Homogeneous Riemannian manifolds with generic Ricci tensor (Q2773305)

A Riemannian homogeneous space \(M\) is called a space with generic Ricci tensor if its Ricci endomorphism \(\rho\) has \(\dim M\) different eigenvalues. The author proves that a simply connected Riemannian homogeneous space \(M\) with generic Ricci tensor is isometric to a Lie group \(G\) with a left-invariant metric.NEWLINENEWLINE Not every geodesic in \(M\) is an orbit of a 1-parameter group of isometries. If \(\nabla_X\rho(X,X) = 0\) for any \(X\in T(M)\), then \(G\) is unimodular. The proof is based on the study of vector fields commuting with the fundamental vector fields on \(M\) corresponding to a transitive isometric action; these vector fields determine a totally geodesic foliation with leaves locally isometric to a Lie group with a left-invariant metric. Generalizing Thurston's example of a symplectic manifold which does not admit any Kähler structure, the author also proves the following theorem: Let \(G\) be a unimodular simply connected Lie group, \(\dim G = 4\) and \(\dim [G,G]\leq 2\). Then, for any left-invariant Riemannian metric on \(G\) there exist two compatible mutually opposite invariant almost Kähler structures. As a corollary, a family of locally homogeneous metrics with generic Ricci tensor on \(T^4\), admitting two compatible mutually opposite locally invariant strictly almost Kähler structures, is constructed.