Invariant Connections with Skew-Torsion and del-Einstein Manifolds

摘要

For a compact connected Lie group G we study the class of bi-invariant affine connections whose geodesics through e is an element of G are the 1-parameter subgroups. We show that the bi-invariant affine connections which induce derivations on the corresponding Lie algebra g coincide with the bi-invariant metric connections. Next we describe the geometry of a naturally reductive space (M = G/K, g) endowed with a family of G-invariant connections del(alpha) whose torsion is a multiple of the torsion of the canonical connection del(c). For the spheres S-6 and S-7 we prove that the space of G(2) (respectively, Spin(7))-invariant affine or metric connections consists of the family del(alpha). Then we examine the "constancy" of the induced Ricci tensor Ric(alpha) and prove that any compact isotropy irreducible standard homogeneous Riemannian manifold, which is not a symmetric space of Type I, is a del(alpha)-Einstein manifold for any alpha is an element of R. We also provide examples of del(+/- 1)-Einstein structures for a class of compact homogeneous spaces M = G/K with two isotropy summands.