Normal bundle of a submanifold

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Definition

Definition for differential manifolds

Let M be a differential manifold (of dimension m) and P a submanifold (of dimension p) of M. Then:

  1. The tangent bundle TM restricts to a vector bundle over P. This is a bundle with fibers having dimension n
  2. The tangent bundle TP is a p-dimensional vector bundle over P. This is a subbundle of the preceding bundle.

The quotient of the first bundle by the second is termed the normal bundle to P in M. It is a vector bundle with fibers having dimension m - p.

Definition for Riemannian manifolds

Let M be a Riemannian manifold and P be a submanifold. We have two bundles:

  1. The tangent bundle TM restricts to a vector bundle over P. This is a bundle with fibers having dimension n
  2. The tangent bundle TP is a p-dimensional vector bundle over P. This is a subbundle of the preceding bundle.

The normal bundle to P in M is defined as the orthogonal complement to bundle (2) in bundle (1).

There is a natural isomorphism between this and the normal bundle defined abstractly for Riemannian manifolds, based on the general fact that a direct sum complement to a subbundle, is isomorphic to the quotient.