Conformal structure on a differential manifold

Definition

Given data

A connected differential manifold $M$.

Definition part

A conformal structure on $M$ associates to $M$ a class of conformally equivalent metrics, viz it associates to $M$ a Riemannian metric that is loose upto conformal equivalence.

Thus, every Riemannian metric gives a conformal structure and every conformal structure arises from a Riemannian metric, but different Riemannian metrics may give rise to the same conformal structure.

Facts

Volume-normalization of a conformal structure

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