Definition:Local Isometry
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Definition
Let $\struct {M, g}$ and $\struct {\tilde M, \tilde g}$ be Riemannian manifolds.
Let $p \in M$ be a point.
Let $\phi : M \to \tilde M$ be a mapping such that for each $p$ there is a neighborhood $U$ such that $\bigvalueat \phi U$ is an isometry onto an open subset of $\tilde M$.
Then $\phi$ is called the local isometry.
This article, or a section of it, needs explaining. In particular: The terminology suggests there is only one such. If that is the case, then it needs to be "with respect to" some entity. Is it in fact a mistake and should it be a local isometry? You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code. |
Sources
- 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Definitions