Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
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Riemannian isometry and local isometry
Definition
An isometry is a diffeomorphism with . A local isometry is a smooth local diffeomorphism with . An isometric immersion is a smooth immersion satisfying that same pullback identity.
Use Pullback of a riemannian metric as a tensor and Diffeomorphisms and local diffeomorphisms of manifolds. Positivity forces injective differential by Pullback of a riemannian metric is riemannian exactly for immersions. In equal dimensions on boundaryless manifolds, the inverse function theorem as applied in the next proposition makes a metric-preserving smooth map a local isometry. At a boundary the definition retains the local-diffeomorphism requirement. An isometric immersion need not have equal source and target dimensions.
Source locator
Lee, Introduction to Smooth Manifolds, 2nd ed., Chapter 13, pp.328–332 and 341–342.