Which smooth compact n-manifolds admit Riemannian metrics of constant Ricci curvature? A direct variational approach sheds some interesting light on this problem, but by no means answers it. This article surveys some recent results concerning both Einstein metrics and the associated variational problem, with the particular aim of highlighting the striking manner in which the 4-dimensional case ...

The equation also appears in investigation of geodesically equivalent metrics. Recall that two metrics on one manifold are geodesically equivalent, if every geodesic of one metric is a reparametrized geodesic of the second metric. Solodovnikov [9] has shown that Riemannian metrics on (n > 3)−dimensional manifolds admitting nontrivial 3-parameter family of geodesically equivalent metrics allow n...

We show that there is a C∞ open and dense set of positively curved metrics on S2 whose geodesic flow has positive topological entropy, and thus exhibits chaotic behavior. The geodesic flow for each of these metrics possesses a horseshoe and it follows that these metrics have an exponential growth rate of hyperbolic closed geodesics. The positive curvature hypothesis is required to ensure the ex...

In this paper, we study a class of Finsler metrics defined by a Riemannian metric and a 1-form on a manifold. We find an equation that characterizes Douglas metrics on a manifold of dimension n ≥ 3.

We discuss the problem of prescribing the mean curvature and conformal class as boundary data for Einstein metrics on 3-manifolds, in the context of natural elliptic boundary value problems for Riemannian metrics.

We prove the Riemannian Penrose Conjecture, an important case of a conjecture [41] made by Roger Penrose in 1973, by defining a new flow of metrics. This flow of metrics stays inside the class of asymptotically flat Riemannian 3-manifolds with nonnegative scalar curvature which contain minimal spheres. In particular, if we consider a Riemannian 3-manifold as a totally geodesic submanifold of a ...

On the manifold M(M) of all Riemannian metrics on a compact manifold M one can consider the natural L-metric as described first by [11]. In this paper we consider variants of this metric which in general are of higher order. We derive the geodesic equations, we show that they are well-posed under some conditions and induce a locally diffeomorphic geodesic exponential mapping. We give a conditio...

Since its introduction by Élie Cartan, the holonomy of a connection has played an important role in differential geometry. One of the best known results concerning holonomy is Berger’s classification of the possible holonomies of Levi-Civita connections of Riemannian metrics. Since the appearance of Berger [1955], much work has been done to refine his list of possible Riemannian holonomies. See...

For each n 2 3, we present a family of Riemannian metrics g on W” such that each Riemannian manifold M” = (IT’, g) has positive bottom of the spectrum of Laplacian A, (M”) > 0 and bounded geometry 1 K 1 < C but M” admits no non-constant bounded harmonic functions. These Riemannian manifolds mentioned above give a negative answer to a problem addressed by Schoen-Yau [ 181 in dimension n > 3.

In Theorem 1, we generalize the results of Szabó [Sz1, Sz2] for Berwald metrics that are not necessary strictly convex: we show that for every Berwald metric F there always exists a Riemannian metric affine equivalent to F . Further, we investigate geodesic equivalence of Berwald metrics. Theorem 2 gives a system of PDE that has a (nontrivial) solution if and only if the given essentially Berwa...