Math 2.0

Differential geometry Riemannian and pseudo-Riemannian manifolds Theorem

Myers–Steenrod theorem

Builds on Riemannian manifoldRiemannian distanceIsometry of Riemannian manifoldsCompact space

Statement

Let (M,g)(M, g) and (N,h)(N, h) be connected Riemannian manifolds with Riemannian distances dgd_g and dhd_h.

  1. If φ ⁣:M→N\varphi \colon M \to N is a surjective map with dh(φ(p),φ(q))=dg(p,q)d_h(\varphi(p), \varphi(q)) = d_g(p, q) for all p,q∈Mp, q \in M, then φ\varphi is smooth and is an isometry of Riemannian manifolds.
  2. The group Isom⁡(M,g)\operatorname{Isom}(M, g) of all isometries of (M,g)(M, g) onto itself, with the compact-open topology, is a Lie group, and its action on MM is smooth.
  3. If MM is compact, then Isom⁡(M,g)\operatorname{Isom}(M, g) is compact.

Remarks

The idea of part 1 is that a distance-preserving map sends minimizing geodesics to minimizing geodesics, so in normal coordinates it acts as a linear isometry, and this forces smoothness. For part 2, note that an isometry of a connected Riemannian manifold is determined by its value and its differential at one point. So Isom⁡(M,g)\operatorname{Isom}(M, g) embeds in the orthonormal frame bundle of MM, and one shows that the image is a closed submanifold. For the round sphere SnS^n, the theorem gives Isom⁡(Sn)=O(n+1)\operatorname{Isom}(S^n) = O(n + 1), a compact Lie group of dimension n(n+1)/2n(n + 1)/2. The theorem is about Riemannian (positive definite) metrics; a pseudo-Riemannian metric defines no distance function.

Going deeper

For a student

Part 1 lets one pass freely between the metric-space view of a Riemannian manifold and its smooth view. Part 2 is the starting point for the study of homogeneous and symmetric spaces. A Riemannian manifold whose isometry group acts transitively is G/HG/H, with GG a Lie group and HH a compact subgroup. The Lie algebra of Isom⁡(M,g)\operatorname{Isom}(M, g) is identified with the complete Killing vector fields of MM.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Compact space, concept, no evidence levelCompact spaceconceptIsometry of Riemannian manifolds, concept, no evidence levelIsometry of Riemannia…conceptRiemannian distance, concept, no evidence levelRiemannian distanceconceptRiemannian manifold, concept, no evidence levelRiemannian manifoldconceptMyers–Steenrod theorem, statement, no evidence levelMyers–Steenrod theoremstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Compact space concept
  2. Isometry of Riemannian manifolds concept
  3. Riemannian distance concept
  4. Riemannian manifold concept
  5. Myers–Steenrod theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Myers%E2%80%93Steenrod_theorem
About this page
TagZ3VRMW8ETQ
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Kindstatement, theorem
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MSC53C20, 22E15
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AboutIsometry of Riemannian manifolds
UsesRiemannian manifold, Riemannian distance, Isometry of Riemannian manifolds, Compact space
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Changed2 November 2026

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