Conjugate points and indices of round-sphere geodesics
= Conjugate points and indices of round-sphere geodesics
{title2=$\operatorname{ind}=(n-1)\#\{j\geq1:j\pi<L\}$}
On the unit round <sphere> $S^n$, a geodesic of speed $L$ on $[0,1]$ has normal <Jacobi equation> $y^{\prime\prime}+L^2y=0$. Its conjugate parameters are $j\pi/L$, each with multiplicity $n-1$. If $L/\pi$ is not an integer, its index is $(n-1)\lfloor L/\pi\rfloor$. For $L=m\pi>0$, its index is $(m-1)(n-1)$ and its nullity is $n-1$.