The Curvature of a Conformally Flat Manifold

The standard N-dimensional sphere is an example of a smooth manifold that is curved but conformally flat. This article calculates the Ricci tensor for an arbitrary N-dimensional conformally flat manifold with arbitrary signature and then shows that the result simplifies as expected when the space is maximally symmetric. The result is used to show that the scalar curvature for a standard sphere of radius σ is N(N-1)/σ^2.

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Revision history

2023-12-16 (clarified the definition of conformally flat)
2023-12-11 (updated a paragraph about notation in the introduction, added a sentence acknowledging that dS and AdS spacetimes are examples)
2023-12-10 (generalized the sphere example to a family of maximally symmetric spaces, added the result for the full Ricci tensor instead of only the diagonal components, fixed an incorrect coefficient in the result for the Ricci tensor, fixed an index-notation collision, updated the abstract)
2023-12-09 (first version)

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updated 2024-05-04

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The Curvature of a Conformally Flat Manifold

Revision history