Conformality is Equivalence Relation on Set of Riemannian Metrics
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Theorem
Let $M$ be a smooth manifold.
Let $\set g$ be the set of Riemannian metrics on $M$.
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Suppose $\set g$ is equipped with the conformality relation.
That is, suppose that for any pair of Riemannian metrics there is a conformal transformation.
Then conformality relation is equivalence relation.
Proof
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Sources
- 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 3$: Model Riemannian Manifolds. Euclidean Spaces