at.algebraic topology - Contractible manifold with boundary

Thursday, 9 December 2010

Given a function $psi:mathbb Rto mathbb R$ ,
set

$Psi=psicircmathrm{dist}_ {partial M}, f=Psicdot(R-mathrm{dist}_ p)$
for some fixed

$R>mathrm{diam}\, M$ .

Further,

$d\,f= (R-mathrm{dist}_ p)cdot d\,Psi-Psicdot d\,mathrm{dist}_ p$
Thus, we may choose smooth increasing

$psi$ ,
such that

$psi(0)=0$
and it is constant outside of little nbhd of

$0$ so that

$Psi$ is smooth.
(It is possible since the function

$mathrm{dist}_ {partial M}$ is smooth and has no critical points in a small neighborhood of

$partial M$ .)
Note that

$d\,Psi$ is positive muliple of

$d\,mathrm{dist}_ {partial M}$ .
Thus

$d_x\,f=0$ means that geodesic from

$x$ to

$p$ goes directly in the direction of minimizing geodesic from

$x$ to

$partial M$ , which can not happen.

Now we can apply Morse theory for $f$ ...

at 12:56

Labels: Mathematics

AnswerDesk