Focal set of a geodesic

Question

In the sphere $S^n$, the focal set of a point is its antipodal point. How about the focal set of a closed geodesic? My guess is that it is another geodesic, but I can neither prove it nor know it is correct.

Answer 1

In $S^n$, the focal set of a closed geodesic is a totally geodesic submanifold of dimension $n-2$, obtained as follows: Let $\gamma$ be a closed geodesic and let $C\subseteq S^n$ be its image. Then $C$ is the intersection of $S^n$ with a $2$-dimensional linear subspace $\Pi\subseteq \mathbb R^{n+1}$, and the focal set of $C$ is the intersection of $S^n$ with the $(n-1)$-dimensional subspace orthogonal to $\Pi$. It's a submanifold isometric to $S^{n-2}$ and disjoint from $C$.

Thus the OP's intuition is correct in the special case $n=3$: The focal set of a closed geodesic in $S^3$ is another geodesic. And Ted Shifrin's intuition is correct when $n=2$: The focal set of a closed geodesic in $S^2$ is a pair of antipodal points. But in other dimensions, the focal set is larger.

One way to prove the claim is to use the relationship between singular values of the exponential map and Jacobi fields. In the case of conjugate points, this is a familiar story in Riemannian geometry: Given $q\in M$, a point $p\in M$ is called a conjugate point of $\boldsymbol q$ if it is a singular value of the restricted exponential map $\exp_p\colon T_pM \to M$, which is equivalent to the existence of a geodesic $\sigma$ from $p$ to $q$ and a nontribial Jacobi field along $\sigma$ that vanishes at $p$ and $q$.

For focal points, the story is similar but perhaps less well known. A focal point of a submanifold $C$ is a singular value of the normal exponential map $\exp|_{NC}\colon NC\to M$, where $NC$ is the normal bundle of $C$ in $M$. In the case that $C$ is totally geodesic, a proof very similar to the one for conjugate points shows that $p\in M$ is a focal point of $C$ if and only if there is a geodesic segment $\sigma\colon[0,b]\to M$ and a Jacobi field $J\colon [0,b]\to TM$ along $\sigma$ satisfying

• $\sigma(0)\in C$;
• $\sigma'(0)$ is orthogonal to $C$;
• $\sigma(b)=p$;
• $J(0)$ is tangent to $C$ and nonzero;
• $D_tJ(0)=0$ (where $D_t$ denotes the covariant derivative along $\sigma$);
• $J(b)=0$.

(When $C$ is not totally geodesic, the condition $D_tJ(0)=0$ has to be replaced by $D_tJ(0) = - W_{\sigma'(0)} (J(0))$, where $W_{\sigma'(0)}$ is the shape operator or Weingarten map in the direction of the normal vector $\sigma'(0)$. I don't know of a source that works this out in full detail, but most of these ideas can be found in Chavel's Riemannian Geometry: A Modern Introduction or Gray's Tubes.)

In $S^n$, because the sectional curvature is identically $1$, it's easy to solve the Jacobi equation. In particular, every Jacobi field that satisfies $D_tJ(0)=0$ is of the form $J(t) = (\cos t)E(t)$ for some vector field $E$ that's parallel along the geodesic. Thus every geodesic that starts orthogonal to $C$ hits a focal point exactly at distance $\pi/2$. A little reflection should convince you that the collection of all such focal points is exactly the $(n-2)$-sphere "orthogonal" to $C$.