Tubular neighbourhood

A neighbourhood of a smooth submanifold $ N $ in a smooth manifold $ M $ that is fibred over $ N $ with fibre $ \mathbf R ^ {d} $, where

$$ d = \mathop{\rm dim} M - \mathop{\rm dim} N. $$

Suppose that in $ M $ a Riemannian metric is chosen and consider segments of geodesics that are normal to $ N $ and start in $ N $. If $ N $ is compact, then there exists an $ \epsilon > 0 $ such that no two segments of length $ \leq \epsilon $ and starting at different points of $ N $ intersect. The union of all such segments of length $ < \epsilon $ is an open neighbourhood $ U $ of $ N $, and is called a tubular neighbourhood of $ N $. It is possible to construct for a non-compact $ N $ a tubular neighbourhood by covering $ N $ with a countable family of compacta and by decreasing $ \epsilon $ as the number of elements of the covering increases. There is a deformation retract $ r: U \rightarrow N $ associating with each point of $ U $ the beginning of a geodesic containing this point. This retract determines a vector bundle with fibre $ \mathbf R ^ {d} $ that is isomorphic to the normal bundle $ \nu $ of the imbedding $ N \rightarrow M $. In this way, the quotient space $ \overline{U}\; / \partial \overline{U}\; $ is homeomorphic to the Thom space of $ \nu $.

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