# Normal bundle of a submanifold

## Definition

### Definition for differential manifolds

Let be a differential manifold (of dimension ) and a submanifold (of dimension ) of . Then:

- The tangent bundle restricts to a vector bundle over . This is a bundle with fibers having dimension
- The tangent bundle is a -dimensional vector bundle over . This is a subbundle of the preceding bundle.

The quotient of the first bundle by the second is termed the **normal bundle** to in . It is a vector bundle with fibers having dimension .

### Definition for Riemannian manifolds

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

- The tangent bundle restricts to a vector bundle over . This is a bundle with fibers having dimension
- The tangent bundle is a -dimensional vector bundle over . This is a subbundle of the preceding bundle.

The normal bundle to in 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.