Almost contact homogeneous manifolds. (English) Zbl 0847.53036

Let \((M, \varphi, \xi, \eta, g)\) be an almost contact manifold. \(M\) is said to be an almost contact homogeneous manifold if a Lie group \(G\) of isometries acts transitively and effectively on \(M\) and \(\varphi\) is invariant under the action of \(G\). \(M\) is homogeneous if and only if there exists a tensor field \(T\) of type (1,2), called an almost homogeneous structure, such that \(\overline \nabla g = \overline \nabla R = \overline T = \overline \nabla \varphi = 0\), where \(\overline \nabla = \nabla - T\) with \(\nabla\) being the Levi-Civita connection on \(M\), and \(R\) is the curvature tensor.
The compact Lie group \(U(n) \times 1\) acts in a natural way on the vector space of tensors with the same symmetries of the almost contact homogeneous structures. D. Chinea, C. Gonzales, and E. Padron decomposed such a vector space into eighteen invariant and irreducible subspaces. The main purpose of this paper is to use this decomposition to obtain some geometrical results about almost contact homogeneous manifolds. Among other results, the complete classification of naturally reductive almost contact manifolds divided into \(2^6\) classes is obtained. The author also shows that a naturally reductive almost cosymplectic manifold is cosymplectic.


53C30 Differential geometry of homogeneous manifolds
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)