Definition:Conformal Transformation between Riemannian Manifolds
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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.
Sources
- 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 3$: Model Riemannian Manifolds. Euclidean Spaces