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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[2]	J.W. Milnor, J.D. Stasheff, "Characteristic classes" , Princeton Univ. Press (1974) MR0440554 Zbl 0298.57008
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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