In preparation for proving the Poincaré-Hopf Theorem, Milnor states in his book (see p. 38, Theorem 1)

For any vector field $v$ on $M$ with only nondegenerate zeros, the index sum $\sum \iota$ is equal to the degree of the Gauss mapping* $g : \partial N_{\epsilon} \rightarrow S^{k-1}$. In particular this sum does not depend on the choice of vector field.

$M \subset \mathbb{R}^{k}$ is in this case a compact submanifold without boundary and $N_{\epsilon} = \{x \in \mathbb{R}^{k}| \exists y \in M: \|x-y\| \leq \epsilon\}$ with $\epsilon > 0$ a $k$-dimensional submanifold enclosing $M$ for $\epsilon$ sufficiently small. In the following proof Milnor considers the squared distance function (I modified the notation a little bit)

$\phi(x,p) := \begin{cases} N_{\epsilon} \times M \rightarrow \mathbb{R}\\ (x,p) \mapsto \|x-p\|^{2} \end{cases}$

and claims that $r(x)$ defined as the function that assigns each $x \in N_{\epsilon}$ the closest point to $x$ on $M$ is well-defined and smooth.

I think this still follows from the combination Lagrange multiplicators + Implicit function theorem, but I'm not able to see why. I know that locally a submanifold can be described by equations, but in general case we don't know enough about these equations to calculate the differential of the (vector-valued) function which stands on the left side after bringing the equations given by the Lagrange-Multiplicator-Method into standard form, and thus the implicit function theorem is not yet applicable.

Moreover, Milnor later defines the vector field $w(x) := (x-r(x))+v(r(x))$ and calculates the differential at a zero $z$ as

$\operatorname{d} w_{z}(h)=\begin{cases} \operatorname{d} v_{z}(h) & h \in TM_{z} \\ h & h \in TM_{z}^{\perp} \end{cases}$

Here we might have to make use of the chain rule, but this requires that we know the differential of $r(x)$, which we would get from the implicit function theorem, too.

I would be grateful if someone can tell if the implicit function theorem is applicable here or how we get the results otherwise.

  • $\begingroup$ So your question is why is r smooth and well defined? Also when you define r, you mean $x\in N_\epsilon$ rather than $x\in M$, right? $\endgroup$ – Tim kinsella May 27 '16 at 5:30
  • $\begingroup$ This is exactly what I'm asking for and yes, there was a little typo. I have corrected it. I'm also interested in the differential $\operatorname{d} r$ to calculate $\operatorname{d} w_{z}$ or how I get it otherwise. $\endgroup$ – Celsius May 27 '16 at 15:45
  • $\begingroup$ So I think Milnor wants you to do problems 11 and 12 at the end of chapter 8. You need some basic picture of the normal bundle of a sub manifold. $\endgroup$ – Tim kinsella May 27 '16 at 22:33
  • $\begingroup$ Yes, that goes in the right direction. The hard work is done in the "product neighborhood theorem" but one crucial thing is left over: I still have a problem to show that the claimed minima of distance in exercise 12 is indeed a minima. It seems that there is a really simple argument for this issue if you already know the idea, but I'm failing to deduce an appropriate inequality oder find a reductio ad absurdum. $\endgroup$ – Celsius Jun 2 '16 at 17:33

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