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

الأصل · EN

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 Rₘ, in terms of Rg 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 Rg tensor vanishes, the formulas coincide with the well-known coordinate expressions for m's connection and curvature tensor.