Let $S_+^{n+1}$ be the open hemisphere of the standard euclidean sphere centered at the north pole and let $M^n$ be a compact, connected and oriented hypersurface of $S_+^{n+1}$. Is it true that if $M$ has nowhere zero Gauss-Kronecker curvature then all its principal curvatures have the same sign?
I heard this follows from the fact that $M$ has an elliptic point, but I don't know why this is true nor it implies my question.