Relationship between geodesics and dimension of spheres?

Question

Let n be an odd positive integer and $U\subset S^{n}\times S^{n}$ be the open subset consisting of all couples (A,B) such that $A\neq -B$. Let $\pi:(S^{n})^{I} \to S^{n}\times S^{n}$ be the fibration defined $\pi(\omega)=(\omega(0),\omega(1))$.I want to show that $\pi$ has a continuous section over U, means a continuous map $s:U\to (S^{n})^{I}$ such that $\pi s$ is the inclusion of U in $S^{n}\times S^{n}$. I found a proof in Farber's article, says that for a couple (A,B) we take s(A,B) is the shortest geodesic connecting A and B. Since n is odd then these geodesics are unique and so the map s is well defined and continuous. My question is: what is the relation between geodesics and dimension of spheres?