Homogeneous berger space and deformations of the SO(3)- structure by its geodesic on 5-dimension Lie groups

An irreducible SO(3)-structure can be defined by means of a symmetric tensor field T of type (0,3) on a manifold M. Definition 1. An SO(3) structure on a 5-dimensional Riemannian manifold (M, g) is a structure defined by means of a rank 3 tensor field T for which the associated linear map X^T _XeEnd(TM), XeTM, satisfies the following condition: (1) symmetricity, i. e. g(X,T _rZ) = g(Z,T _YX) = g(X,T _ZY), (2) the trace tr(T _X) = 0, (3) for any vector field ХеТМ, Tx X = g(XX)X. In any tangent space, it is possible to choose an adapted basis {e ₁, e ₂, e ₃, e ₄, e ₅} in which metrics g and tensor Т have the canonical form g _v = and =2 1(6(e2)2+ 4)2 - 2(e1)2 - 3(e2)2 - 3(e5)2)+ +e ((e ) -(e ) ) + ^V3e e e . Her, {e , e , e , e , e } is the dual coframe. Polarising the expression yields components of T: ^111 _ , ^122 _ , ^144 _ , ^133 _ 2, ^155 _ t433 _ 2 , 455 _ 2 , 235 _ 2 . Thus, an irreducible SO(3)-structure on a manifold is a Riemannian structure g and a tensor field T possessing properties (1) - (3). Theorem 1. The stabilizer of T _iJk is an irreducible SO(3) embedded into O(5). Since the stabilizer T _ijk is an irreducible SO(3), its orbit under the action of O(5) is a 7-dimension homogeneous space O(5)/SO(3). A homogeneous Berger space M = SO(5)/SO(3) is topologically equivalent to an S fiber bundle over S . With respect to the biinvariant scalar product (A,B) _ -1-tr(AB) on SO(5), a decomposition of the Lie algebra so(5) into a direct sum so(5) = so(3) + V of the Lie algebra and ad(SO(3)) of an invariant space V has been obtained. Examples of deformations of the structural tensor T by geodesics g _t of the homogeneous space SO(5)/SO(3) are considered, the covariant divergence of the obtained structure tensor is calculated, and the property of nearly integrability is investigated.

Keywords

Authors

References