Parallelism of Distributions and Geodesics on F(2K + S, S

International Journal of Contemporary Mathematical Sciences
Vol. 9, 2014, no. 11, 515 - 522
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/ijcms.2014.4668
Parallelism of Distributions and Geodesics on
F (2K + S, S)-Structure Lagrangian Manifolds
Abhishek Singh, Ramesh Kumar Pandey and Sachin Khare
Department of Mathematics, B.B.D. University,
Lucknow-226005, Uttar Pradesh, India
c 2014 Abhishek Singh, Ramesh Kumar Pandey and Sachin Khare. This is
Copyright an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
In this paper, authors have shown that if an almost product structure
P on the tangent space of a 2n− dimensional Lagrangian manifold E is
defined and the F (2K + S, S)− structure on the vertical tangent space
TV (E) is given, then it is possible to define the similar structure on the
horizontal subspace TH (E) and also on the tangent space T (E) of E.
Linear connections on the Lagrangian F (2K +S, S)− structure manifold
E are also discussed. Certain other interesting results like geodesics in
E are also studied.
Mathematics Subject Classification: 53D12, 53D35
Keywords: Lagrangian manifold, distribution, parallelism, geodesics
1
Introduction
Let M be an n− dimensional and E be a 2n− dimensional differentiable manifold and let η = (E, π, M ) be the vector bundle with π(E) = M .Suppose
U is a coordinate neighborhood in M with local coordinates (x1 , x2 , ......xn ).
The induced coordinates in π −1 (U ) are (xi , y α ), 1 ≤ i ≤ n, 1 ≤ α ≤ n [13].
The canonical basis for tangent space Tu (E) at u ∈ π −1 (U ) is { ∂x∂ i , ∂y∂α } or
1
simply {∂i , ∂α } where ∂i = ∂x∂ i etc. If (xh , y α ) be coordinates of a point in the
intersecting region π −1 (U ) ∩ π −1 (U 0), we can write
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Abhishek Singh, Ramesh Kumar Pandey and Sachin Khare
1
1
xi = xi (xi )
(1)
1
∂xα α
y
(2)
∂xα
It is easy to prove for another canonical basis in the intersecting region.
1
yα =
∂i1 =
∂xi
∂i
∂xi1
(3)
∂α1 =
∂y α
∂α
∂y α1
(4)
We denote by T (E) the tangent space of E spanned by {∂i , ∂α } and its
subspaces by TV (E) and TH (E) spanned by {∂α } and {∂i } respectively. Obviously
T (E) = TV (E) ⊕ TH (E)
(5)
and
dimTV (E) = dimTH (E) = n
Let us suppose that the Riemannian material structure on T (E) is given by
[15]
G = gij (xi , y α )dxi ⊗ dxj + gab (xi , y α )δy a ⊗ δy b
(6)
where,
gij (xi , y α ) = gij (xi )
and
gab = 12 ∂a∂bL(xi , y α )
where, L(xi , y α ) the Lagrange function. We call such a manifold as Lagrangian
manifold [15].
If X ∈ T (E), we can write
i
(7)
X = X ∂i + X α ∂α
The automorphism P : χ(T (E)) → χ(T (E))
defined by
i
P X = X ∂i + X α ∂α
(8)
is a natural almost product structure on T (E) i.e. P 2 = I, I unit tensor
field. If v and h are the projection morphisms of T (E) onto TV (E) and TH (E)
respectively, then
P0 h = v 0 P
(9)
2
The F(2K+S, S)-structure
If on the vertical space TV (E), there exists a non-null tensor field Fv of type
(1, 1) satisfying
Fv2K+S + FvS = 0
(10)
Parallelism of distributions and geodesics
517
where K is a fixed integer greater than or equal to 1 and S is a fixed odd
integer greater than or equal to 1.The rank of (Fv ) = r = constant, we say that
TV (E) admits F (2K + S, S)− structure. Let us call Fv as Lagrange vertical
structure on TV (E).
Theorem 2.1 If Lagrange vertical structure Fv is defined on the vertical
space TV (E), it is possible to define similar structure on the horizontal subspace
TH (E) with the help of the almost product structure of T (E).
Proof Let us put
Fh = P Fv P
(11)
then Fh is a tensor field of type (1, 1) on TH (E).
Also
Fh2 = (P Fv P )(P Fv P ) = P Fv2 P
as P is an almost product structure on T (E).
Fh3 = (P Fv P )(P Fv P )(P Fv P ) = P Fv3 P
−−−−−−−−−−−−−−
2K+S

F
= (P Fv P )(P Fv P ) − − − − − 2K + S(times)


 h
⇒ Fh2K+S = (P Fv2K+S P )





(12)
&
2K
F2K
h = P Fv P
so
FhS = P FvS P
(13)
from (12) & (13)
(
Fh2K+S + FhS = P (Fv2K+S + FvS )P
=0
(14)
by virtue of (10).
Thus, Fh gives F (2K + S, S)− structure on TH (E).
Theorem 2.2 If Lagrange vertical F (2K + S, S)− structure Fv of rank
r be defined on TV (E), the similar type of structure can be defined on the
enveloping space T (E) with the help of projection morphism of T (E).
Proof Since Lagrange structure Fv is defined on TV (E), the Lagrange
horizontal structure Fh is induced on TH (E) by theorem (10). If v and h are
projection morphisms of TV (E) and TH (E) on T (E), let us put
F = Fh h + Fv v
(15)
518
Abhishek Singh, Ramesh Kumar Pandey and Sachin Khare
As hv = vh = 0 and h2 = h, v 2 = v, we have
F 2 = Fh2 h + Fv2 v
Similarly F 3 = Fh3 h + Fv3 v and so on.
Thus
F 2K+S + F S = (Fh2K+S + FhS )h + (Fv2K+S + FvS )v
=0
by (10), (11) and (14).
Hence
F 2K+S + F S = 0.
Since rank(Fv ) = rank(Fh ) = r, hence
rank(F ) = 2r.
On T (E) with F (2K + S, S)− structure of rank 2r, let us define operators
m = I + F 2K
(16)
Then it is easy to show that
`2 = `, m2 = m, ` + m = I, `m = m` = 0.
Hence the operators ‘` ’ and ‘m’ when applied to the tangent space are
complementary projection operators [12].
3
Parallelism of distributions
Let E be 2n− dimensional Lagrangian manifold. For F (2K + S, S)− structure
on T (E), let L and M be the complementary distributions corresponding to
¯ and f
complementary projection operators ‘`’ and ‘m’. Let ∇
∇ be defined as
follows
¯ X Y = `∇X (`Y ) + m(∇X mY )
∇
(17)
and
f
∇
XY
= `∇`X (`Y ) + m∇mX (mY ) + `[mX, `Y ] + m[`X, mY ]
(18)
¯ and f
it can be shown easily that ∇
∇ are linear connections on E.
Definition3.1 The distribution L is called ∇− parallel if for all X ∈
L, Y ∈ T (E) the vector field ∇Y X ∈ L.
Definition 3.2 The distribution L will be said ∇− half parallel if for all
X ∈ L, Y ∈ T (E), (∆F )(X, Y ) ∈ L where
(∆F )(X, Y ) = F ∇X Y − F ∇Y X − ∇F X Y + ∇Y (F X)
(19)
Parallelism of distributions and geodesics
519
Definition 3.3 We call the distribution L as ∇− anti half parallel if for
all X ∈ L, Y ∈ T (E), (∆F )(X, Y ) ∈ M.
We now prove the following theorems.
Theorem 3.1 On the F (2K + S, S)− structure manifold, the distributions
¯ as well as ∇
˜ parallel.
L and M are ∇
Proof. Since `m = m` = 0, hence from (17) and (18), we have
¯ X Y = m∇X (mY )
m∇
¯ X Y = 0 Therefore
If Y ∈ L, mY = 0 so m∇
¯
∇X Y ∈ L. Hence for Y ∈ L, X ∈ T (E)
¯ X Y ∈ L. So L is ∇−
¯
⇒∇
parallel.
Similarly for X ∈ T (E), Y ∈ L
˜ X Y = m∇mx mY + m[`X, mY ] = 0 as mY = 0.
∇
˜ X Y ∈ L. Hence L is ∇−
˜ parallel.
So ∇
¯ and ∇
˜ parallelism of M can also be proved.
In a similar manner, ∇
Theorem 3.2 On the F (2K + S, S)− structure manifold, the distributions
¯ are equal.
L and M are ∇− parallel if and only if ∇ and ∇
Proof. If L, M are ∇− parallel then ∀X, Y ∈ T (E), m∇X (`Y ) = 0 and
`∇X (mY ) = 0.
Therefore, since ` + m = I,
∇X (`Y ) = `∇X (`Y ) and ∇X mY = m∇X (mY )
So
¯ XY
∇X Y = `∇X (`Y ) + m∇X (mY ) = ∇
¯
Hence ∇ = ∇.
The converse of the theorem can be proved easily.
Theorem 3.3 On the F (2K + S, S)− structure manifold, E, the distribu¯ anti half parallel if for all X ∈ M, Y ∈ T (E)
tion M is ∇−
¯ Y (F X) = m∇F X mY.
m∇
Proof. Since F m = mF = 0, hence in view of the equation (19) for connection
¯
∇
¯ Y F X − m∇
¯ FXY
m(∆F )(X, Y ) = m∇
(20)
520
Abhishek Singh, Ramesh Kumar Pandey and Sachin Khare
In view of the equation (17) , we have
¯ F X Y = `∇F X (`Y ) + m∇F X (mY )
∇
¯ F X Y = m∇F X (mY ) as `m = 0, m2 = m
m∇
¯ Y (F X) − m∇F X (mY )
m(∆F )(X, Y ) = ∇
As (∆F )(X, Y ) ∈ L so m(∆F )(X, Y ) = 0. Thus
¯ Y (F X) = m∇F X (mY ),
m∇
which proves the proposition.
4
Geodesics on the Lagrangian manifold
Let γ be a curve in E with tangent T. Then γ is called geodesic with respect
to connection ∇ if ∇T T = 0.
¯ if
Theorem 4.1 A curve γ will be geodesic with respect to connection ∇
the vector fields
∇T T − ∇T (mT ) ∈ M and ∇T (mT ) ∈ L.
¯ hence ∇
¯ T T = 0. On
Proof. Since γ is geodesic with respect to connection ∇,
making use of the equation (17), the above equation assumes the following
form
`∇T (`T ) + m∇T (mT ) = 0.
Since ` + m = I, we can write the above equation as
`∇T (I − m)T + m∇T (mT ) = 0
or
`∇T T − `∇T (mT ) + m∇T (mT ) = 0.
Therefore
`(∇T T − ∇T (mT )) = 0
and m∇T (mT ) = 0. Hence ∇T T − ∇T (mT ) ∈ M and ∇T (mT ) ∈ L, which
proves the proposition.
Theorem 4.2 The (1, 1) tensor field ‘ `’ and ‘m’ are always covariantly
¯
constants with respect to connection ∇.
Proof. ∀X, Y ∈ T (E), we have
Parallelism of distributions and geodesics
¯ X `)(Y ) = ∇
¯ X (`(Y ) − `∇
¯ X Y.
(∇
521
(21)
Making use of (17) we get
¯ X `)(Y ) = `∇X (`2 Y ) + m∇X (m`Y ) − `{`∇X `Y + m∇X mY }
(∇
Since `2 = `, m2 = m, `m = m` = 0, we get
¯ X `)(Y ) = `∇X (`Y ) − `∇X `Y = 0.
(∇
So, ‘` ’ is covariantly constant. The fact that ‘m’ is covariantly constant can
be proved analogously.
Acknowledgment. The authors are thankful to Professor S. Ahmad Ali
Dean School of Applied Sciences, B.B.D. University, Lucknow, providing suggestions for the improvement of this paper.
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Received: June 11, 2014

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