Characteristics of riemannian manifolds and Pseudo-Riemannian metric

In the present paper we characterize the Riemannian manifolds, pseudo-Riemannian metric and several applications of Riemannian manifolds have been investigated. We discuss linear connections, connectors, torsion and space of all covariant derivatives on Riemannian manifolds and also geometry of geodesics structure of Riemann manifold. In a pseudo-Riemann manifold there is a torsion-free covariant derivative which is compatible with the Riemann metric. Further characterize the geodesic distance conformal metrics, sectional curvature relations to vector analysis in three dimensions of Riemannian and pseudo-Riemannian manifold and several theorems are investigated.