Focal point free geodesics are locally length minimizing (Jost Exercise 4.2) I'm working on the following Exercise from Jost's Riemannian Geometry and Geometric Analysis (Exercise 4.2).

Let $M$ be a submanifold of the Riemannian manifold $N$, $c:[a,b]\to N$ geodesic with $c(a)\in M$, $\dot{c}(a)\in (T_{c(a)}M)^\perp$. For $\tau\in(a,b]$, $c(\tau)$ is called a focal point of $M$ along $c$ if there exists a nontrivial Jacobi field $X$ along $c$ with $X(a)\in T_{c(a)}M$, $X(\tau)=0$.
  Show:
  
  
*
  
*If $M$ has no focal point along $c$, then for each $\tau\in(a,b)$, $c$ is the unique shortest connection to $c(\tau)$ when compared with all sufficiently close curves with initial point in $M$.
  

I initially tried solving this by generalizing the techniques used for showing the analogous result for conjugate points.
For part (1) I began by showing that the focal points of $M$ are exactly the critical values of $\exp^\perp$, where $\exp^\perp$ is the exponential map restricted to the normal bundle, $TM^\perp$. Using this and the fact that we are assuming there are no focal points of $M$ along $c$ we have that $\exp^\perp_{c(a)}$ is of maximal rank along any radial curve $t\mapsto tv$ for $t\in[0,1]$ with $v\in T_{c(a)}M^\perp$. By the inverse function theorem we see that for every $t$ that $\exp_{c(a)}^\perp$ is a diffeomorphism in a suitable neighborhood of $tv\in T_{c(a)}M^\perp$. By compactness, we can cover $\{tv~:~0\leq t\leq 1\}$ by finitely many neighborhoods $\Omega_i$ for $i=1,\dots,k$. Now define $U_i:=\exp_{c(a)}^\perp\Omega_i$. 
Here's where I'm getting stuck: I would like to find sufficiently small $\varepsilon > 0$ such that if $d(g(a),c(a))<\varepsilon$, $g(\tau)=c(\tau)$, and $d(g(t),c(t))<\varepsilon$ for all $t\in[a,\tau]$, then $g([t_{i-1},t_{i}])\subseteq U_i$. How do I find such an $\varepsilon$? I'm not sure how to deal with the issue that $g$ can have different initial conditions in $M$. Any ideas or hints would be extremely helpful. Maybe there's a more efficient way to go about proving this result altogether! 
 A: Let us begin with the definition of a focal point.
Definition.$\label{def}$ Let $(M,g)$ be a Riemannian manifold, and $N\subset M$ a submanifold. Then a focal point of $N$ is a point $q\in M$ such that there exists a geodetic $\gamma\colon[0,l]\to M$, with $\gamma(0)=p\in N$, $\gamma'(0)\in(T_{N,p})^{\perp}$, $\gamma(l)=q$, and a non-trivial Jacobi field $J$ along $\gamma$, satisfying


*

*$J(0)\in T_{N,p}$,

*$J'(0)+S_{\gamma'(0)}(J(0))\in(T_{N,p})^{\perp}$,

*$J(l)=0$,


where $S$ is the linear operator on $T_{N,p}$ given by the second fundamental form of $N\subset M$, also called shape operator or Weingarten map.
For instance, the set of focal points (the focal set) of $S^2\subset\mathbb{R}^3$ is the centre of the sphere, and that of $S^1\subset S^2$ is the union of the two antipodal points "at the centre" of the circumference. These points are exactly those points one would expect a "focal point" to be, that is points where geodesic normal to the submanifold concur. This fact is well explained by the following preposition:
Proposition. (M.P.Do Carmo, Riemannian Geometry,  4.4)
The point $q\in M$ is a focal point of $N\subset M$ if and only if it is a critical value of the map $\exp^{\perp}:(T_N)^{\perp}\to M$.
Here, the map $\exp^{\perp}$ denotes the map
$$
\exp^{\perp}(p,v):=\exp_p(v),\qquad p\in N,\ v\in(T_{N,p})^{\perp},
$$
where $\exp$ is the exponential map of $M$. Notice that $\exp^{\perp}$ is very similar to $\exp$ (in fact $\dim(T_N)^{\perp}=\dim M$), with the difference that the former moves along the submanifold. From this point of view one sees that the focal points are indeed those point in which the map $\exp^{\perp}$ is injective no longer, and thence there are more than one geodesic intersecting that point. 
What about the problem now? 
When we are in the case $\dim N =2$, $N\subset M=\mathbb{R}^3$ (assume completeness for the sake of simplicity), the fact that the statement is true seems intuitive: fixing a point $p$, focal points associated to it are the centres of the spheres with curvature equal to one of the principal curvatures at $p$. One can then link the point $p$ with the first (in one direction) of the focal points $q$ through a normal geodesic (a line) $c$ (let us say $c(0)=p$, $c(l)=q$). Now, for the values $t\in(0,l)$, the spheres with centre $c(t)$ that include the point $p$ have curvature strictly greater than that of the greatest principal curvature at $p$, and hence in a neighbourhood of $p$ in $N$ there are no points at a lesser distance from $c(t)$ than $p$.
Now, we would like to transpose this line of reasoning to an arbitrary submanifold $N$ of an arbitrary manifold $M$, yet that would be fairly hard since we haven't got the euclidean structure on $M$, and therefore we cannot determine the curvature of a geodesic sphere (at least, not as easily as in the euclidean case). So, let us take another path.
What we want to prove is that, given a normal geodesic $c\colon[0,l]\to M$ as before, each curve $\gamma(t)$ with $\gamma(0)\in U$ and $\gamma(l)=c(l)$ is longer than $c$, where $U$ is a neighbourhood of $c(0)=p$ in $N$. Clearly, we just need to prove it in the case in which $\gamma$ is a geodesic. So, we can choose a Jacobi field $X$ along $c$ such that


*

*$X(l)=0$,

*$X(0)\in T_{N,p}$,

*$g(X(0),X(0))=1$.


This is because we want this Jacobi field to be associated to a variation $H(s,t)=c_s(t)$ such that $c_s(0)\in N$, defined for $s\in(-\epsilon,\epsilon)$, for some $\epsilon>0$. Let $X_s(t)$ be $d_{(s,t)}H\cdot\frac{\partial}{\partial s}$ ($X_0=X$).
Recall now that, from the second variation formula and given that $X\perp c'$, we have
$$
\begin{align*}
\frac{d^2}{ds^2}L(c_s)|_{s=0}&=[g(\frac{\nabla}{ds}X_0(t),c_0'(t))]^l_0+\int^l_0g(X',X')-R(X,c',X,c')dt\\
&=-g(\frac{\nabla}{ds}X_0(0),c'(0))-g(X(0),X'(0)).
\end{align*}
$$
We just need to show that this is positive. First off, given that $X$ is a Jacobi field, $||X(0)||_g=1$ and $||X(l)||_g=0$, we have $X'(0)=-\frac{1}{l}X(0)$. Hence, we have
$$
\frac{d^2}{ds^2}L(c_s)|_{s=0}=\frac{1}{l}-g(\frac{\nabla}{ds}X_0(0),c'(0)).
$$
Now, if $c(l)$ were a focal point, we would like to say that we have
$$
\frac{d^2}{ds^2}L(c_s)|_{s=0}=0,
$$
for we expect the lenght of the geodesics to be constant (or nearly) in a neighbourhood of $t=0$. We can prove that this is indeed true. Suppose $c(\lambda)$ is the "first" focal point (if there is none we can say that $\lambda=+\infty$, which in the following expressions has its usual meaning of a limit), then we have got a Jacobi field $X$ like in the definition, and we can choose it so that $||X(0)||_g=1$. We can see then that
\begin{align*}
\frac{d^2}{ds^2}L(c_s)|_{s=0}&=-g(\frac{\nabla}{ds}X_0(0),c'_0(0))-g(X(0),X'(0))\\
&=-(\frac{\partial}{\partial s}g(X_s(0),c'_s(0))-g(X_0(0),\frac{\nabla}{ds}c'_s(0)))-g(X(0),X'(0)),
\end{align*}
but from the very definitions we have that $g(X_s(0),c'_s(0))=0$ for every $s$, and $\frac{\nabla}{ds}c'_s(0))=-S_{c'(0)}(X(0))$. Hence,
$$
\frac{d^2}{dt^2}L(c_t)|_{t=0}=-g(X(0),X'(0)+S_{c'(0)}(X(0)))=0.
$$
Due to this, we have
$$
g(\frac{\nabla}{ds}X_0(0),c'_0(0))=\frac{1}{\lambda}.
$$
Notice that this value depends only on the structure of the submanifold $N$ and the point $c(0)$, and not on the point $c(l)$. So we have (considering the variation with respect to $c(l)$)
$$
\frac{d^2}{ds^2}L(c_s)|_{s=0}=\frac{1}{l}-\frac{1}{\lambda}
$$
and, therefore,
$$
\frac{d^2}{ds^2}L(c_s)|_{s=0}>0\qquad\textrm{if }l<\lambda,
$$
meaning that in a neighbourhood of $c(0)$ along a curve there is no shortest line to $c(l)$ than the geodesic $c$, as we were set to prove. Moreover, we can see that this is no longer true if $l\ge\lambda$. Using a compactness argument we can see that there exists a neighbourhood of $c(0)$ in $N$ such that the results are valid.
