Conjugate points

Remarks

Example: on the unit sphere $S^n$, let $\gamma(t)$ be a unit-speed great circle and $E$ a parallel unit field normal to $\gamma'$. The Jacobi field $J = \sin(t) E$ vanishes at $t = 0$ and $t = \pi$, so $\gamma(0)$ and $\gamma(\pi)$, which are antipodal, are conjugate; there are $n - 1$ independent choices of $E$, so the multiplicity is $n - 1$.

Non-example: in $\R^n$ with the Euclidean metric, the Jacobi fields along $\gamma(t) = p + tv$ are $J(t) = A + tB$ with constant vectors $A, B$. Such a field vanishing at two times is zero, so no two points are conjugate.

Equivalently, with $p = \gamma(0)$ and $\gamma(t) = \exp_p(tv)$, the point $\gamma(t_2)$ is conjugate to $p$ along $\gamma$ if and only if the differential of the exponential map $\exp_p$ is singular at $t_2 v$. The multiplicity is at most $n - 1$, where $n = \dim M$.