Fixed point theorem on spheres

Solution 1:

I think, what you gave is (a sketch of) the simplest proof, unless you know about the Lefshetz fixed point theorem. Using the latter, then the argument goes like this: $H_k(S^n)$ is nonzero only in dimensions $0$ and $n$ and $H_0(S^n)\cong H_n(S^n)\cong Z$. The action of $f$ on $H_0$ is trivial and the action on $H_n$ is by multiplication by $d=\deg(f)$. The Lefschetz number of $f$ then equals $$ \Lambda_f= (-1)^0 + (-1)^n (d)= 1+ d(-1)^n. $$ This number is nonzero unless $$ d= (-1)^{n+1} $$ as required. If $\Lambda_f\ne 0$ then $f$ has a fixed point (this is the Lefschetz fixed point theorem).

As for your solution, do not bother with geodesic flow, that's unnecessarily complicated; you can use linear algebra instead. For $t\in [0,1]$ and $x, y\in S^n$ non-antipodal, define $$ z_t= \frac{tx+(1-t)y}{|tx+(1-t)y|}\in S^n. $$ This is a point on the shortest arc of the great circle connecting $x$ and $y$. The rest you can figure out yourself.

Solution 2:

Best solution: Notice that $\|f(x)-(a \circ f) (x)\| < 2$ for every $x.$ But then a problem above this one in the book, asks to prove that such maps must be homotopic (smoothly so if maps themselves are smooth.) Then your computation goes thru with nothing on conscience!