Tubular neighbourhood
A neighbourhood of a smooth submanifold in a smooth manifold
that is fibred over
with fibre
, where
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Suppose that in a Riemannian metric is chosen and consider segments of geodesics that are normal to
and start in
. If
is compact, then there exists an
such that no two segments of length
and starting at different points of
intersect. The union of all such segments of length
is an open neighbourhood
of
, and is called a tubular neighbourhood of
. It is possible to construct for a non-compact
a tubular neighbourhood by covering
with a countable family of compacta and by decreasing
as the number of elements of the covering increases. There is a deformation retract
associating with each point of
the beginning of a geodesic containing this point. This retract determines a vector bundle with fibre
that is isomorphic to the normal bundle
of the imbedding
. In this way, the quotient space
is homeomorphic to the Thom space of
.
An analogue of the notion of a tubular neighbourhood can also be introduced for topological manifolds (where one has to consider locally flat imbeddings, [2]).
References
[1] | R. Thom, "Quelques propriétés globales des variétés différentiables" Comm. Math. Helv. , 28 (1954) pp. 17–86 |
[2] | R.C. Kirby, L.C. Siebenmann, "Foundational essays on topological manifolds, smoothings, and triangulations" , Princeton Univ. Press (1977) |
Comments
Tubular neighbourhoods were introduced by H. Whitney in his treatment of differentiable manifolds (see [a2] for some history).
References
[a1] | M.W. Hirsch, "Differential topology" , Springer (1976) pp. Chapt. 5, Sect. 3 |
[a2] | J. Dieudonné, "A history of algebraic and differential topology: 1900–1960" , Birkhäuser (1989) pp. Chapt. III |
Tubular neighbourhood. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tubular_neighbourhood&oldid=15283