Normal bundle
of a submanifold
The vector bundle consisting of tangent vectors to the ambient manifold that are normal to the submanifold. If 
is a Riemannian manifold, 
is an (immersed) submanifold of it, 
and 
are the tangent bundles over 
and 
(cf. Tangent bundle), then the normal bundle 
of 
is the subbundle in 
consisting of the vectors 
, 
, that are orthogonal to 
.
With the help of normal bundles one constructs, for example, tubular neighbourhoods of submanifolds (cf. Tubular neighbourhood). The normal bundle over 
, regarded up to equivalence, does not depend on the choice of the Riemannian metric on 
, since it can be defined without recourse to the metric as the quotient bundle 
of the tangent bundle 
restricted to 
by the vector bundle 
. Somewhat more general is the construction of the normal bundle of an arbitrary immersion (cf. Immersion of a manifold) 
of differentiable manifolds:
 |
Similarly one defines the normal bundle 
of a non-singular algebraic subvariety 
in a non-singular algebraic variety 
or that of an analytic submanifold 
in an analytic manifold 
; it is an algebraic (or analytic) vector bundle over 
of rank 
. In particular, if 
, then 
is isomorphic to the restriction to 
of the bundle over 
that determines the divisor 
.
When 
is an analytic subspace of an analytic space 
, the normal bundle of 
is sometimes defined as the analytic family of vector spaces 
dual to the conormal sheaf 
(see Normal sheaf). For applications of normal bundles to the problem of contractibility of submanifolds see Exceptional analytic set; Exceptional subvariety.
References
| [1] | A.L. Onishchik, "Pseudoconvexity in the theory of complex spaces" J. Soviet Math. , 14 : 3 (1980) pp. 1363–1406 Itogi Nauk. i Tekhn. Algebra. Topol. Geom. , 15 (1977) pp. 93–156 Zbl 0449.32020 |
| [2] | J.W. Milnor, J.D. Stasheff, "Characteristic classes" , Princeton Univ. Press (1974) MR0440554 Zbl 0298.57008 |
| [3] | V.A. Rokhlin, D.B. Fuks, "Beginner's course in topology. Geometric chapters" , Springer (1984) (Translated from Russian) MR759162 |
| [4] | M.W. Hirsch, "Differential topology" , Springer (1976) MR0448362 Zbl 0356.57001 |
| [5] | I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001 |
Comments
References
| [a1] | N.E. Steenrod, "The topology of fibre bundles" , Princeton Univ. Press (1951) MR0039258 Zbl 0054.07103 |
Normal bundle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Normal_bundle&oldid=14667