We numerically calculate Perelman’s entropy for a variety of canonical metrics on CP-bundles over products of Fano Kähler-Einstein manifolds. The metrics investigated are Einstein metrics, Kähler-Ricci solitons and quasi-Einstein metrics. The calculation of the entropy allows a rough picture of how the Ricci flow behaves on each of the manifolds in question.

In 1984 C.Shibata has dealt with a change of Finsler metric which is called a β-change of metric [12]. For a β-change of Finsler metric, the differential oneform β play very important roles. In 1985 M.Matsumoto studied the theory of Finslerian hypersurfaces [6]. In there various types of Finslerian hypersurfaces are treated and they are called a hyperplane of the 1st kind, a hyperplane of the 2...

We show that by taking a certain scaling limit of a Euclideanised form of the Plebanski–Demianski metrics one obtains a family of local toric Kähler–Einstein metrics. These can be used to construct local Sasaki–Einstein metrics in five dimensions which are generalisations of the Y p,q manifolds. In fact, we find that these metrics are diffeomorphic to those recently found by Cvetic, Lu, Page an...

We elaborate an unified geometric approach to classical mechanics, Riemann–Finsler spaces and gravity theories on Lie algebroids provided with nonlinear connection (N–connection) structure. There are investigated the conditions when the fundamental geometric objects like the anchor, metric and linear connection, almost sympletic and related almost complex structures may be canonically defined b...

We study a class of two-dimensional Finsler metrics defined by a Riemannian metric α and a 1-form β. We characterize those metrics which are Douglasian or locally projectively flat by some equations. In particular, it shows that the known fact that β is always closed for those metrics in higher dimensions is no longer true in two-dimensional case. Further, we determine the local structures of t...

We study the existence of projectable G-invariant Einstein metrics on the total space of G-equivariant fibrations M = G/L → G/K, for a compact connected semisimple Lie group G. We obtain necessary conditions for the existence of such Einstein metrics in terms of appropriate Casimir operators, which is a generalization of the result by Wang and Ziller about Einstein normal metrics. We describe b...

We locally classify all SO(3)-invariant four-dimensional pseudo-Finsler Berwald structures. These are Finslerian geometries which closest to (spatially, or SO(3))-spherically symmetric pseudo-Riemannian ones — and serve as ansatz find solutions of Finsler gravity equations generalize the Einstein equations. that there exist five classes non-pseudo-Riemannian (i.e. non-quadratic in velocities) S...

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...