In this paper, we study left-invariant Einstein-like metrics on the compact Lie group G. Assume that there exist two subgroups, H⊂K⊂G, such G/K is a compact, connected, irreducible, symmetric space, and isotropy representation of G/H has exactly inequivalent, irreducible summands. We prove left metric ⟨·,·⟩t1,t2 G defined by first equation, must be an A-metric. Moreover, groups do not admit non...

We study three Möbius invariant metrics, and three affine invariant analogs, all of which are bilipschitz equivalent to the Poincaré hyperbolic metric. We exhibit numerous illustrative examples.

This paper is a review of recent results on integrable nonholonomic geodesic flows of left–invariant metrics and leftand right–invariant constraint distributions on compact Lie groups.

The starting point for constructing all known examples of compact manifolds with positive (or even quasi-positive) curvature is the fact that bi-invariant metrics on compact Lie groups are nonnegatively curved. In order to generalize this fundamental starting point, we address the question: given a compact Lie group G, classify the left-invariant metrics on G which have nonnegative curvature. T...

The Yamabe invariant Y(M) of a smooth compact manifold is roughly the supremum of the scalar curvatures of unit-volume constant-scalar-curvature Riemannian metrics g on M . (To be precise, one only considers those constant-scalar-curvature metrics which are Yamabe minimizers, but this technicality does not, e.g. affect the sign of the answer.) In this article, it is shown that many 4-manifolds ...

Let Gm,n be the Grassmann space of m-dimensional subspaces of F. Denote by θ1(X ,Y), . . . , θm(X ,Y) the canonical angles between subspaces X ,Y ∈ Gm,n. It is shown that Φ(θ1(X ,Y), . . . , θm(X ,Y)) defines a unitarily invariant metric on Gm,n for every symmetric gauge function Φ. This provides a wide class of new metrics on Gm,n. Some related results on perturbation and approximation of subs...