Tubular neighbourhood

A neighbourhood of a smooth submanifold in a smooth manifold that is fibred over with fibre , where

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