Normal Bundle Theorem
Jump to navigation
Jump to search
Theorem
Let $\tilde M$ be an $m$-dimensional Riemannian manifold.
Let $M \subseteq \tilde M$ be an immersed or embedded $n$-dimensional submanifold with or without boundary.
Let $\valueat {T \tilde M} M$ be the ambient tangent bundle.
Let $NM$ be the normal bundle of $M$.
Then $NM$ is a rank-$\paren {m - n}$ smooth vector subbundle of $\valueat {T \tilde M} M$.
Proof
This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. 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 {{ProofWanted}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
Sources
- 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Methods for Constructing Riemannian Metrics