If $M$ is a $2n$-dimensional simply connected compact oriented Riemannian manifold, then isometries on $M$ preserving orientation have a fixed point

Question

These are original conditions on $M$, $M$ is a $2n$-dimensional compact orientable Riemannian manifold with positive sectional curvature, $\phi:M\rightarrow M$ is an isometry that preserves orientation.

Using Synge's theorem, we know that $M$ is simply-connected, $\phi$ is a degree $1$ map and $\phi_*:H_k(M)\rightarrow H_k(M)$ is an isomorphism, but this is still far from using Lefschetz's fixed point theory, and I don't have any idea on this problem.