On homogeneous manifolds endowed withalmost product structure Текст научной статьи по специальности «Физика»

Научни трудове на Съюза на учените в България - Пловдив Серия В. Техника и технологии, том XIII., Съюз на учените, сесия 5 - 6 ноември 2015 Scientific Works of the Union of Scientists in Bulgaria-Plovdiv, series C. Technics and Technologies, Vol. XIII., Union of Scientists, ISSN 1311-9419, Session 5 - 6 November 2015.

ВЪРХУ ХОМОГЕННИ МНОГООБРАЗИЯ СНАБДЕНИ СЪС СТРУКТУРА НА ПОЧТИ ПРОИЗВЕДЕНИЕ Атанаска Георгиева, Георги Костадинов, Христо Мелемов

ПУ „Паисий Хилендарски", Факултет по Математика и Информатика,

Пловдив

ON HOMOGENEOUS MANIFOLDS ENDOWED WITHALMOST PRODUCT STRUCTURE AtanaskaGeorgieva, GeorgiKostadinov, HristoMelemow

PU „PaisiiHilendarski", Faculty of Mathematics and Informatics, Plovdiv

Abstract: The spaces admitting transitively acting Lie group of transformations arise in differential geometry, as well as differential equations and physics. Of a great importance are the invariant structures on these spaces. We consider almost product structures i.e. tensor field

P of type (1,1), such that P2 = id and the corresponding decompositions of the space. Keywords: Lie-group, reductive homogeneous space, invariant almost product structure

^Preliminaries

We consider a smooth Riemannien manifold (M, g) endowed with a transitive action of a Lie group G . We suppose that G acts effectively on M . Let K be the isotropy subgroup of a reference point o G M, and g and k be the corresponding Lie algebras, so we have M = G / K. A homogeneous space M = G / K with an invariant metric is said to be

naturally reductive if there exists reductive decomposition g m, [k, m] ' m , such that

([5, T L, U=(S, T U L) o-1)

for any S, T, U g m. Here (•, ■) denotes the induced metric on m and subscript m the projection onto m with respect to the reductive decomposition. An almost product structure

48

on M is a smooth tensor field P of type (1,1), such that P2 = id . The metric tensor g and the tensor P satisfy the condition (PX, PY) = (X, 7).

There arise two orthogonal distributions on M corresponding to the eigenval-ues 1 and -1 of P . Thus the structure P defines an orthogonal decomposition m = m, © m2 . The almost product structure P on M = G / K is said to be invariant if : àd (k) °P = P °ad (k) holds. Here ad ( k ) means the linear isotropic representation of k .

We note that the structure P as well as the invariant distributions and vector fields will be identified with their values at the point l = K e G/K .

If A, B, C, ... are vectors in m, X, Y, Z,... are in and S, T, U,... are arbitrary vectors at o e M, then the following relations are satisfied, [2].

([a BL, C) = (a [B cD, (X 7 L., Z) = {X, 7, ZL., (1.2)

([A4, BL. , X) = (A [B XL, ) ([X, 7L, , ) = (X, [7, AL

Lemma. Let (G / K, l^, ^, P ) be naturally reductive space with invariant almost product

structure and g = k " m = k m, mz is the corresponding decomposi-tion. The following assertions hold:

i) If [m ,m ]œ m ©k, (i = 1, 2) then [, m; ]k = k is a subalgebra of k and [ m;, m; ]k © m; = g; is a subalgebra of g

ii) The distributions (m;) = {a* (X) : a e G; X e mj, (i = 1,2) are geodesically parallel.

iii) If [mj, m; ]œ mi ©k, and [mj, m2]œ k, then k; = [mj, mj], are ideals in k kj n k2 = 0 and g = [m;, m; ] ©m;, are ideals in g (i = 1, 2)

Proof:

i) Let A, Bern,, E e k

[E, [4 Bl ] = [E, [4 B]] - [^ [4 B]„ ] = - [4 [B E]] * [B [^ A]] -- [E, [a4,

= - [A, [B, E]]k - [B, [E, A]]k - ([A, [B, E]] + [B, [A, E]] + [E, [A, B]]) = = [A,[E,B]]k +[B,[A,E]]k e []

ii) The geodesics of the canonical connection of M = G / K are the orbits of 1-parameter subgroups, corresponding to the elements ofm.

iii) From (1.1) and (1.2) by using Jacobi identity we get the following relation

([[S, T]k U] , ^ + ([[S, TL , U]m , V) = ([[U, V ]k ,S] , T) ^[[U, V ]m ,S] , ^ , C1"4)

where S, T, U, V are arbitrary elements of m = © m2.

Now, if we set S = A e m1, T = X e m2 in accordance with the condition iii) we conclude

that [[A, X]k , U] = 0 . From the fact that G acts effectively this implies [A, X= 0, i.e.

[m ,m2 ]k = 0 . Thus, [m ,m2 = 0 and it is easy to show that kk and g are ideals. The Lemma is proved.

2. Main Results

Theorem 1. Let Gj and G2 be the subgroups of G corresponding to the sub-algebras gt and g2, respectively. Then

a) the homogeneous spaces Gj / (Gj n K) and G2 / (G2 n K) are totally geodesic subspaces of the space M = G / K .

b) if m and m2 satisfy the conditions iii) of the previous Lemma, then there exist a decomposition of the space G / K in a product (local) of the form

G / K = GJ K x G2/ K2 where K1 = G1 n K, K2 = G2 n K . Proof:

a) Since the geodesics in M = G / K are the orbits of the 1- parameter subgroups the assertion follows from i) of the Lemma.

b) From iii) of the Lemma it follows that Gj and G2 are normal subgroups. Then the proof follows from [1], ch. X, ph. 5. The Theorem is proved.

Let the subgroup K is not a maximal reductive subgroup of G .

Theorem 2. Let K с H с G be Lie subgroups and the following decomposi-tion of g is fulfilled: g = k Фf Ф n = h©m,where

f ©n = m, h = k © f, \kf ] с f, [hn] с n, \ff ] с f ©k .

Then, H / K is a totally geodesic submanifold of M = G / K . Proof : The assertion follows from Th. 1, a).

Example. Let M be a two point homogeneous Riemannian space and TM be the ber bundle of unit spheresequiptwithSasakimetric. M isasymmetricspaceofrank 1.Thenwegetthedecomposition

g = k© m ©m,, satisfying: [ \kmt ]с mt, [mm ]с k, [k©m ,m2 ]с m2and

ToSn = mt. So, we obtain that the spheres at any point are totally geodesic submanifolds, [3].

3. Subspaces and factor spaces

Let (M = G / K, (.,.), P) be a naturally reductive almost product space with the

decomposition g = k © m, m = m © m2, and G' с G be a closed subgroup of the Lie group

G . The homogeneous space N = G' / (G ' nK) with the decomposition g = k ' ©nt © n2

where k' = g nk, nt = g nmt, n, = g n m, is said to be a subspace of the almost product manifold M .

Suppose that [nt, nt ] с k' © nt. Then g" = [nt, nt ] + nx is a subalgebra of g . Further,

if the condition [n, n2 ] с k" is faithful, then from iii) of the Lemma we conclude that g" is an ideal of g with the corresponding normal subgroup G" с G. Thus, we get an almost product

manifold N1 = G" / (G " n K). The set of G" orbits in M = G / H is said to be a factor space and we have M / N1 = G / KG" with the decomposition g =(k+ nt)©n ©m, where n[ = / nt :

In accordance with the previous results we may formulate the next theorem.

Theorem 3. Let (M = G / K, (,), P) be a homogeneous almost product manifold with reductive decomposition g = k © m1 © m2 . There exists (1:1)- correspondence between

the sets of the subspaces of M and the pairs {(n,n2) : nt c mt , n2 c m2} satisfying the following conditions

[n.]m cn © ,[i,n ]cn, (i; j = 1; 2)

ACKNOWLEDGEMENT. This work is partially supported by project NI15-FMI-004 of the Scienti c Research Fund, Plovdiv University, Bulgaria.

References:

[1] Kobayashi S., Nomizu K., Foundations of Di erential Geometry, Interscience Publ. , vol. 2, 1963.

[2] Balashchenko V., Naturally reductive almost product manifolds, Di erential Geometry and Application, 1999, 10-14.

Kostadinov G., Submanifolds in unit spheres ber bundle of two-point homo-geneous space, Comptes rendus de FAcademie bulgare des Science, vol. 42, Atanaska Georgieva, Georgi Kostadinov, Hristo Melemov

[3] No 1, 1989, pp 39-41.