Introduction to Riemannian Manifolds by John M. Lee

Author:John M. Lee
Language: eng
Format: epub, pdf
ISBN: 9783319917559
Publisher: Springer International Publishing

Proposition 7.29

(Conformal Transformation of the Levi-Civita Connection). Let (M, g) be a Riemannian or pseudo-Riemannian n-manifold (with or without boundary), and let be any metric conformal to g. If and denote the Levi-Civita connections of g and , respectively, then

(7.42)

where the gradient on the right-hand side is that of g. In any local coordinates, the Christoffel symbols of the two connections are related by

(7.43)

where is the ith component of .

Proof.

Formula (7.43) is a straightforward computation using formula (5.​10) for the Christoffel symbols in coordinates, and then (7.42) follows by expanding everything in coordinates and using (7.43).

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