Theorem
Tubular neighborhood theorem
Every embedded submanifold has a neighborhood modeled on a neighborhood of the zero section in its normal bundle.
Statement
Let be an embedded submanifold of a smooth manifold. The tubular neighborhood theorem states that admits a tubular neighborhood: some open neighborhood of the zero section in the normal bundle is diffeomorphic to an open neighborhood of in , and the diffeomorphism restricts to the inclusion on the zero section. No compactness hypothesis on is needed; for noncompact , the allowable radius in the normal directions may vary from point to point.
Construction idea
Choose a Riemannian metric on , identify the normal bundle with the orthogonal complement of in , and apply the exponential map to normal vectors. The inverse function theorem gives the required model near each zero vector. A locally finite shrinking argument produces a single open neighborhood on which this map is injective Hirsch, Chapter 4.
Consequences
After choosing a fiberwise star-shaped domain, the tubular model deformation retracts onto . It therefore gives a controlled neighborhood with the homotopy type of the submanifold. The theorem is also the starting point for the Thom isomorphism, collar-like constructions away from boundaries, and isotopy extension arguments.
Scope and refinements
The theorem guarantees existence but not a canonical neighborhood: different metrics and shrinkings give different embeddings. Relative and equivariant versions require compatibility with extra subsets or group actions. For manifolds with boundary, neatness or other boundary conditions are normally imposed so that the tubular model respects the boundary.
References
- John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Springer, 2012. DOI record. Relevant: Chapter 10, tubular neighborhood theorem.
- Morris W. Hirsch, Differential Topology, Springer, 1976. DOI record. Relevant: Chapter 4, tubular neighborhoods.