# How to understand conjugate points on a Riemannian manifold?

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 $J(0)=J(t)=0$.

$$\newcommand{\ga}{\gamma}$$

Jacobi fields measure how geodesics that start at the same point but with different velocities spread apart from each other.

Let $$p \in M, v,w \in T_PM$$. We consider a $$1$$-parameter family of geodesics:

$$\ga_s(t)=\exp_p(t(v+sw)).$$

Recall that $$\ga(t)=\exp_p(tv)$$ is the unique geodesic passing through $$p$$ (at $$t=0$$) with velocity $$v$$. Thus, $$\ga_s$$ is the unique geodesic passing through $$p$$ (at $$t=0$$) with velocity $$v+sw$$. (We are slowly changing the initial velocity of our geodesic).

$$J(t)=\frac{\partial \ga_s(t)}{\partial s}|_{s=0} \in T_{\exp_p(tv)}M=T_{\ga(t)}M$$ is a vector field along the initial geodesic $$\ga$$.

Note that $$J(0)=0$$. (since all our geodesics start at the same point $$p$$ at $$t=0$$). Moreover, every Jacobi field along $$\ga$$ satisfying $$J(0)=0$$ can arise from such a family of geodesics.

$$\|J(t)\|$$ measures the rate of "spreading apart" of the geodesics at time $$t$$, when their initial velocity at $$p$$ change.

Hence, if $$q=\ga(t_0)$$ is conjugate to $$\ga(0)=p$$ along $$\ga$$, this means that there is a continuous family of geodesics starting from $$p$$ which "almost" meet at $$q$$. (They meet at $$q$$ only "up to first order").

Clearly, if $$\ga_s(t_0)=q$$ for every $$s$$ (that is all the geodesics in the family meet at $$q$$ at time $$t_0$$) then $$J(t_0)=0$$, but the reverse implication is false in general.

• What do you mean by "almost meet?" In general, the family of geodesics won't all intersect at $q$, but then in what sense do they almost meet? – Mathmank Feb 9 '16 at 21:49

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 $p$