TITLE: How to understand conjugate points on a Riemannian manifold?
QUESTION [7 upvotes]: I'm having trouble grasping what it means for two points to be conjugate on a Riemannian manifold. Could someone provide a geometric or intuitive explanation for this? 
For clarification: given a geodesic $\gamma: [0,a] \to M$, the point $\gamma(t)$ is conjugate to $p=\gamma(0)$ if there exists a Jacobi field $J$, not identically zero, along $\gamma$ such that $J(0)=J(t)=0$.

REPLY [3 votes]: Let $p,q$ two points, and $c$ a path between them. The energy functional is a nice function on the space $\Omega_p^q$of paths from $p$ to $q$, and a geodesic $c$ is just a critical point of this functional. If you think of this $\Omega $ as a   manifold, a Jacobi Field exists iff the second derivative $E"$ of $E$ is degenerate, and this Jacobi field is an element of the kernel of $E"$.   A good way to produce this is to consider a one parameter family of geodesics $c_t$ such that at $t=0$, the geodesic is non degenerate with Morse index $i$ and at $t=t_0$ with Morse index $i+1$. Between, some geodesic must be degenerate. For instance, take a simple closed geodesic of positive index. Let $p$ be some point on $c$, and $c_t$ be the arc of $c$ between $p$ and a point at the distance $t$. If $t$ is small, the index is still $0$, and if $t=t_0$ is the length of $c$ the index is $1$, so somewhere in between there is a conjugate point to $p$