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Let $M$ be a complete riemannian manifold, $L$ a smooth submanifold of $M$ and $\gamma$ a geodesic with $\gamma'(0)$ normal to $L$. A focal point of $L$ is a critical value of the normal exponential $\exp:\nu L\to M$. So the question is: is the set of focal points $\gamma(t)$ of $L$ discrete?

I know that's related to the Morse index theorem, but all the versions of it I've found are too general and do not contain a conclusion about finiteness.

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  • $\begingroup$ Finite dimensions? In infinite dimensions really strange thhings can happen with focal and conjugate points. $\endgroup$ Nov 21, 2013 at 7:59
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    $\begingroup$ In finite dimensions the answer is yes, it is discrete. The way I know how to prove this involves at looking at everything from the symplectic or contact viewpoint and using the properties of the differential of the geodesic flow of a Riemannian or Finsler metric on the bundle of Lagrnagian planes. If no simpler answers are forthcoming. I'll write that up later. $\endgroup$ Nov 21, 2013 at 8:03

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I don't think that what I will now write is the best or quickest answer to your question, but maybe it is worthwhile to set it down because the point of view is a bit of useful folklore.

First let me translate the problem from Morse theory to Hamiltonian dynamics:

1. Instead of the submanifold $L$ consider its conormal bundle $\nu(L) \subset T^* M$ defined as the set of all covectors bases at some point of $L$ that vanish on the tangent space to $L$. The conormal bundle is a Lagrangian submanifold.

2. Consider the geodesic flow $\phi_t : T^*M \setminus O \rightarrow T^*M \setminus O$ on the slit cotangent bundle. Consider also the flow $$ D\phi_t : T(T^*M \setminus O) \rightarrow T(T^*M \setminus O) $$ obtained as the differential of the geodesic flow.

3. Last, but not least, consider the bundle of tangent Lagrangian planes over $T^*M \setminus O$ that I'll denote by $\lambda(T^*M \setminus O)$.

The flow $D\phi_t$ induces a flow on $\lambda(T^*M \setminus O)$ that I'll denote by the same symbol. This flow has an important "twist" condition: if $p_x$ is a point in $T^* M \setminus O$ and $\phi_t(p_x)$ is the integral curve passing through it, consider the vertical Lagrangian tangent plane $V_{\phi_t(p_x)} \subset T_{\phi_t(p_x)}(T^* M \setminus 0)$ and the curve of Lagrangian planes in the (symplectic) vector space $T_{p_x} (T^*M \setminus 0)$ defined by $$ t \mapsto D\phi_{-t}(V_{\phi_t(p_x)}) =:\Lambda(t). $$

4. The twist condition satisfied by geodesic flows for Riemannian and Finsler metrics (among other non-degenerate Hamiltonians) implies that if $t$ and $t'$ are sufficiently close, then $\Lambda(t)$ and $\Lambda(t')$ are transverse.

5. Now we have to translate focal points into this language: Consider $p_x$ on the conormal bundle of $L$ (at the point $x$) and the orbit $\phi_t(p_x)$ $(t \in \mathbb{R})$.

Proposition. The point $y$ obtained by projecting $\phi_s(p_x)$ onto $M$ is a focal point if the Lagrangian plane $\Lambda(s) \subset T_{p_x}(T^*M \setminus O)$ does not intersect the (Lagrangian) tangent plane of the conormal bundle $\nu(L)$ transversely at this point. The dimension of the intersection is the multiplicity of the focal point.

The twist condition immediately gives you that the set of focal points along the geodesic obtained by projecting $\phi_t(p_x)$ on $M$ is discrete.

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If you vary the normal geodesic along $L$ (of codimension 1, say), the first focal points sometimes trace out another surface (with singularities) which is called "Wiedersehensfläche" (see again surface). This topic has been treated by Blaschke, Chern, Berger, ...

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In infinite dimension, the distribution of conjugate (or focal) points depends on the Fredholmness of the exponential map. If the exponential map is not Fredholm, in which case you have two types of conjugate points (monoconjugate/epiconjugate), you may have accumulation. For a discussion of this topic, see reference: Biliotti, Leonardo(I-UPM); Exel, Ruy(BR-FSC); Piccione, Paolo(BR-SPL-IMS); Tausk, Daniel V.(BR-SPL-IMS) On the singularities of the exponential map in infinite dimensional Riemannian manifolds. (English summary) Math. Ann. 336 (2006), no. 2, 247–267.

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