One of the classical problems in differential geometry is the investigation of closed manifolds which admit Riemannian metrics with given lower bounds for the sectional or the Ricci curvature and the study of relations between the existence of such metrics and the topology and geometry of the underlying manifold. Despite many efforts during the past decades, this problem is still far from being...

Abstract We classify Einstein metrics on $$\mathbb {R}^4$$ R 4 invariant under a four-dimensional group of isometries including principal action the Heisenberg group. consider which are either Ricci-flat or negative Ricci curvature. show that all ...

We investigate left-invariant Hitchin and hypo flows on 5-, 6and 7-dimensional Lie groups. They provide Riemannian cohomogeneity-one manifolds of one dimension higher with holonomy contained in SU(3), G2 and Spin(7), respectively, which are in general geodesically incomplete. Generalizing results of Conti, we prove that for large classes of solvable Lie groups G these manifolds cannot be comple...