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Michael Albanese
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Given a function $\psi:\mathbb R\to \mathbb R$, set $$\Psi=\psi\circ\mathrm{dist}_ {\partial M},\ \ \ \ \ f=\Psi\cdot(R-\mathrm{dist}_ p)$$ for some fixed $R>\mathrm{diam}\\, M$$R>\mathrm{diam}\ M$.

Further, $$d\\,f= (R-\mathrm{dist}_ p)\cdot d\\,\Psi-\Psi\cdot d\\,\mathrm{dist}_ p$$ Thus$$d\,f = (R-\mathrm{dist}_ p)\cdot d\,\Psi-\Psi\cdot d\,\mathrm{dist}_ p$$

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$$d\,\Psi$ is positive muliple of $d\\,\mathrm{dist}_ {\partial M}$$d\,\mathrm{dist}_ {\partial M}$. Thus $d_x\\,f=0$$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 $f$...

Given a function $\psi:\mathbb R\to \mathbb R$, set $$\Psi=\psi\circ\mathrm{dist}_ {\partial M},\ \ \ \ \ f=\Psi\cdot(R-\mathrm{dist}_ p)$$ for some fixed $R>\mathrm{diam}\\, M$.

Further, $$d\\,f= (R-\mathrm{dist}_ p)\cdot d\\,\Psi-\Psi\cdot d\\,\mathrm{dist}_ p$$ 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 $f$...

Given a function $\psi:\mathbb R\to \mathbb R$, set $$\Psi=\psi\circ\mathrm{dist}_ {\partial M},\ \ \ \ \ f=\Psi\cdot(R-\mathrm{dist}_ p)$$ for some fixed $R>\mathrm{diam}\ M$.

Further, $$d\,f = (R-\mathrm{dist}_ p)\cdot d\,\Psi-\Psi\cdot d\,\mathrm{dist}_ p$$

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 $f$...

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Anton Petrunin
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Given functionsa function $\phi,\psi:\mathbb R\to \mathbb R$$\psi:\mathbb R\to \mathbb R$, set $$\Phi=\phi\circ\mathrm{dist}_ {\partial M},\ \ \ \Psi=\psi\circ\mathrm{dist}_ {\partial M} $$ $$f=\Psi\cdot(R-\mathrm{dist}_ p)+\Phi$$$$\Psi=\psi\circ\mathrm{dist}_ {\partial M},\ \ \ \ \ f=\Psi\cdot(R-\mathrm{dist}_ p)$$ for some fixed $R>\mathrm{diam}\\, M$.

Further, $$d\\,f=(R-\mathrm{dist}_ p)\cdot d\\,\Psi +d\\,\Phi-\Psi\cdot d\\,\mathrm{dist}_ p$$$$d\\,f= (R-\mathrm{dist}_ p)\cdot d\\,\Psi-\Psi\cdot d\\,\mathrm{dist}_ p$$ Thus, we may choose smooth increasing $\phi$ and   $\psi$, such that $\phi(0)=\psi(0)=0$$\psi(0)=0$ and both $\phi$, $\psi$ areit is constant outside of little nbhd of $0$ so that $\Phi$ and $\Psi$ areis 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 $d\\,\Phi$ and $d\\,\Psi$ bothis 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 $f$...

Given functions $\phi,\psi:\mathbb R\to \mathbb R$, set $$\Phi=\phi\circ\mathrm{dist}_ {\partial M},\ \ \ \Psi=\psi\circ\mathrm{dist}_ {\partial M} $$ $$f=\Psi\cdot(R-\mathrm{dist}_ p)+\Phi$$ for some fixed $R>\mathrm{diam}\\, M$.

Further, $$d\\,f=(R-\mathrm{dist}_ p)\cdot d\\,\Psi +d\\,\Phi-\Psi\cdot d\\,\mathrm{dist}_ p$$ Thus, we may choose smooth increasing $\phi$ and $\psi$, such that $\phi(0)=\psi(0)=0$ and both $\phi$, $\psi$ are constant outside of little nbhd of $0$ so that $\Phi$ and $\Psi$ are 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 $d\\,\Phi$ and $d\\,\Psi$ both 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 $f$...

Given a function $\psi:\mathbb R\to \mathbb R$, set $$\Psi=\psi\circ\mathrm{dist}_ {\partial M},\ \ \ \ \ f=\Psi\cdot(R-\mathrm{dist}_ p)$$ for some fixed $R>\mathrm{diam}\\, M$.

Further, $$d\\,f= (R-\mathrm{dist}_ p)\cdot d\\,\Psi-\Psi\cdot d\\,\mathrm{dist}_ p$$ 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 $f$...

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Anton Petrunin
  • 45k
  • 14
  • 135
  • 299

Given functions $\phi,\psi:\mathbb R\to \mathbb R$, set $$\Phi=\phi\circ\mathrm{dist}_ {\partial M},\ \ \ \Psi=\psi\circ\mathrm{dist}_ {\partial M} $$ $$f=\Psi\cdot(R-\mathrm{dist}_ p)+\Phi$$ for some fixed $R>\mathrm{diam}\\, M$.

Further, $$d\\,f=(R-\mathrm{dist}_ p)\cdot d\\,\Psi +d\\,\Phi-\Psi\cdot d\\,\mathrm{dist}_ p$$ Thus, we may choose smooth increasing $\phi$ and $\psi$, such that $\phi(0)=\psi(0)=0$ and both $\phi$, $\psi$ are constant outside of little nbhd of $0$ so that $\Phi$ and $\Psi$ are 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 $d\\,\Phi$ and $d\\,\Psi$ both 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 $f$...