Normal Bundle Theorem

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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




Sources