to take is
Wab
= C ub
vc ∂wa
∂xc
+ uc
wa ∂vb
∂xc
+ uc
vb ∂wa
∂xc
+ ua
vc ∂wb
∂xc
+ uc
wb ∂va
∂xc
+ uc
va ∂wb
∂xc
(88)
Where C is another constant. With this component interpretation, the following coordinate
free version of definition 36 can be proposed.
Definition 38. Given u, v, w ∈ ΓTM, a metric g ∈ Γ M and a general connection on
M, (u ◦ v ◦ w)
... ∈ ΓΛ0M is such that
u ◦ v ◦ w
... = 2 g u, vw + g v, uw + g w, uv (89)
− g u, T (v, w) + g v, T (u, w) + g w, T (u, v)
32
33. 5.3 Third Order Vectors & the Connection, a Scalar
This definition is justified by considering the coordinate expression of W
... and is the starting
point of the next lemma. The right hand side has been written in this way so that the torsion
and torsion free parts of W
... are clear. Later, this definition will be rewritten in a simpler
form.
Lemma 39. Take a metric g ∈ Γ M and let a third order vector field W ∈ ΓT3M have
components given by (87), (88) and arbitrary Wa. Then
Wabc
Γcab + Wab
gab = 2 g u, vw + g v, uw + g w, uv (90)
− g u, T (v, w) + g v, T (u, w) + g w, T (u, v)
Proof.
Wabc
Γcab + Wab
gab = A ua
vb
wc
+ uc
va
wb
+ ub
vc
wa
+ B ub
va
wc
+ ua
vc
wb
+ uc
vb
wa
Γcab
+ C ub
vc ∂wa
∂xc
+ uc
wa ∂vb
∂xc
+ uc
vb ∂wa
∂xc
+ ua
vc ∂wb
∂xc
+ uc
wb ∂va
∂xc
+ uc
va ∂wb
∂xc
gab
= gcd A Γd
abua
vb
wc
+ Γd
abuc
va
wb
+ Γd
abub
vc
wa
+ B Γd
abub
va
wc
+ Γd
abua
vc
wb
+ Γd
abuc
vb
wa
+ C ud
ve ∂wc
∂xe
+ ue
wc ∂vd
∂xe
+ ue
vd ∂wc
∂xe
+ uc
ve ∂wd
∂xe
+ ue
wd ∂vc
∂xe
+ ue
vc ∂wd
∂xe
= gcd A wc
( uv)d
− ue ∂vd
∂xe
+ uc
( vw)d
− ve ∂wd
∂xe
+ vc
( wu)d
− we ∂ud
∂xe
+ B wc
( vu)d
− ve ∂ud
∂xe
+ vc
( uw)d
− ue ∂wd
∂xe
+ uc
( wv)d
− we ∂vd
∂xe
+ C ud
ve ∂wc
∂xe
+ ue
wc ∂vd
∂xe
+ ue
vd ∂wc
∂xe
+ uc
ve ∂wd
∂xe
+ ue
wd ∂vc
∂xe
+ ue
vc ∂wd
∂xe
= Ag(w, uv) + Ag(u, vw) + Ag(v, wu) + Bg(w, vu) + Bg(v, uw) + Bg(u, wv)
+ gcd Cue
wc ∂vd
∂xe
− Awc
ue ∂vd
∂xe
+ Cuc
ve ∂wd
∂xe
− Auc
ve ∂wd
∂xe
+ Cue
vc ∂wd
∂xe
− Bue
vc ∂wd
∂xe
+ Cud
ve ∂wc
∂xe
− Bud
we ∂vc
∂xe
+ Cvd
ue ∂wc
∂xe
− Avd
we ∂uc
∂xe
+ Cwd
ue ∂vc
∂xe
− Bwd
ve ∂uc
∂xe
Taking A = B = C = 1.
= g(w, uv) + g(u, vw) + g(v, wu) + g(w, vu) + g(v, uw) + g(u, wv)
+ gcd ud
ve ∂wc
∂xe
− we ∂vc
∂xe
+ vd
ue ∂wc
∂xe
− we ∂uc
∂xe
+ wd
ue ∂vc
∂xe
− ve ∂uc
∂xe
= g(w, uv) + g(u, vw) + g(v, wu) + g(w, vu) + g(v, uw) + g(u, wv)
+ g([v, w], u) + g([u, w], v) + g([u, v], w)
= g(u, vw + wv + [v, w]) + g(v, wu + uw + [u, w]) + g(w, uv + vu + [u, v])
= g u, 2 vw − T (v, w) + g v, 2 uw − T (u, w) + g w, 2 uv − T (u, v)
= 2 g u, vw + g v, uw + g w, uv − g u, T (v, w) + g v, T (u, w) + g w, T (u, v)
33
34. 5.3 Third Order Vectors & the Connection, a Scalar
Since during the proof it is taken that A = B = C = 1, notice that the coefficient Wabc originally
assumed to be cyclicly symmetric, actually turns out to be fully symmetric. That is to say,
invariant under all permutations of abc. Now that this coordinate free result has been shown, it
was previously mentioned that definition 38 can be rewritten in a more elegant fashion. Simply
by rearranging the right hand side of (89) and dividing by 2 one has the following.
Definition 40. Given u, v, w ∈ ΓTM, a metric g ∈ Γ M and a general connection
on M, (u ◦ v ◦ w)
... ∈ ΓΛ0M is such that
1
2
u ◦ v ◦ w
... = g u, v ◦ w : + g v, u ◦ w : + g w, u ◦ v : (91)
This result shows that there is indeed some sort of link between third order vectors combining
with the connection, and second order vectors combining with the connection. The existence
of such a coordinate free relationship only strengthens the claim that an inductive definition
for combining arbitrary order vectors and the connection is possible. Using coordinates alone
it would be almost impossible to spot this relationship.
In this section, suggestions have been made for coordinate and coordinate free definitions which
demonstrate how first, second and third order vectors can be combined with the connection
to form both scalar, U
... and vector quantities, (U : )e. Working with these low orders, a
relationship between subsequent orders has been found by expressing U
... in terms of (U : )e
for various U ∈ ΓT2M. Next, the most important results of the research are discussed and
their possible applications to physics considered.
34
35. 6 Analysis & Discussion
This section will act as an overall review of the main results presented in the report so far, which
are believed to be original. There will also be a more in depth discussion about the possible
physical applications of the work.
6.1 Discussion of Results
Beginning first of all with the vectorial quantity (U : )c for U ∈ ΓT2M.
Result 1. (Lemma). Given a second order vector field U ∈ ΓT2M such that U = Ua∂a +
Uab
2 ∂2
ab and a general connection on M,
(U : )c
=
Uab
2
Γc
ab + Uc
(92)
is a vector quantity.
Result 2. (Definition). Given v, w ∈ ΓTM and a general connection on manifold M,
(v ◦ w) : ∈ ΓTM is such that
(v ◦ w) : = vw −
1
2
T (v, w) (93)
Going from the coordinate, to the coordinate free definition in the second order case was straight-
forward after noticing the link between the symmetry of the coefficient Uab and the torsion.
The fact that Uab must equal Uba meant that Uab could not just be proportional to vawb, but
had to be proportional to vawb + vbwa. In the coordinate expression (result 1), this term is
multiplied by a Christoffel symbol Γc
ab. For a general connection of course, Γc
ab = Γc
ba hence
the definition of (v ◦ w) : is forced to contain a torsion term. Notice that for a torsion free
connection, (v ◦ w) : reduces to the covariant derivative of w in the direction of v.
Now (V : )e for V ∈ ΓT3M is considered.
Result 3. (Lemma). Given a third order vector field V ∈ ΓT3M such that V = V a∂a +
V ab
2 ∂2
ab + V abc
6 ∂3
abc and a general connection on M,
(V : )e
=
V abc
6
Γd
abΓe
cd + ∂cΓe
ab +
V ab
2
Γe
ab + V e
(94)
is a vector quantity.
Result 4. (Definition). Given u, v, w ∈ ΓTM and a torsion free connection 0 on M,
(u ◦ v ◦ w) : 0 ∈ ΓTM is such that
(u ◦ v ◦ w) : 0
= 0
u
0
vw −
1
3
R(u, v)w −
1
3
R(u, w)v (95)
A different set of methods were required to extract a coordinate free definition from (V : )e,
due to the complexity of the symmetric expansion of V abc. From f-linearity requirements alone,
35
36. 6.1 Discussion of Results
it was shown that (u ◦ v ◦ w) : must consist of terms f-linear in all arguments, along with
three terms of the form T ( −−, −). Using the first Bianchi identity and equations (68) and
(69), it was found that two of the f-linear terms must be curvature tensors. This was enough to
fully define (u ◦ v ◦ w) : in the case of a torsion free connection, result 4. This expression ties
together the concepts of curvature and third order vectors, a relationship which is believed has
not been recognised before. Due to lack of time and methods, the exact form of (u ◦ v ◦ w) :
for a general connection was not found. All known identities that (u ◦ v ◦ w) : should satisfy
were exhausted calculating the first six terms. Looking back to definition 29, this left 6 unknown
coefficients after symmetry considerations.
As well as combining higher order vectors with the connection to form vectorial quantities, it
was also found that it is possible to construct scalars. For this kind of object, the notation
W
... was introduced. The case where W ∈ ΓT2M was found trivially.
Result 5. (Lemma). Given a second order vector field W ∈ ΓT2M such that W = Wa∂a +
Wab
2 ∂2
ab, a metric g ∈ Γ M and a general connection on M,
W
... =
Wab
2
gab (96)
is a scalar quantity.
Result 6. (Definition). Given v, w ∈ ΓTM, a metric g ∈ Γ M and a general connection
on M, (v ◦ w)
... ∈ ΓΛ0M is such that
(v ◦ w)
... = g(v : , w : ) (97)
This was the first expression to be found relating vectors of subsequent orders combining with
the connection. By brute force calculation directly from the coordinate definition, third order
analogues of results 5 and 6 were found.
Result 7. (Lemma). Given a third order vector field W ∈ ΓT3M such that W = Wa∂a +
Wab
2 ∂2
ab + Wabc
6 ∂3
abc and a general connection on M,
W
... = Wabc
Γcab + Wab
gab (98)
is a scalar quantity.
Result 8. (Definition). Given u, v, w ∈ ΓTM, a metric g ∈ Γ M and a general connec-
tion on M, (u ◦ v ◦ w)
... ∈ ΓΛ0M is such that
1
2
u ◦ v ◦ w
... = g u, v ◦ w : + g v, u ◦ w : + g w, u ◦ v : (99)
Results 6 and 8 are perhaps the most important to come out of the research. Both not only show
that it is possible to move between W : and W
... , but more importantly relate first/second
36
37. 6.2 Physical Applications
order vectors combining with the connection and second/third order vectors combining with
the connection respectively. As has already been highlighted, this points to a possible inductive
definition which combines arbitrary order vectors and the connection.
All eight of these results have arisen from a natural relationship between the connection and
the higher order vector components. It has been shown that both the Christoffel symbols
and higher order vector components are in general non-tensorial. Take the specific example of
U ∈ ΓT2M with first component, Ua. Under a change of coordinate frame, both transform
with a piece which is tensorial and an additional non-linear piece, dependant on second order
derivatives of each coordinate function. Combining the two together in the right way has the
effect of cancelling out the additional, non-tensorial term. It was explained in section 3.3 that
the fundamental reason for this cancellation is their dual jet space relationship.
6.2 Physical Applications
The study of higher order vectors is fairly abstract, yet it has been shown that combining them
with the connection leads to relationships between them and useful, measurable geometric
quantities. Covariant derivatives, curvature and torsion lend themselves well to the study of
gravity, where the nature of the space in question has direct consequence in the theory. General
relativity for example has the geodesic deviation equation. This equation states that the only
way gravity can be ‘measured’ is to look at the curvature of the manifold in which a test particle
moves[12]. It is natural then to expect, that it may be possible to express some equations from
Einstein’s theory, in terms of these new coordinate free objects. In lemma 41, the condition
which a vector must satisfy in order for it to be Killing is rewritten. A vector u ∈ ΓTM is
Killing if Lug = 0, that is to say that the Lie derivative of the metric in the direction of u is
zero[12]. Every Killing vector corresponds to a conserved quantity in the spacetime described
by g, energy or momentum for example[12]. It is straightforward to show assuming metric
compatibility and using the Leibniz rule that
Lug = 0 =⇒ u g(v, w) = g [u, v], w + g v, [u, w] (100)
For all v and w.
Lemma 41. Consider first order vectors u, v, w ∈ ΓTM, a metric g ∈ Γ M and a metric
compatible connection on M. u is a Killing vector if
u g(v, w) =
1
2
[u, v] ◦ w
... +
1
2
[u, w] ◦ v
... (101)
Proof. Beginning with equation (100). u is a Killing vector if for all v and w it satisfies
u g(v, w) = g [u, v], w + g v, [u, w] (102)
Now consider the following.
1
2
u ◦ v ◦ w
... −
1
2
v ◦ u ◦ w
... = g u, v ◦ w : + g v, u ◦ w : + g w, u ◦ v :
− g v, u ◦ w : − g u, v ◦ w : − g w, v ◦ u :
= g w, u ◦ v : − g w, v ◦ u :
= g w, [u, v] : = g w, [u, v]
37
38. 6.2 Physical Applications
Then immediately by relabelling.
1
2
u ◦ w ◦ v
... −
1
2
w ◦ u ◦ v
... = g v, [u, w]
Substituting these two expressions directly into equation (102) gives
u g(v, w) =
1
2
u ◦ v ◦ w
... −
1
2
v ◦ u ◦ w
... +
1
2
u ◦ w ◦ v
... −
1
2
w ◦ u ◦ v
...
=
1
2
u ◦ v ◦ w − v ◦ u ◦ w
... +
1
2
u ◦ w ◦ v − w ◦ u ◦ v
...
=
1
2
[u, v] ◦ w
... +
1
2
[u, w] ◦ v
...
This is a nice result which involves both of the new second and third order scalar objects.
A possible application of the vectorial objects (U : )c in a similar area of physics, are to new
cosmological models. The method for doing such modelling usually begins with the construction
of a Lagrangian, which is then integrated to obtain the action. The equations which define the
physical laws of the universe in question, are obtained by finding the stationary points of the
action. In theory, the Lagrangian contains all of the necessary information for a complete
description of the physical system. For a given universe, it is sensible to require that the
Lagrangian be invariant under Lorentz group transformations. This assures that any equations
of motion respect special relativity. The requirement is satisfied by the following Lagrangian
which yields Maxwell’s equations in a vacuum[16].
LMaxwell = −
1
2
dA ∧ dA + A ∧ J (103)
Where A is the electromagnetic potential 1-form and J is the 4-current 1-form. The advantage
of using coordinate free language to write down this Lagrangian is that Lorentz invariance is
automatically built in. With this in mind, the following cosmological Lagrangian featuring
U ∈ ΓT2M such that U = v ◦ w for v, w ∈ ΓTM, can be suggested.
LT2M = κ1d(U : ) ∧ d(U : ) + κ2(U : ) ∧ (U : ) (104)
The first term is dynamical and the second corresponds to the field mass, each have a coupling
of κ1 and κ2 respectively. This is in complete analogy with the Lagrangian for a massive scalar
field given in (118). In accordance with equation (103), wedging each of the two forms must
give an overall 4-form. This can be achieved by setting (U : ) to be a 1-form on M. The
manifold M is 4-dimensional, which means that (U : ) is in fact a 3-form on M. The degrees
therefore add correctly when the two forms are wedged together. It is straightforward to check
that having (U : ) as a 1-form ensures that the dynamical term is also an overall 4-form. A
more detailed discussion of exterior calculus can be found in appendix section A.
By writing down this Lagrangian, second order vectors are being viewed as possible new sources
of matter. Looking back to the coordinate free result, result 2, this could be seen as a fairly
reasonable suggestion. The expression is written in terms of curvature and torsion, both of
which are quantities which play a central role in general relativity and Einstein-Cartan theory
respectively. The Einstein-Cartan model of gravity is similar to general relativity but with non-
zero torsion. It is believed that torsion may feature in a theory of gravity in order to capture
38
