We show that a four-dimensional complete gradient shrinking Ricci soliton with positive isotropic curvature is either a quotient of S4 or a quotient of S3 × R. This gives a clean classification result removing the earlier additional assumptions in [14] by Wallach and the second author. The proof also gives a classification result on gradient shrinking Ricci solitons with nonnegative isotropic c...

We show an analogue of the Lorentzian splitting theorem for weighted Lorentz-Finsler manifolds: If a Berwald spacetime nonnegative Ricci curvature satisfies certain completeness and metrizability conditions includes timelike straight line, then it necessarily admits one-dimensional family isometric translations generated by gradient vector field Busemann function. Moreover, our formulation in t...

In this paper, we study Kropina metrics with isotropic scalar curvature. First, obtain the expressions of Ricci curvature tensor and Then, characterize on by analysis.

We investigate the notions of a connection of Finsler type and of Berwald type on the first jet bundle J1π of a manifold E which is fibred over IR. Such connections are associated to a given horizontal distribution on the bundle π0 1 : J 1π → E, which in particular may come from a time-dependent system of second-order ordinary differential equations. In order to accomodate three existing constr...

in this paper, we study a class of finsler metrics which contains the class of p-reducible andgeneral relatively isotropic landsberg metrics, as special cases. we prove that on a compact finsler manifold,this class of metrics is nothing other than randers metrics. finally, we study this class of finsler metrics withscalar flag curvature and find a condition under which these metrics reduce to r...

Let (Mn, g), n ≥ 4, be a compact simply-connected Riemannian manifold with nonnegative isotropic curvature. Given 0 < l ≤ L, we prove that there exists ε = ε(l, L, n) satisfying the following: If the scalar curvature s of g satisfies l ≤ s ≤ L and the Einstein tensor satisfies |Ric − s n g| ≤ ε then M is diffeomorphic to a symmetric space of compact type. This is a smooth analogue of the result...