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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.
Depends on
Used by
- A local isometry from a complete connected manifold has geodesically complete target image Corollary
- Bonnet-Myers fundamental group is finite Corollary
- Diameter rigidity from toponogov under a sectional lower bound Corollary
- Riemannian isometries preserve length and distance Corollary
- Simply connected complete nonpositively curved manifolds have unique geodesics between points Corollary
- A flat torus showing simple connectedness is needed for global exp injectivity Example
- Killing jacobi fields from rotations Example
- Projective-plane curvature via a hemisphere Example
- Toponogov comparison on a round sphere Example
- Arcs joining two points of a connected submanifold avoiding finitely many points Lemma
- Closed hyperbolic three-manifold groups contain no Z-squared subgroup Lemma
- Local isometries send geodesics to geodesics Lemma
- Flat torus model geometry Proposition
- Killing fields restrict to Jacobi fields along geodesics Proposition
- Riemannian isometries form a group and local isometries are local diffeomorphisms Proposition
- A complete local isometry is a covering map Theorem
- A Riemannian manifold is flat iff it is locally isometric to Euclidean space Theorem
- Cartan hadamard Theorem
- Cheng maximal diameter rigidity Theorem
Dependency tree · two levels
10 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- John M. Lee, Introduction to Smooth Manifolds, second edition (standard reference, not scraped)