Differential Geometry

Element of differential geometry Riemannian geometry No Riemannian geometry

Element of differential geometry and applications
to probability and statistics

J´rˆme Lapuyade-Lahorgue
eo

eo 1/ 49
Element of differential geometry and applications to probability and statistics


Plan

1 Element of diﬀerential geometry
2 Riemannian geometry
Deﬁnitions
Distance and geodesic curves
Information geometry
Gradient, Laplacian and Brownian motions on manifolds
3 No Riemannian geometry
Notion of connexion
The Riemannian case and Levi-Civita connexion
Second-order derivative, torsion and curvature

eo 2/ 49


Main deﬁnitions

Let M a C 1 -diﬀerentiable manifold.
The parametrisation:

θ ∈ Θ ⊂ Rn → ϕ(θ) ∈ M; (1)

eo 3/ 49


Main deﬁnitions


θ ∈ Θ ⊂ Rn → ϕ(θ) ∈ M; (1)

The directional derivative of f : M → R at P along a curve
γ : R → M:
f (γ(t)) − f (γ(0))
lim , (2)
t→0 t
where γ(0) = P.

eo 3/ 49


Main deﬁnitions


θ ∈ Θ ⊂ Rn → ϕ(θ) ∈ M; (1)

The directional derivative of f : M → R at P along a curve
γ : R → M:
f (γ(t)) − f (γ(0))
lim , (2)
t→0 t
where γ(0) = P.
A tangent vector of M at P: An application which associates to f
a directional derivative.
The set of tangent vectors at P is denoted TP (M) and (v .f )(P)
the directional derivative at P for γ(0) = v .
˙

eo 3/ 49


Main definitions

Vector field: An application which associates at θ a tangent vector
at P = ϕ(θ). The set of vector field, called tangent space, is
denoted T (M)
The derivative V .f of f along the vector map V is an application
which associates at θ the derivative (V (θ).f )(P);
k-differential form: An application which associates at θ a
k-multilinear, antisymmetric form from TP (M) × . . . × TP (M) to
R. The set of k-differential forms is k (M).

eo 4/ 49


Properties of the tangent space

We have: dim T (M) = n, and a base is given by:

∂
, (3)
∂θi
the application which associates the derivative along the curve
t → ϕ(θ1 , . . . , θi −1 , t, θi +1 , . . . , θn ).

eo 5/ 49


Operations on diﬀerential forms

k k
We have: dim (M) = Cn and:

α= αi1 ,...,ik ωi1 ,...,ik , (4)
1≤i1 <...<ik ≤n

k
where ωi1 ,...,ik is the base of (M).
∂ ∂
A possible base is such that ωi1 ,...,ik ∂θi1 , . . . , ∂θik = 1. For k = 1
and such a base, ωj is denoted dθj .

eo 6/ 49



Exterior product

The exterior product is the application:
k l k+l
∧: × → ,

such that:
α → α ∧ β is linear;
α ∧ β = (−1)kl β ∧ α.
We have ωi1 ,...,ik = dθi1 ∧ . . . ∧ dθik , where (dθ1 , . . . , dθn ) is the
∂ ∂
dual base of ∂θ1 , . . . , ∂θn .

eo 7/ 49



Integral of diﬀerential form

For the form ω = fdθ1 ∧ . . . ∧ dθn :

ω= f (θ)dθ1 . . . dθn , (5)
V ϕ−1 (V)

where V ⊂ M has a no empty interior, and = 1 or −1 depending
on the orientation. If doesn’t depend on f , the manifold is
orientable.
Measurable manifolds: The form fdθ1 ∧ . . . ∧ dθn such that
|f |dθ1 . . . dθn is the Lebesgue measure on M is called volume form.

eo 8/ 49



Restriction of diﬀerential form

Let S ⊂ M a k-dimensional sub-manifold and
k
α= fi1 ,...,ik dθi1 ∧ . . . ∧ dθin ∈ (M). Its restriction
1≤i1 <...<ik
k
α|S ∈ (S) is:

D(θi1 , . . . , θik )
α|S = fi1 ,...,ik ((ϕ−1 ◦ψ)(u1 , . . . , uk )) du1 ∧. . .∧duk ,
D(u1 , . . . , uk )
(6)
where is the orientation of S,
ψ : (u1 , . . . , uk ) → ψ(u1 , . . . , uk ) ∈ S a parametrization of S and
D(θi1 , . . . , θik )
the Jacobian of ϕ−1 ◦ ψ.
D(u1 , . . . , uk )
eo 9/ 49



Integral of k-form

k
The integral of α ∈ (M) is:

α= α|S , (7)
S S

where S is a k-dimensional sub-manifold.

eo 10/ 49



Border of a manifold and exterior derivative

Deﬁnition
Let f and g two functions from E to F , f and g are k-homotopic
if there exists an application:

H : [0, 1]k × E → F , with [0, 1]0 = {0, 1} . (8)

such (u1 , . . . , uk ) → H(u1 , . . . , uk , x) continuous,
H(0, . . . , 0, x) = f (x) and H(1, . . . , 1, x) = g (x).

Two topological spaces E and F has the same k-homotopy if there
exists two functions f : E → F and g : F → E such g ◦ f and f ◦ g
are k-homotopic to the respective identity functions.
eo 11/ 49




Example
Two sets with diﬀerent number of connected componants can
not be k-homotopic;
The sets R and {x} are 1-homotopic;
R and the circle S 1 are not 1-homotopic but 0-homotopic;
The sphere S 2 and the torus T 2 are not 1-homotopic but
2-homotopic;
Two homeomorph sets have the same k-homotopy.

eo 12/ 49




Deﬁnition
The border application is an application δ which associates to a
n-dimensional connexe manifold M:
If the manifold has the same n-homotopy than S n , δM = ∅;
If the manifold is homeomorph to the hypercube or an
half-space of Rn , δM is the image of the set of faces or the
delimitation of the half-space by the homeomorphism;
In other cases, the manifold has a hole, the border is the
union of the border of the manifold completed without the
hole and the border of the hole.

eo 13/ 49




We have the following properties:
δM is either empty or dim δM = dim M − 1;
δδM = ∅;
If M is a star-shaped open set, then δM = ∅ implies that M
is a border;
The set Kerδ/Imδ is called De Rham homology.

eo 14/ 49




The exterior derivative is an application d such
k
d (M) ⊂ k+1 (M) such that for any S ⊂ M,
k
k + 1-dimensional sub-manifold and for any α ∈ (M):

dα = α(Green-Stockes formula) (9)
S δS

There exists a unique linear d such (9) holds and:
n
∂f
df = dθj ;
∂θj
j=1
d(f α) = df ∧ α + fdα (Leibnitz rule).
eo 15/ 49



Orientation of the border
Let V = ϕ([0, 1]n ) a n-dimensional sub-manifold such δV = ∅ and
consider:
α = fdθ1 ∧ . . . ∧ θj−1 ∧ θj+1 ∧ . . . ∧ θn (10)
We have:

α = j [f (θj = 1) − f (θj = 0)] dθ1 . . . dθj−1 dθj+1 . . . dθn
δV [0,1]n−1
∂f
= j dθ1 . . . dθn
[0,1]n ∂θj

= j (−1)j−1 dα. (11)
V

So j = (−1)j−1 .
eo 16/ 49



Properties of the exterior derivative

d is R-linear;
n
∂f
df = dθj ;
∂θj
j=1
d ◦ d = 0;
d(α ∧ β) = dα ∧ β + (−1)|α| α ∧ dβ (Leibnitz rule).
Remark: The set Ker d/Im d is called the De Rham cohomology.

eo 17/ 49



Interior derivative

Let V a vector ﬁeld, the interior derivative ιV is the unique linear
application from k to k−1 such that:
ιV f = 0;
ιV (df ) = V .f ;
ιV (α ∧ β) = ιV α ∧ β + (−1)|α| α ∧ ιV β (Leibnitz rule).

eo 18/ 49


Lie derivative

Integral curves and flows

Let V be a vector field. An integral curve t → θ(t) with origine
P = ϕ(θ(0)) is a solution of:

˙
V (θ(t)) = θ(t), (12)

with:
n
˙
θ= ˙ ∂ .
θj (13)
∂θj
j=1

The flow (φt )t∈R of V is a set of applications φt which associates
to P ∈ M the point Q = ϕ(θ(t)), where θ is the integral curve
with origine P.

eo 19/ 49


Lie derivative

Properties of the ﬂow

φs ◦ φt = φt ◦ φs = φs+t ;
φ0 = IdM .

eo 20/ 49


Lie derivative

Properties of the flow

φs ◦ φt = φt ◦ φs = φs+t ;
φ0 = IdM .

We define the pullback φ∗ , for P = ϕ(θ) and Q = φt (P):
t
If W (θ) ∈ TP (M) then (φ∗ W )(θ) ∈ TQ (M);
t
If α(θ) defined on TP (M) × . . . × TP (M), then (φ∗ α)(θ)
t
defined on TQ (M) × . . . × TQ (M).

eo 20/ 49


Lie derivative

The Lie derivative

The pullbacks are respectively deﬁned by:
φ∗ f = f ◦ φt ;
t
(φ∗ W )(θ)(f ) = W ((ϕ−1 ◦ φt ◦ ϕ)(θ))(φ∗ f );
t t
(φ∗ α)(θ)(φ∗ W1 (θ), . . . , φ∗ Wk (θ)) =
t t t
α((ϕ−1 ◦ φt ◦ ϕ)(θ))(W1 (θ), . . . , Wk (θ)).

The Lie derivative is then deﬁned as:
φ∗ T − φ0 T
t
∗
LV (T ) = lim , (14)
t→0 t
where T = f , W or α.
eo 21/ 49


Lie derivative

Properties of the Lie derivative

The Lie derivative satisfies:
T → LV T is linear;
LV (f ) = V .f : extends the directional derivative;
LV (fW ) = (V .f )W + f LV (W ): Leibnitz rule for vector fields;
LV (W .f ) = LV (W ).f + W .LV (f ): composition of
directional derivative;
LV (α ∧ β) = LV (α) ∧ β + α ∧ LV (β): Leibnitz rule for
differential form.

eo 22/ 49


Lie derivative

Lie derivative of vector ﬁeld: LV (W ) is the unique vector ﬁeld
such that:
LV (W ).f = V .(W .f ) − W .(V .f ). (15)

Proof.

V .(W .f ) = LV (W .f ) because W .f is a function,
= LV (W ).f + W .LV (f ),
= LV (W ).f + W .(V .f ). (16)

eo 23/ 49


Lie derivative

Lie derivative of diﬀerential form: LV α is given by the
Green-Ostrogradski formula:

LV α = (d ◦ ιV )α + (ιV ◦ d)α. (17)

Example
If ω ∈ n (M), LV ω = divV ω. We have the classical
Green-Ostrogradski formula:

divV ω = ιV ω. (18)
V δV

eo 24/ 49


Definitions

In the first part, we were studying topology and measure theory on
a manifold.
From now, we will study geometry.
A first mean to provide a manifold with a geometry: to define it as
a Riemannian manifold.

A Riemannian manifold is a manifold such that each tangent space
TP (M) is provided with an inner product. We denote:

∂ ∂
gi ,j (θ) = (θ), (θ) , (19)
∂θi ∂θj
∂
and ei = ∂θi .
In Riemannian manifolds, the volume form is
√
ω = det G dθ1 ∧ . . . ∧ dθn .
eo 25/ 49


Deﬁnitions

Example (The sphere S 2 )
A parametrisation and the associated base are:
 
 x = sin(θ) cos(ϕ)  eθ = cos(θ) cos(ϕ)ex + cos(θ) sin(ϕ)ey
y = sin(θ) sin(ϕ) − sin(θ)ez
z = cos(θ) eϕ = − sin(θ) sin(ϕ)ex + sin(θ) cos(ϕ)ey
 

If the inner product is inheritated from R3 then:

1 0
G= , (20)
0 sin2 θ

and the volume form is ω = sin θdθ ∧ dϕ.

eo 26/ 49



Riemannian geometry and geodesic curves

The distance between two points P = ϕ(θ (1) ) and Q = ϕ(θ (2) ) is
the minimum of:
t (2)
˙ ˙
gi ,j (θ(t))θi (t)θj (t)dt, (21)
t (1) i ,j

where t → θ(t) describes the set of curves of M.
˙ ˙ ˙
Let us denote L(θ, θ) = i ,j gi ,j (θ)θi θj , the minimum is reached
for t → θ(t), called geodesic curve, solution of:

∂L ˙ d ∂L ˙
(θ(t), θ(t)) − (θ(t), θ(t)) = 0. (22)
∂θk dt ˙
∂ θk
eo 27/ 49



Riemannian geometry and geodesic curves

The Euler-Lagrange equation can be expressed as:

¨ ˙ ˙ ∂gi ,k (θ(t)) 1 ∂gi ,j (θ(t))
gi ,k (θ(t))θi (t)+ θi (t)θj (t) − = 0.
∂θj 2 ∂θk
i i ,j

eo 28/ 49



Geometry of the set of probability densities

We study parameterized set of probability distribution.
At each θ ∈ Θ ⊂ Rk , we associate a probability density
y → p(y ; θ):
b
p(y ; θ)dy = 1 and Pθ (Y ∈ [a, b]) = p(y ; θ)dy ;
a
For each fixed y , θ → p(y ; θ) is differentiable.
The Riemannian metric is chosen in order to the volume form is
the prior distribution such that an infinite sample of p(y ; θ)
provides the maximum of information.

eo 29/ 49




θ ∈ Θ a parameter that we want to estimate, y → p(y ; θ) the
corresponding density and y1:n = (y1 , . . . , yn ) a sample of p(y ; θ).
The Bayesian inference consists in:
1 A prior knowledge on θ: p(θ);
p(y1:n ; θ)p(θ)
2 y1:n brings posterior knowledge: p(θ|y1:n ) = ,
p(y1:n )
where:
p(y1:n ) = p(y1:n ; θ)p(θ)dθ. (23)

eo 30/ 49




Statistical inference is like observing the parameter through a noisy
channel:
Noisy channel
H(Θ) H(Y1:n |Θ) H(Y1:n )

H(Θ) = − log(p(θ))p(θ)dθ: prior information at input;

H(Y1:n ) = − log(p(y1:n ))p(y1:n )dy1:n : total information
from observation at output;
H(Y1:n |Θ) = − log(p(y1:n ; θ))p(y1:n ; θ)p(θ)dθdy1:n :
information added from noise of the channel (randomness);
eo 31/ 49



Jeffreys’ priors and Fisher metric

Finally: H(Y1:n ) − H(Y1:n |Θ) is the remaining information on θ.
Asymptotically: When N big enough, this quantity is maximal for:

p(θ) ∝ det IY (θ) Jeffreys prior. (24)

where:
∂ ∂
IY (θ)i ,j = E log p(Y ; θ) × log p(Y ; θ) Fisher information.
∂θi ∂θj
(25)
One can show that the Jeffreys prior corresponds to the volume
form, consequently G = IY (θ).

eo 32/ 49



Distance between probability distributions

Let D(θ1 , θ2 ) the distance between two distributions p(y ; θ1 ) and
ˆN
p(y ; θ2 ) and θML = arg max p(y1 , . . . , yN ; θ) the maximum
likelihood estimator. We show that:

ˆN 1
lim D(θML , θ0 ) = N 0, in distribution, (26)
N→+∞ N

where θ0 is the true parameter.
Utility: In interval estimation, the conﬁdent interval
ˆN
θ : D(θ, θML) < doesn’t depend on the true parameter.

eo 33/ 49



The gradient and Laplace-Beltrami operator

Let f be a function from M to R, there exists a unique vector
field, called gradient of f , denoted gradf such that:

gradf , V = df (V ), (27)

for all vector fields V .

The Laplace-Beltrami operator of f is defined as:

∆f = divgradf . (28)

eo 34/ 49



Denote g i ,j the coeﬃcients of G −1 , we have:

∂f ∂f
gradf = G −1 e1 + . . . + en ;
∂θ1 ∂θ1
 
n n
∂Vj 1 ∂gk,i 
divV =  + Vj g i ,k ;
∂θj 2 ∂θj
j=1 i ,k=1
n n
∂2f ∂g i ,j ∂f
∆f = g i ,j +
∂θi ∂θj ∂θj ∂θi
j=1 i =1
 
n n
1 ∂f  ∂gk,i 
+ g l,j g i ,k .
2 ∂θl ∂θj
l=1 i ,k=1

eo 35/ 49



Martingals, local martingals and semi-martingals

Let (Ω, A, P) a probability space.
A continuous time process (Mt ) is a martingal if
E [Mt |σ((Mu )u≤s )] = Ms for any s ≤ t;
A continuous time process (Mt ) is a local martingal if there
exists an increasing sequence of stopping times (Tn ) such that
the processes (Mt∧Tn ) are martingals;
A predictible process (At ) is a process such that for any
ω ∈ Ω, there exists a Radon measure µ(ω) such that
At (ω) − As (ω) = µ(ω) (]s, t]);
A semi-martingal is the sum of a local martingal and a
predictible process.
eo 36/ 49



Stochastic integral and diﬀerential forms

If M and N are two semi-martingals, we deﬁne the Itˆ integral:
o

t n−1
2
f (Ms )dNs =L lim f (Mtk )(Ntk+1 − Ntk );
0 n→+∞
k=0
(29)
t
Pt = 0 f (Ms )dNs will be denoted dPt = f (Ms )dNs .

eo 37/ 49



Itˆ formula
o

Let M and N two local martingals, there exists a unique predictible
process denoted M, N such that MN − M, N is a local
martingal.
If M and N are semi-martingals, M, N is the predictible process
associated to their respective local martingal parts.
Let M = (M (1) , . . . , M (n) ) ∈ Rn a semi-martingal and f a function
from Rn to R, then:
n n
∂f (j) 1 ∂2f
df (Mt ) = (Mt )dMt + (Mt )d M (i ) , M (j) .
∂mj 2 ∂mi ∂mj t
j=1 i ,j=1

eo 38/ 49



Brownian motion in a manifold

A Brownian motion in Rn is the martingal Gaussian process
(B (1) , . . . , B (n) ) such that B (i ) , B (i ) = t and B (i ) , B (j) = 0.
A function f from M to R is solution of the Heat equation if:
1 ∂f
∆f + = 0. (30)
2 ∂t
The Brownian motion M = ϕ(Θ) on M is such that, if f is
∂ ∂
solution of the Heat equation and ∂θ1 , . . . , ∂θn orthonormal
base:
n
∂f
df (Mt , t) = dΘj . t
∂θj
j=1

eo 39/ 49



Expression of the Brownian motion
n
∂ ∂2
If ∆ = ai + bi ,j , then:
∂θi ∂θi ∂θj
i =1 i ,j

k
(i ) ai (i ) (l)
dΘt = dt + αl dBt , (31)
2
l=1

where B (l) is Brownian motion on R and:
 (1) (1)   (1) (k)   
α1 . . . αk α1 . . . α1 b1,1 . . . b1,k
 . . . × . . . = . . . .
 . . .
. .   .
. . .
. .   .
. . .
. . 
.
(k) (k) (1) (k) bk,1 . . . bk,k
α1 . . . αk αk . . . αk
eo 40/ 49


Notion of connexion

Notion of connexion

If (e1 , . . . , en ) is a base of the tangent space Lei (ej ) = 0, however
the tangent space is not constant.
The change of the tangent spaces defines the geometry of the
manifold. For this, we define another mean to derive vector fields:
the connexion:
V is linear;
V (fW ) = (V .f )W + f V (W ) (Leibnitz rule);
fV (W ) =f V (W ).

eo 41/ 49


Notion of connexion

The Christoffel coefficients

The Christoffel coefficients Γk,j determines the connexion:
i

n
ei (ej ) = Γik,j ek ,
k=1

so we have:
  
n n n
∂Wk
V (W ) =  Vi  + Wj Γk,j  ek .
i
∂θi
k=1 i =1 j=1

eo 42/ 49



The Levi-Civita connexion

The geodesics are the integral curves of the vector ﬁeld V such
that:
V (V ) = 0. (32)
In Riemannian case: A connexion is Riemannian if the previous
equation is equivalent to the Euler-Lagrange equation, we show:
∂gl,k 1 ∂gl,j
gi ,k Γil,j = − . (33)
∂θj 2 ∂θk
k

A connexion without torsion is a connexion such that Γk,j = Γk .
i j,i
The Levi-Civita connexion is the unique Riemannian without
torsion connexion.
eo 43/ 49



If is the Levi-Civita connexion, we have:

i 1 ∂gl,k ∂gj,k ∂gl,j
gi ,k Γl,j = + − .
2 ∂θj ∂θl ∂θk
k

The Levi-Civita is the unique without torsion connexion such that:

ek . ei , ej = ek ei , ej + ei , ek ej .

eo 44/ 49



Second-order derivative

(v .f )(P) is correctly deﬁned, because depends only on v = γ(0).
˙
However v .(V .f ) depends also on γ (0), where γ integral curve of
¨
V.
We deﬁne the second order derivative of f as:
2
( v f )(P) = v .(V .f )(P),

eo 45/ 49



Second-order derivative

(v .f )(P) is correctly defined, because depends only on v = γ(0).
˙
However v .(V .f ) depends also on γ (0), where γ integral curve of
¨
V.
We define the second order derivative of f as:
2
( v f )(P) = v .(V .f )(P),

where V is the vector field associated to geodesic such P = γ(0)
and v = γ(0).
˙
Remark: The second derivative depends on the geometry.

eo 45/ 49




We deﬁne the second-order derivative:
2 2
Vf :θ→( V (θ) f )(ϕ(θ)).

We have:
2
Vf = V .(V .f ) − ( V V ).f

Remark: It coincides with the classical second-order derivative if V
is geodesic vector ﬁeld.

eo 46/ 49



We deﬁne:
2
V ,W f = V .(W .f ) − ( V W ).f ;
2
V ,W Z = V( W Z) − VW
Z.
(34)

The torsion of a connexion is deﬁned as:

T (V , W ) = VW − WV − LV W .

We have:
2 2
V ,W f − W ,V f = −T (V , W ).f , (35)
the torsion says that the second-order derivatives don’t commute
for functions.
eo 47/ 49



The curvature of a without torsion connexion is deﬁned as:

R(V , W , Z ) = V( W Z) − W( V Z) − LV W Z .

We have:
2 2
V ,W Z − W ,V Z = R(V , W , Z ), (36)
the curvature says that the second-order derivatives don’t
commute for vector ﬁelds.
In a three dimensional space:

R(V , W , Z ).U = −K Vol(V , W )Vol(Z , U), (37)

K is called Gauss curvature.

eo 48/ 49



No riemannian set of probability distributions

The study of no Riemannian or information geometry with torsion
has been by S. I. Amari and H. Nagaoka in Methods of Information
Geometry

eo 49/ 49

Differential Geometry

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