the main objective of this paper is to find the necessary and sufficient condition of a given finslermetric to be einstein in order to classify the einstein finsler metrics on a compact manifold. the consideredeinstein finsler metric in the study describes all different kinds of einstein metrics which are pointwiseprojective to the given one. this study has resulted in the following theorem tha...

In Finsler geometry, there are infinitely many models of constant curvature. The Funk metrics, the Hilbert-Klein metrics and the Bryant metrics are projectively flat with non-zero constant curvature. A recent example constructed by the author is projectively flat with zero curvature. In this paper, we introduce a technique to construct non-projectively flat Finsler metrics with zero curvature i...

‎In this paper we study Finsler metrics with orthogonal invariance‎. ‎We‎ ‎find a partial differential equation equivalent to these metrics being locally projectively flat‎. ‎Some applications are given‎. ‎In particular‎, ‎we give an explicit construction of a new locally projectively flat Finsler metric of vanishing flag curvature which differs from the Finsler metric given by Berwald in 1929.

The well-known Funk metric F (x, y) is projectively flat with constant flag curvature K = −1/4 and the Hilbert metric Fh(x, y) := (F (x, y) + F (x,−y))/2 is projectively flat with constant curvature K = −1. These metrics are the special solutions to Hilbert’s Fourth Problem. In this paper, we construct a non-trivial R-flat spray using the Funk metric. It is then an inverse problem in the calcul...

In this article, the aim is to introduce a para-Sasakian manifold with a canonical paracontact connection. It is shown that φ−conharmonically flat , φ−W2 flat and φ−pseudo projectively flat para-Sasakian manifolds with respect to canonical paracontact connection are all η−Einstein manifolds. Also, we prove that quasi-pseudo projectively flat para-Sasakian manifolds are of constant scalar curvat...

In the context of uniformisation problems, we study projective varieties with klt singularities whose cotangent sheaf admits a projectively flat structure over smooth locus. Generalising work Jahnke-Radloff, show that torus quotients are only semistable and extremal Chern classes. An analogous result for nef normalised sheaves follows.

Theorem 1. Let T 2 be the two dimensional torus. Then for any positive integer m there is a complete torsion free projectively flat connection, ∇, on T 2 such that for any point p ∈ T 2 there is a point q ∈ T 2 with the property that any broken ∇-geodesic between p and q has at least m breaks. Moreover if T 2 is viewed as a Lie group in the usual manner, this connection is invariant under trans...