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

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.