nLab
conformal map

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Definition

A smooth function f:X 1X 2f \;\colon\; X_1 \to X_2 between two Riemannian manifold (X 1,g 1)(X_1,g_1), (X 2,g 2)(X_2,g_2) is called conformal if it preserves the conformal geometry induced by the Riemannian metrics g 1g_1 and g 2g_2, hence if there is a smooth function r:X 1r\;\colon\; X_1 \to \mathbb{R} such that

f *g 2=rg 1. f^\ast g_2 = r g_1 \,.

If X 1=X 2X_1 = X_2 and ff is a diffeomorphism, then this is also called a conformal transformation.

Examples

References

Last revised on May 1, 2017 at 06:46:02. See the history of this page for a list of all contributions to it.