Let $h:S^{n-1}\to S^{n-1}$ be $C^{\infty}$ map. How to prove that a function $F: S^{n-1}\times[0,1]\to S^{n-1}$ given by
$F(v,t)=(\cos{\pi t})v+(\sin{\pi t})h(v)$
is proper $C^{\infty}$ map?
Let $h:S^{n-1}\to S^{n-1}$ be $C^{\infty}$ map. How to prove that a function $F: S^{n-1}\times[0,1]\to S^{n-1}$ given by
$F(v,t)=(\cos{\pi t})v+(\sin{\pi t})h(v)$
is proper $C^{\infty}$ map?
Assuming $F(v,t)\in S^{n-1}$ there is not much to show. Suppose $A$ is compact in $S^{n-1}$, so $A$ is closed. The map $F$ is obviously continuous, so $F^{-1}(A)$ is closed, and since $S^{n-1}\times [0,1]$ is compact $F^{-1}(A)$ is compact, so $F$ is proper.
The fact that $F$ is $C^{\infty}$ is obvious, too, it's the composition of $C^{\infty}$ maps.
One may ask why $F(v,t)\in S^{n-1}$, though. Is $\langle h(v),v \rangle = 0$?