List of Formulas in Riemannian Geometry - Under A Conformal Change

Under A Conformal Change

Let be a Riemannian metric on a smooth manifold, and a smooth real-valued function on . Then

is also a Riemannian metric on . We say that is conformal to . Evidently, conformality of metrics is an equivalence relation. Here are some formulas for conformal changes in tensors associated with the metric. (Quantities marked with a tilde will be associated with, while those unmarked with such will be associated with .)

Note that the difference between the Christoffel symbols of two different metrics always form the components of a tensor.

Here is the Riemannian volume element.

Here is the Kulkarni–Nomizu product defined earlier in this article. The symbol denotes partial derivative, while denotes covariant derivative.

Beware that here the Laplacian is minus the trace of the Hessian on functions,

Thus the operator is elliptic because the metric is Riemannian.

If the dimension, then this simplifies to

We see that the (3,1) Weyl tensor is invariant under conformal changes.

Let be a differential -form. Let be the Hodge star, and the codifferential. Under a conformal change, these satisfy