Regular value set

The theorem of regular value is a result of the differential topology . In English this sentence is called Submersion Theorem . With the help of the theorem it is possible to constructively find submanifolds for a differentiable manifold.

sentence

Let and be differentiable manifolds and let it be a differentiable map . In addition, let be a regular value of . Then the crowd ${\displaystyle M}$ ${\displaystyle N}$ ${\displaystyle f\colon M\to N}$ ${\displaystyle n}$ ${\displaystyle f}$

${\displaystyle U:=f^{-1}(n)=\{m\in M\mid f(m)=n\}}$

a closed , differentiable submanifold of . Then applies to the tangent space ${\displaystyle M}$

${\displaystyle T_{m}U=\operatorname {Kern} \,\,(Df(m))}$

where denotes the differential of in point . ${\displaystyle Df(m)}$ ${\displaystyle f}$ ${\displaystyle m}$

If is finite-dimensional, then for the codimension of ${\displaystyle N}$ ${\displaystyle U}$

${\displaystyle \operatorname {codim} (U)=\dim(N)}$.

This follows from the statement about the tangent space. If it is also finite dimensional, the dimension of can be calculated using the formula ${\displaystyle M}$ ${\displaystyle M}$

${\displaystyle \operatorname {codim} (U)=\dim(M)-\dim(U)}$

to calculate.

example

With the help of the theorem one can show that the -dimensional unit sphere is a submanifold of . Be it ${\displaystyle n}$ ${\displaystyle \mathbb {S} ^{n}}$ ${\displaystyle \mathbb {R} ^{n+1}}$

${\displaystyle f\colon \mathbb {R} ^{n+1}\to \mathbb {R} }$defined by . ${\displaystyle f(x)=\|x\|^{2}=\sum _{i=1}^{n+1}x_{i}^{2}}$

Then applies . All that remains is to show that 1 is a regular value. You can see this through ${\displaystyle \mathbb {S} ^{n}=f^{-1}(1)}$

${\displaystyle Df(x)=D\|x\|^{2}=2x^{t}}$.

The operator stands for the matrix transposition . The term becomes zero for only . For everyone else , rank applies ${\displaystyle (.)^{t}}$ ${\displaystyle x=0}$ ${\displaystyle 2x^{t}}$ ${\displaystyle x\in \mathbb {R} ^{n+1}}$

${\displaystyle \operatorname {Rg} (2x^{t})=1}$.

So in particular the differential is for surjective and thus is a real submanifold. ${\displaystyle \|x\|=1}$ ${\displaystyle \mathbb {S} ^{n}}$

literature

• Konrad Königsberger : Analysis. Volume 2. 3rd revised edition. Springer, Berlin et al. 2000, ISBN 3-540-66902-7 , pp. 118f.
• R. Abraham, JE Marsden, T. Ratiu: Manifolds, Tensor Analysis and Applications (= Applied Mathematical Sciences 75). Springer, New York NY 1988, ISBN 0-387-96790-7 .