Definition:Conformal Transformation between Riemannian Manifolds

Definition
Let $\struct {M, g}$ and $\struct {\tilde M, \tilde g}$ be Riemannian manifolds.

Let $\phi : M \to \tilde M$ be a diffeomorphism.

Let $f \in \map {C^\infty} M$ be a positive smooth real function.

Suppose $\phi$ pulls $\tilde g$ back to a metric that is conformal to $g$:


 * $\exists f \in \map {C^\infty} M : f > 0 : \phi^* \tilde g = f g$

Then $\phi$ is called a conformal transformation.