Normal bundle

of a submanifold

The vector bundle consisting of tangent vectors to the ambient manifold that are normal to the submanifold. If $X$ is a Riemannian manifold, $Y$ is an (immersed) submanifold of it, $T _ {X}$ and $T _ {Y}$ are the tangent bundles over $X$ and $Y$( cf. Tangent bundle), then the normal bundle $N _ {Y/X}$ of $Y$ is the subbundle in $T _ {X} \mid _ {Y}$ consisting of the vectors $u \in T _ {X,y }$, $y \in Y$, that are orthogonal to $T _ {Y,y}$.

With the help of normal bundles one constructs, for example, tubular neighbourhoods of submanifolds (cf. Tubular neighbourhood). The normal bundle over $Y$, regarded up to equivalence, does not depend on the choice of the Riemannian metric on $X$, since it can be defined without recourse to the metric as the quotient bundle $T _ {X} \mid _ {Y} / T _ {Y}$ of the tangent bundle $T _ {X}$ restricted to $Y$ by the vector bundle $T _ {Y}$. Somewhat more general is the construction of the normal bundle of an arbitrary immersion (cf. Immersion of a manifold) $f: Y \rightarrow X$ of differentiable manifolds:

$$N _ {Y/X} = \ f ^ { * } T _ {X} / T _ {Y} .$$

Similarly one defines the normal bundle $N _ {Y/X}$ of a non-singular algebraic subvariety $Y$ in a non-singular algebraic variety $\overline{X}\;$ or that of an analytic submanifold $Y$ in an analytic manifold $X$; it is an algebraic (or analytic) vector bundle over $Y$ of rank $\mathop{\rm codim} Y$. In particular, if $\mathop{\rm codim} Y = 1$, then $N _ {Y/X}$ is isomorphic to the restriction to $Y$ of the bundle over $X$ that determines the divisor $Y$.

When $Y$ is an analytic subspace of an analytic space $( X, {\mathcal O} _ {X} )$, the normal bundle of $Y$ is sometimes defined as the analytic family of vector spaces $N _ {Y/X} \rightarrow Y$ dual to the conormal sheaf $N _ {Y/X} ^ {*}$( see Normal sheaf). For applications of normal bundles to the problem of contractibility of submanifolds see Exceptional analytic set; Exceptional subvariety.

References

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[4]	M.W. Hirsch, "Differential topology" , Springer (1976) MR0448362 Zbl 0356.57001
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[a1]	N.E. Steenrod, "The topology of fibre bundles" , Princeton Univ. Press (1951) MR0039258 Zbl 0054.07103