Expressing the curvature tensor and connection of a given metric in terms of those of another metric

TL;DR

This paper derives formulas expressing the connection and curvature tensor of a second metric on a Riemannian manifold in terms of an original metric's curvature and derivatives, generalizing coordinate expressions.

Contribution

It provides generalized formulas for the connection and curvature of one metric in terms of another, extending coordinate expressions to a more abstract setting.

Findings

Formulas relate $m$'s connection and curvature to $g$'s curvature and derivatives.

Special case recovers classical coordinate expressions when $g$ is Euclidean.

Generalizes coordinate formulas to abstract Riemannian manifolds.

Abstract

Let $(M,g)$ be a Riemannian manifold, and $m$ be a second metric on $M$. We give expressions of $m$'s associated connection, and Riemann curvature tensor $R_m$, in terms of $R_g$ and certain combinations of covariant derivatives of $m$ (with respect to the Levi-Civita connection associated with $g$). The formulas turn out to be generalizations of the coordinate expressions. Coordinate expression formulas can be recovered from ours by setting $g$ as the Euclidean metric induced by a given coordinate chart. As the covariant derivative induced by $g$ becomes the ordinary partial derivative and the $R_g$ tensor vanishes, the formulas coincide with the well-known coordinate expressions for $m$'s connection and curvature tensor.