This document discusses recurrent and symmetric properties of curvature tensors on Hsu-structure manifolds. Some key points:
- It defines different types of recurrence for curvature tensors K, W, C, L, V on a Hsu-structure manifold, including (1)-recurrence, (12)-recurrence, and Ricci-(1)-recurrence.
- It establishes relationships between different types of recurrent Hsu-structure manifolds. It proves that a P-(1)-recurrent manifold is also P-(2)-recurrent if the condition in equation (2.5) holds.
- It defines symmetric recurrence, where the recurrence parameter A(T1) is equal to 0.
- It
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On Twisted Paraproducts and some other Multilinear Singular IntegralsVjekoslavKovac1
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Stochastic Gravity in Conformally-flat SpacetimesRene Kotze
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Wits Node Seminar: Dr Sunandan Gangopadhyay (NITheP Stellenbosch)
TITLE: Path integral action of a particle in the noncommutative plane and the Aharonov-Bohm effect
On Twisted Paraproducts and some other Multilinear Singular IntegralsVjekoslavKovac1
Presentation.
9th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial, June 12, 2012.
The 24th International Conference on Operator Theory, Timisoara, July 3, 2012.
Stochastic Gravity in Conformally-flat SpacetimesRene Kotze
The National Institute for Theoretical Physics, and the Mandelstam Institute for Theoretical Physics, School of Physics, would like to invite to its coming talk in the theoretical physics seminar series, entitled:
"Stochastic Gravity in Conformally-flat Spacetimes"
to be presented by Prof. Hing-Tong Cho (Tamkang University, Taiwan)
Abstract: The theory of stochastic gravity takes into account the effects of quantum field fluctuations onto the classical spacetime. The essential physics can be understood from the analogous Brownian motion model. We shall next concentrate on the case with conformally-flat spacetimes. Our main concern is to derive the so-called noise kernels. We shall also describe our on-going program to investigate the Einstein-Langevin equation in these spacetimes.
Dates: Tuesday, 17th February 2015
Venue: The Frank Nabarro lecture theatre, P216
Time: 13.20 - 14.10 - TODAY
Research Inventy : International Journal of Engineering and Scienceresearchinventy
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
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Construindo negócios Web 2.0 que apostam na Experiência do UsuárioUTFPR
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In this article we prove the strong convergence result for a pair of nonself mappings using Jungck S- iterative scheme in Convex metric spaces satisfying certain contractive condition. The results are the generalization of some existing results in the literature
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1) i) Explain how to derive the representation of the Cartesian coordinates x,y,z
in terms of the spherical coordinates ρ, θ, φ to obtain
(0.1) r =< x = ρsin(φ)cos(θ), y = ρsin(φ)sin(θ), z = ρcos(φ) > .
What are the conventional ranges of ρ, θ, φ?
ii) Conversely, explain how to express ρ, sin(θ), cos(θ), cos(φ), sin(φ) as
functions of x,y,z.
iii) Consider the spherical coordinates ρ,θ, φ. Sketch and describe in your own
words the set of all points x,y,z in x,y,z space such that:
a) 0 ≤ ρ ≤ 1, 0 ≤ θ < 2π, 0 ≤ φ ≤ π b) ρ = 1, 0 ≤ θ < 2π, 0 ≤ φ ≤ π,
c) 0 ≤ ρ < ∞, 0 ≤ θ < 2π, φ = π
4
, d) ρ = 1, 0 ≤ θ < 2π, φ = π
4
,
e) ρ = 1, θ = π
4
, 0 ≤ φ ≤ π. f) 1 ≤ ρ ≤ 2, 0 ≤ θ < 2π, π
6
≤ φ ≤ π
3
.
iv) In a different set of Cartesian Coordinates ρ, θ, φ sketch and describe in your
own words the set of points (ρ, θ, φ) given above in each item a) to f). For example
the set in a) in x,y,z space is a ball with radius 1 and center (0,0,0). However, in
the Cartesian coordinates ρ, θ, φ the set in a) is a rectangular box.
2) [Computation and graphing of vector fields]. Given r =< x,y,z > and the
vector Field
(0.2) F(x,y,z) = F(r) =< 1 + z,yx,y >,
1
FINAL PROJECT, MATH 251, FALL 2015 2
i) Draw the arrows emanating from (x,y,z) and representing the vectors F(r) =
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is an ellipse. Draw the arrows emanating from (x(t),y(t),z(t)) and representing
the vector values of dr(t)
dt
, F(r(t)) = F(x(t),y(t),z(t)) . Let θ(t) be the angle
between the arrows representing dr(t)
dt
and F(r(t)) . First draw a 5 raw table
recording t, (x(t),y(t),z(t)), dr(t)
dt
, F(r(t)), cos(θ(t)) for the points (x(t),y(t),z(t))
corresponding to t = 0,π
4
, 3π
4
, 5π
4
, 7π
4
. Then draw the arrows.
iii) Given the surface
r(θ,φ) =< x = 2sin(φ)cos(θ), y = 2sin(φ)sin(θ), z = 2cos(φ) >,0 ≤ θ < 2π, 0 ≤ φ ≤ π,
in parametric form. Use trigonometric formulas to show that the following iden-
tity holds
x2(θ,φ) + y2(θ,φ) + z2(θ,φ) ≡ 22.
iv) Draw the arrows emanating from (x(θ,φ),y(θ,φ),z(θ,φ)) and representing the
vectors ∂r(θ,φ)
∂θ
× ∂r(θ,φ)
∂φ
, F(r(θ,φ)) = F(x(θ,φ),y(θ,φ),z(θ,φ)) . Let α(θ,φ) be
the angle between the arrows representing ∂r(θ,φ)
∂θ
× ∂r(θ,φ)
∂φ
and F(r(θ,φ)) . First
draw a table with raws and columns recording (θ,φ),(x(θ,φ),y ...
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1. Lata Bisht, Sandhana Shanker/ International Journal of Engineering Research and
Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue6, November- December 2012, pp.1323-1328
On Recurrent Hsu-Structure Manifold
Lata Bisht * And Sandhana Shanker **
HOD, Department of Applied Science, BTKIT, Dwarahat, Almora, Uttarakhand, India-263653
**
Assist. Prof., Department of Applied Science, BTKIT, Dwarahat, Almora, Uttarakhand, India-
263653
ABSTRACT
In this Paper, we have defined recurrence Then { , g} is said to give to Vn - metric Hsu-
and symmetry of different kinds in Hsu-structure structure and Vn is called a metric Hsu-structure
manifold. Some theorems establishing relationship manifold.
between different kinds of recurrent Hsu- The curvature tensor K, a vector -valued tri-linear
structure manifold have been stated and proved. function w.r.t. the connexion D is given by
Furthermore, theorems on different kinds of
recurrent and symmetric Hsu-structure manifold K ( X , Y , Z ) D X DY Z DY D X Z D[ X ,Y ] Z ,
involving equivalent conditions with respect to (1.3)
various curvature tensors have also been where
discussed.
Keywords : Hsu-structure manifold , Curvature [ X , Y ] DX Y DY X .
tensor ,Recurrent Parameter.
The Ricci tensor in Vn is given by
Ric (Y , Z ) (C1 K )(Y , Z ).
1
I. INTRODUCTION
If on an even dimensional manifold Vn, n (1.4)
= 2m of differentiability class C∞, there exists a 1
where by (C1 K )(Y , Z ) , we mean the contraction of
vector valued real linear function , satisfying
K ( X ,Y , Z ) with respect to first slot.
2 ar In , For Ricci tensor, we also have
(1.1)a X a X , for arbitrary vector field X.
r
(1.1)b
Ric(Y , Z ) Ric(Z ,Y ),
(1.5)a
Where X X , 0 ≤ r ≤ n and 'r' is an integer and
Ric(Y , Z ) g (r (Y ), Z ) g (Y , r (Z )),
(1.5)b
' a ' is a real or imaginary number.
(C1 r ) R.
1
Then { } is said to give to Vn a Hsu-structure
defined by the equations (1.1) and the manifold Vn is (1.5)c
called a Hsu-structure manifold.
The equation (1.1)a gives different structure for Let W, C, L and V be the Projective, conformal,
different values of ' a ' and 'r'. conharmonic and concircular curvature tensors
respectively given by
If r 0 , it is an almost product structure. 1
W ( X ,Y , Z ) K ( X ,Y , Z ) [ Ric(Y , Z ) X
If a 0 , it is an almost tangent structure. (n 1)
If r 1 and a 1 , it is an almost product
structure. Ric( X , Z )Y ].
If r 1 and a 1 , it is an almost complex
(1.6)
structure.
If r =2 then it is a GF-structure which includes
C X ,Y , Z K X ,Y , Z RicY , Z X
π-structure for a 0 ,
1
an almost complex structure for a i , n2
an almost product structure for a 1 ,
an almost tangent structure for a 0 . Ric X , Z Y g Y , Z r ( X ) g X , Z r (Y )}
Let the Hsu-structure be endowed with a metric R
[ g (Y , Z ) X g ( X , Z )Y ].
tensor g, such that g ( X , Y ) a g ( X , Y ) 0 (n 1)( n 2)
r
(1.2)
(1.7)
1323 | P a g e
2. Lata Bisht, Sandhana Shanker/ International Journal of Engineering Research and
Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue6, November- December 2012, pp.1323-1328
1
L( X , Y , Z ) K ( X , Y , Z ) [ Ric(Y , Z ) X or equivalently
(n 2)
Ric( X , Z )Y g ( X , Z )r (Y ) g (Y , Z )r ( X )].
a r P X , Y , Z , T1 P X , Y ,T1 , Z
(1.8) a P X , T , Y , Z a AT P X , Y , Z .
r
1
r
1
(2.2)b
R
V ( X ,Y , Z ) K ( X ,Y , Z ) [ g (Y , Z ) X
n(n 1) Definition (2.3): A Hsu-structure manifold is said to
be (123)-recurrent in P, if
g ( X , Z )Y ].
a r P X , Y , Z , T1 P X , T1 , Y , Z
(1.9)
A manifold is said to be recurrent, if
a r P X , Y , T1 , Z a r P X , Y , Z , T1
(K )( X , Y , Z , T ) A(T1 ) K ( X , Y , Z ).
(1.10) a AT P X , Y , Z ,
r
1
(2.3)a
The recurrent manifold is said to be symmetric, if
A(T1 ) 0, in the equation (1.10). or equivalently
The manifold is said to be Ricci-recurrent, if
a r P X , Y , Z , T1 a r P X , T1 , Y , Z
(Ric)(Y , Z , T ) A(T1 ) Ric(Y , Z ),
(1.11)a
P X , Y , T1 , Z a P X , Y , Z , T1
r
or
(r )(Y , T ) A(T1 )r (Y ),
a r AT1 P X , Y , Z ,
(1.11)b (2.3)b
or
or equivalently
(R)(T ) A(T1 ) R.
(1.11)c
a r P X , Y , Z , T1 a r P X , T1 , Y , Z .
The Ricci-recurrent manifold is said to be symmetric,
a P X , Y , T1 , Z P X , Y , Z , T1
r
a AT P X , Y , Z .
if r
A(T1 ) 0, in equation (1.11). 1
(2.3)c
Note: Similarly (2), (3), (4), (23), (24), (13), (14),
II. RECURRENCE AND SYMMETRY OF (34), (124), (134), (234), and (1234) recurrence in P
DIFFERENT KINDS: can also be defined.
Let P, a vector-valued tri-linear function, be
any one of the curvature tensors K, W, C, L or V. Definition (2.4): A Hsu-structure manifold is said to
Then we will define recurrence of different kinds as be Ricci-(1)-recurrent, if
follows:
a r Ric Y , Z , T1 Ric Y , T1 , Z .
Definition (2.1): A Hsu-structure manifold is said to a r AT1 RicY , Z , T1 .
be (1) - recurrent in P, if
a r P X , Y , Z , T1 P X ,T1 , Y , Z (2.4)
a r AT1 P X , Y , Z . Note: Similarly Ricci-(2)-recurrent can also be
defined.
(2.1)
Definition (2.5): A recurrent Hsu-structure manifold
Where A(T1 ) is a non-vanishing C∞ function. is said to be P-symmetric and Ricci-symmetric
Definition (2.2): A Hsu-structure manifold is said to
if A T1 0 .
be (12)- recurrent in P, if
a r P X , Y , Z , T1 P X ,T1 , Y , Z Theorem (2.1): A P-(1)-recurrent Hsu-structure
a P X , Y , T1 , Z a AT1 P X , Y , Z ,
r r
manifold is P-(2)-recurrent Hsu-structure manifold
for the same recurrence parameter iff
(2.2)a
1324 | P a g e
3. Lata Bisht, Sandhana Shanker/ International Journal of Engineering Research and
Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue6, November- December 2012, pp.1323-1328
P X , T1 , Y , Z P X , Y , T1 , Z .
a r AT1 P X , Y , Z .
(2.5) (2.10)
Necessary condition: If P-(1)-recurrent Hsu- If the manifold is P-(12)-recurrent then from equation
structure manifold is P-(2)-recurrent Hsu-structure (2.2)a, we have
manifold then
a r P X , Y , Z , T1 P X , T1 , Y , Z
r
a P X , Y , Z , T1 P X , T1 , Y , Z a P X , Y , T1 , Z a AT1 P X , Y , Z .
r r
a r AT1 P X , Y , Z a r P X , Y , Z , T1 (2.11)
P X , Y , T1 , Z a r AT1 P X , Y , Z . Using equation (2.9) in equation (2.11) , we get the
P X , T , Y , Z P X , Y , T , Z .
equation (2.10). Hence the theorem.
1 1 Note: Similarly it can shown that a P-(12)-recurrent
Sufficient condition: Hsu-structure manifold is P-(2)-recurrent for the
If P X , T , Y , Z P X , Y , T , Z ,
same recurrence parameter, provided
a r P X , T1 , Y , Z 0.
1 1
then P-(1)-recurrent Hsu-structure manifold is P-(2)-
recurrent Hsu-structure manifold. (2.12)
Note: Similarly a P-(1)-recurrent Hsu-structure Theorem (2.4): A P-(123)-recurrent Hsu-structure
manifold is P-(3)-recurrent Hsu-structure manifold manifold is P-(1)-recurrent for the same recurrence
parameter, provided
for the same recurrence parameter iff
P X , T1 , Y , Z P X , Y , Z , T1 , a r P X , Y , T1 , Z a r P X , Y , Z , T1 0.
(2.6) (2.13)
and
Proof: Barring Y and Z in equation (2.1), we get
A P-(2)-recurrent Hsu-structure manifold is P-(3)-
recurrent Hsu-structure manifold for the same a r P X , Y , Z , T1 P X , T1 , Y , Z
recurrence parameter iff
P X , Y , T1 , Z P X , Y , Z , T1 .
r
a AT1 P X , Y , Z .
(2.14)
(2.7) In equation (2.3)a if
Theorem (2.2): A Ricci-(1)-recurrent Hsu-structure
manifold is Ricci-(2)-recurrent Hsu-structure
a r P X , Y , T1 , Z a r P X , Y , Z , T1 0
manifold for the same recurrence parameter iff, .
Ric Y , T1 , Z Ric Y , Z , T1 . Then we get the equation (2.14). Hence the theorem.
Note: Similarly we can prove that a P-(123)-recurrent
Necessary condition: If Ricci-(1)-recurrent Hsu- Hsu-structure manifold is P-(2)-recurrent,
structure manifold is Ricci-(2)-recurrent Hsu- if
structure manifold then
a r Ric Y , Z , T1 Ric Y , T1 , Z
a r P X , T1 , Y , Z a r P X , Y , Z , T1 0,
a r AT1 RicY , Z a r RicY , Z , T1 and P-(3)-recurrent , if
Ric Y , Z , T1 a r AT1 RicY , Z .
r
r
a P X , T1 , Y , Z a P X , Y , T1 , Z 0
Ric Y , T , Z Ric Y , Z , T .
,
1 1 for the same recurrence parameter.
Sufficient condition: Theorem (2.5): A P-(123)-recurrent Hsu-structure
If Ric Y , T , Z Ric Y , Z , T ,
manifold is P-(12)-recurrent for the same recurrence
1 1 parameter, provided
(2.8)
then Ricci-(1)-recurrent Hsu-structure manifold is
a r P X , Y , Z , T1 0.
Ricci-(2)-recurrent Hsu-structure manifold. (2.15)
Proof: Barring Z in equation (2.2)a , we get
Theorem (2.3): A P-(12)-Recurrent Hsu-structure
manifold is P-(1)-recurrent for the same recurrence a r P X , Y , Z , T1 P X , T1 , Y , Z
a P X , Y , T , Z a AT P X , Y , Z
parameter, provided r r
P X , Y , T1 , Z 0.
r 1 1 .
a
(2.16)
(2.9) Using equation (2.15) in equation (2.3)a , we get
Proof: Barring Y in equation (2.1) , we get
equation(2.16). Hence the theorem.
a r P X , Y , Z , T1 P X , T1 , Y , Z Note: Similarly we can prove that P-(123)-recurrent
Hsu structure manifold is P-(13)-recurrent
a r P X , Y , T1 Z 0,
1325 | P a g e
4. Lata Bisht, Sandhana Shanker/ International Journal of Engineering Research and
Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue6, November- December 2012, pp.1323-1328
and P-(23)-recurrent , if
V X , T1 , Y , Z a r AT1 V X , Y , Z .
a P X , T1 , Y , Z 0,
r
(2.21)
for the same recurrence parameter. If a (1)-recurrent Hsu-structure manifold is
Theorem (2.6): In a (1)-recurrent Hsu-structure conformal-(1) - recurrent and conharmonic (1)-
manifold, if any two of the following conditions hold recurrent for the same recurrence parameter then
for the same recurrence parameter, then the third also from equation (2.21), we get
holds:
a) It is conformal (1)-recurrent,
a r V X , Y , Z , T1 V X , T1 , Y , Z
b) It is conharmonic (1)-recurrent, a AT1 V X , Y , Z .
r
c) It is concircular (1)-recurrent. (2.22)
Which shows, that the manifold is concircular-(1)-
Proof: From equation (1.6), (1.7) and (1.8), we have recurrent.
C X , Y , Z L X , Y , Z K X , Y , Z
n The proof of the remaining two cases follows
similarly.
n2
V X , Y , Z .
Theorem (2.7): In a (1)-symmetric Hsu-structure
manifold, if any two of the following conditions hold
(2.17) for the same recurrence parameter, then third also
Barring X in equation (2.17) ,we get holds:
n 2 K X ,Y , Z
a) It is conformal (1)-symmetric,
n
C X ,Y , Z L X ,Y , Z b) It is conharmonic (1)-symmetric,
c) It is concircular (1)-symmetric.
V X , Y , Z . Proof: The statement follows from the theorem (2.6)
and definition (2.5).
(2.18) Theorem (2.8): In a (12)-recurrent Hsu-structure
Multiplying equation (2.18) by AT1 , barring X and manifold, if any two of the following conditions hold
then using equation (1.1)a in the resulting equation, for the same recurrence parameter, then the third also
we get holds:
a r AT1 C X , Y , Z a r AT1 L X , Y , Z a) It is conformal (12)-recurrent,
b) It is conharmonic (12)-recurrent,
na r AT1
K X , Y , Z V X , Y , Z . c) It is concircular (12)-recurrent.
n 2 Proof: Barring X and Y in equation (2.17), we get
(2.19)
Differentiating equation (2.18) w.r.t. T1, using
C X ,Y , Z L X ,Y , Z n
n2
K X ,Y , Z
equation (2.18) and then barring X in the resulting
equation, we get
V X ,Y , Z .
a r C X , Y , Z , T1 C X , T1 , Y , Z
(2.23)
Multiplying equation (2.23) by AT , barring X
a L X , Y , Z , T L X , T , Y , Z
1
r
1 1 and then using equation (1.1)a in the resulting
n
n 2
a K X ,Y , Z ,T K X ,T ,Y , Z equation,Cwe get, Z a AT LX ,Y , Z
r
1
a AT X , Y
1 r
1
r
1
na AT1
r
a V X , Y , Z , T1 V X , T1 , Y , Z .
r
n 2
K X ,Y , Z V X ,Y , Z .
(2.20) (2.24)
Subtracting equation (2.19) from (2.20) , we have
a r C X , Y , Z , T1 C X , T1 , Y , Z Differentiating equation (2.23) w.r.t T1 , using
equation (2.23) and then barring X in the resulting
a AT C X , Y , Z a L X , Y , Z , T1
r r equation, we get
L X , T1 , Y , Z a r AT1 L X , Y , Z
a r C X , Y , Z , T1 C X , T1 , Y , Z
a C X , Y , T , Z a LX , Y , Z , T
r r
n
n 2
a r K X , Y , Z , T1 K X , T1 , Y , Z L X , T , Y , Z a L X , Y , T , Z
1
1
r
1
1
a r AT1 K X , Y , Z a r V X , Y , Z , T1
n
n 2
a r K X , Y , Z , T1 K X , T1 , Y , Z
1326 | P a g e
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Vol. 2, Issue6, November- December 2012, pp.1323-1328
a r K X , Y , T1 , Z a r V X , Y , Z , T1 Multiplying equation (2.28) by AT1 , barring X and
then using equation (1.1)a in the resulting equation ,
we get
V X , T1 , Y , Z a rV X , Y , T1 , Z .
a r AT1 C X , Y , Z a r AT1 L X , Y , Z
na AT
(2.25)
K X , Y , Z V X , Y , Z .
r
Subtracting equation (2.24) from equation (2.25), we 1
get n 2
a r C X , Y , Z , T1 C X , T1 , Y , Z (2.29)
a r C X , Y , T1 , Z a r AT1 C X , Y , Z Differentiating equation (2.28) w.r.t T1 , using
a L X , Y , Z , T1 L X , T1 , Y , Z
r
equation (2.28) and then barring X in the resulting
equation, we get
a L X , Y , T , Z a AT LX , Y , Z
r
1
r
1
a r C X , Y , Z , T1 C X , T1 , Y , Z
n
n 2
a K X ,Y , Z ,T K X ,T ,Y , Z
r
1 1
a r C X , Y , T1 , Z a r C X , Y , Z , T1
a r K X , Y , T1 , Z a r A(T1 ) K X , Y , Z
a r L X , Y , Z , T1 L X , T1 , Y , Z
a LX , Y , T , Z a LX , Y , Z , T
r r
1 1
a r V X , Y , Z , T1 V X , T1 , Y , Z
a V X , Y , T , Z a AT V X , Y , Z .
r r n
a r K X , Y , Z , T1 K X , T1 , Y , Z
n 2
1 1
(2.26)
If a (12)-recurrent Hsu-structure manifold is
conformal (12)-recurrent and conharmonic (12)-
recurrent for the same recurrence parameter then
a r K X , Y , T1 , Z a r K X , Y , Z , T1
from equation (2.26) , we get
a r V X , Y , Z , T1 V X , T1 , Y , Z a r V X , Y , Z , T1 V X , T1 , Y , Z
a rV X , Y , T1 , Z a r AT1 V X , Y , Z a V X , Y , T , Z a V X , Y , Z , T
r
1
r
1
(2.27) .
Which shows, that the manifold is concircular-(12)- (2.30)
recurrent. Subtracting equation (2.29) from equation (2.30), we
The proof of the remaining two cases follows get
similarly.
Theorem (2.9): In a (12)-symmetric Hsu-structure
a r C X , Y , Z , T1 C X , T1 , Y , Z
manifold, if any two of the following conditions hold a C X , Y , T , Z a C X , Y , Z , T
r
1
r
1
for the same recurrence parameter, then third also
holds:
a) It is conformal (12)-symmetric, a r AT1 C X , Y , Z a r L X , Y , Z , T1
L X , T , Y , Z a LX , Y , T , Z
b) It is conharmonic (12)-symmetric, r
c) It is concircular (12)-symmetric. 1 1
Proof: The statement follows from the theorem (2.8)
and definition (2.5).
Theorem (2.10): In a (123)-recurrent Hsu-structure
a r L X , Y , Z , T1 a r AT1 L X , Y , Z
manifold, if any two of the following conditions hold
for the same recurrence parameter, then the third also
n
n 2
a r K X , Y , Z , T1 K X , T1 , Y , Z
holds:
a) It is conformal (123)-recurrent,
b) It is conharmonic (123)-recurrent,
c) It is concircular (123)-recurrent.
Proof: Barring X, Y and Z in equation (2.17), we get
a r K X , Y , T1 , Z a r K X , Y , Z , T1
C X ,Y , Z L X ,Y , Z n 2 K X ,Y , Z
n
a r A(T1 ) K X , Y , Z a r V X , Y , Z , T1
V X , Y , Z . V X , T , Y , Z a V X , Y , T , Z
1
r
1
(2.28)
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6. Lata Bisht, Sandhana Shanker/ International Journal of Engineering Research and
Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue6, November- December 2012, pp.1323-1328
a r V X , Y , Z , T1 a r AT1 V X , Y , Z Manifold", Journal of Tensor Society, Vol.23,
2009, pp. 79-104.
. [11] U.C. De, N. Guha and D. Kamila, "On
(2.31) Generalized Ricci recurrent manifolds", Tensor
If a (123)-recurrent Hsu-structure manifold is (N.S.) (56), 1995, pp. 312-317.
conformal (123)-recurrent and conharmonic (123)-
recurrent for the same recurrence parameter then
from equation (2.31), we get
a r V X , Y , Z , T1 V X , T1 , Y , Z
a rV X , Y , T1 , Z a rV X , Y , Z , T1
a r AT1 V X , Y , Z .
(2.32)
This shows, that the manifold is concircular (123)-
recurrent Hsu-structure manifold.
The proof of the remaining two cases follows
similarly.
Theorem (2.11): In a (123)-symmetric Hsu-structure
manifold, if any two of the following conditions hold
for the same recurrence parameter, then third also
holds:
a) It is conformal (123)-symmetric,
b) It is conharmonic (123)-symmetric,
c) It is concircular (123)-symmetric.
Proof: The statement follows from the theorem (2.8)
and definition (2.5).
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