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So we know that given a Riemannian manifold $(M,g)$ we can calculate its Riemannian curvature $R$. This covariant $4$-tensor field then satisfies some important symmetries \begin{gather*} R_{ijkl} = - ...

Pg.248 of "Textbook in Tensor Calculus and Differential Geometry" by Prasun Nayak. Let us suppose that $\lambda_{h|}^i$ is not a unit vector and therefore, the mean curvature $M_h$ in this ...

Given a Riemannian manifold $(M,g)$ is it possible to calculate the distance between two points on this manifold. Is it possible the inverse? That means: given a formula of the distance, for example: ...