commutative properties of four-dimensional generalized symmetric pseudo-riemannian manifolds were considered. specially, in this paper, we studied skew-tsankov and jacobi-tsankov conditions in 4-dimensional pseudo-riemannian generalized symmetric manifolds.

I. Introduction-a quick historical survey of geodesic flows on negatively curved spaces. II. Preliminaries on Riemannian manifolds A. Riemannian metric and Riemannian volume element B. Levi Civita connection and covariant differentiation along curves C. Parallel translation of vectors along curves D. Curvature E. Geodesics and geodesic flow F. Riemannian exponential map and Jacobi vector fields...

Commutative properties of four-dimensional generalized symmetric pseudo-Riemannian manifolds were considered. Specially, in this paper, we studied Skew-Tsankov and Jacobi-Tsankov conditions in 4-dimensional pseudo-Riemannian generalized symmetric manifolds.

In this paper, we prove that a non-Riemannian isotropic Berwald metric or a non-Riemannian (α,β) -metric admits no concurrent vector fields. We also prove that an L-reducible Finsler metric admitting a concurrent vector field reduces to a Landsberg metric.In this paper, we prove that a non-Riemannian isotropic Berwald metric or a non-Riemannian (α,β) -metric admits no concurrent vector fi...

Suppose that Σ = ∂M is the n-dimensional boundary of a connected compact Riemannian spin manifold (M, 〈 , 〉) with nonnegative scalar curvature, and that the (inward) mean curvature H of Σ is positive. We show that the first eigenvalue of the Dirac operator of the boundary corresponding to the conformal metric 〈 , 〉H = H 〈 , 〉 is at least n/2 and equality holds if and only if there exists a non-...

If M is the underlying smooth oriented 4-manifold of a Del Pezzo surface, we consider the set of Riemannian metrics h on M such that W(ω, ω) > 0, where W is the self-dual Weyl curvature of h, and ω is a non-trivial self-dual harmonic 2-form on (M,h). While this open region in the space of Riemannian metrics contains all the known Einstein metrics on M , we show that it contains no others. Conse...