PropositionStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10

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Riemannian isometries form a group and local isometries are local diffeomorphisms

Statement

Isometries of a fixed Riemannian manifold form a group. A metric-preserving smooth map between equal-dimensional boundaryless Riemannian manifolds is a local diffeomorphism.

Facts & Assumptions

Given: Isometries of $(M, g)$ , and a smooth $F : (P, g_{P}) \to (Q, g_{Q})$ with $F^{*} g_{Q} = g_{P}$ and equal dimensions for the second assertion.

[F1]

Riemannian isometry and local isometry: An isometry $F : (M, g) \to (N, h)$ is a diffeomorphism with $F^{*} h = g$ . A local isometry is a smooth local diffeomorphism with $F^{*} h = g$ . An isometric immersion is a smooth immersion satisfying that same pullback identity. Use def-pullback-riemannian-metric and def-diffeomorphism-and-local-diffeomorphism-of-manifolds. Positivity forces injective differential by prop-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.

[F2]

Pullback of covariant tensors is smooth and functorial: If $F : M \to N$ is smooth and $T$ is a smooth covariant tensor field on $N$ , then $F^{*} T$ is a smooth covariant tensor field on $M$ . Moreover, $({id}_{M})^{*} T = T, (G \circ F)^{*} T = F^{*} (G^{*} T)$ for every composable smooth map $G$ .

[F3]

The smooth inverse function theorem on manifolds: Let $F : M \to N$ be a smooth map and let $p \in M$ . If $d F_{p} : T_{p} M \to T_{F (p)} N$ is an isomorphism, then there are open neighbourhoods $U$ of $p$ and $V$ of $F (p)$ such that $F ∣_{U} : U \to V$ is a diffeomorphism.

Proof

technique · direct

1.1

Identity preserves $g$ , and if $a^{*} g = b^{*} g = g$ then $(b \circ a)^{*} g = a^{*} b^{*} g = g$ . For an isometry $a$ , $(a^{- 1})^{*} g = (a^{- 1})^{*} a^{*} g = (a \circ a^{- 1})^{*} g = g$ . Composition of diffeomorphisms is associative, so these identities give the group laws.

F1F2given

2.1

If $d F_{p} v = 0$ , then $g_{P} (v, v) = g_{Q} (d F_{p} v, d F_{p} v) = 0$ , hence $v = 0$ . Equal finite dimensions make $d F_{p}$ an isomorphism. The smooth inverse function theorem on the boundaryless manifolds supplies a diffeomorphism on a neighbourhood of each point, exactly the local-diffeomorphism conclusion. This includes dimension zero.

F1F3given∎

Source locator

Lee, Introduction to Smooth Manifolds, 2nd ed., Chapter 13, pp.328–332 and 341–342.

Depends on

Used by

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Sources

John M. Lee, Introduction to Smooth Manifolds, second edition (standard reference, not scraped)