normal bundle

Let $X$ be an immersed submanifold of $M$, with immersion $i:X\to M$. Then we can restrict the tangent bundle of $M$ to $N$ or more properly, take the pullback $i^{*}TM$. This, as an vector bundle over $X$ should contain a lot of information about the embedding of $X$ into $M$. But there is a natural injection $TX\to i^{*}TM$, and the subbundle which is the image of this only has information on the intrinsic properties of $X$, and thus is useless in obtaining information about the embedding of $X$ into $M$. Instead, to get information on this, we take the quotient $i^{*}TM/TX=NX$, the normal bundle of$X$. The normal bundle is very strongly dependent on the immersion $i$. If $E$ is any vector bundle on $X$, then $E$ is the normal bundle for the embedding of $X$ into $E$ as the zero section.

The normal bundle determines the local geometry of the embedding of $X$ into $M$ in the following sense: In $M$, there exists an open neighborhood $U\supset X$ which is diffeomorphic to $NX$ by a diffeomorphism taking $X$ to the zero section.