Normal bundle of a submanifold

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.