We introduce an evolution equation which deforms metrics on 3-manifolds with sectional curvature of one sign. Given a closed 3-manifold with an initial metric with negative sectional curvature, we conjecture that this flow will exist for all time and converge to a hyperbolic metric after a normalization. We shall establish a monotonicity formula in support of this conjecture. Note that in contr...

A Finsler metric is of sectional flag curvature if its flag curvature depends only on the section. In this article, we characterize Randers metrics of sectional flag curvature. It is proved that any non-Riemannian Randers metric of sectional flag curvature must have constant flag curvature if the dimension is greater than two. 0. Introduction Finsler geometry has a long history dated from B. Ri...

In this paper‎, ‎the‎ ‎authors prove that a strictly Kähler-Berwald manifold with‎ ‎nonzero constant holomorphic sectional curvature must be a‎ Kähler manifold‎. 

We establish a new algebraic characterization of sectional curvature bounds $\sec\geq k$ and $\sec\leq using only terms in the Weitzenb\"ock formulae for symmetric $p$-tensors. By introducing analogue Kulkarni-Nomizu product, we provide simple formula such terms. also give an application Bochner technique to closed $4$-manifolds with indefinite intersection form $\sec>0$ or $\sec\geq0$, obtaini...

We characterize the natural diagonal almost product (locally product) structures on the tangent bundle of a Riemannian manifold. We obtain the conditions under which the tangent bundle endowed with the determined structure and with a metric of natural diagonal lift type is a Riemannian almost product (locally product) manifold, or an (almost) para-Hermitian manifold. We find the natural diagona...

Michor and Mumford have shown that the distances between planar curves in the simplest metric (not involving derivatives) are identically zero. We derive geodesic equations and a formula for sectional curvature for conformally equivalent metrics. We show if the conformal factor depends only on the length of the curve, the metric behaves like an L metric, the sectional curvature is not bounded f...