Tubular environment

In mathematics , the tubular environment or tube environment is a frequently used technical aid in differential topology .

Tube environment of a curve in the plane

Tube environment of a curve in a non-orientable surface

Theorem of the tubular neighborhood

Let it be a differentiable manifold and a compact differentiable submanifold . Then there is an environment of in with the following property: ${\displaystyle M}$ ${\displaystyle N\subset M}$ ${\displaystyle U(N)}$ ${\displaystyle N}$ ${\displaystyle M}$

There is a fiber bundle with total space , base and fiber diffeomorphic too ${\displaystyle U(N)\to N}$ ${\displaystyle U(N)}$ ${\displaystyle N}$

${\displaystyle B^{k}=\{x\in \mathbb {R} ^{k}\ |\ |x|<1\},k=dim(M)-dim(N)}$.

Furthermore is the zero cut of this fiber bundle. ${\displaystyle N\subset U(N)}$

This environment is called the tube environment of , it is only clearly defined except for isotopy . ${\displaystyle U(N)}$ ${\displaystyle N}$

See also

• Collar area

literature

• James R. Munkres: Elementary differential topology. Lectures given at Massachusetts Institute of Technology, Fall 1961. Revised edition. In: Annals of Mathematics Studies , No. 54. Princeton University Press, Princeton NJ 1966

Web links

• Schnürer: Differential Topology (PDF) Theorem 1.16