As a consequence of the formula for the first variation of the energy of a curve, we have the following known characterization of geodesics.

A piecewise differentiable curve $c:[0,1]\to M$ is a geodesic if and only if, for every proper variation $f$ of $c$, we have $\frac{dE}{ds}(0)=0.$

But I'm trying to prove a slight modification of this result, which is given by:

Let $N$ be a closed submanifold of $M$ and a piecewise differentiable curve $c:[0,1]\to M$ such that $c(0),c(1)\in N$. If $\frac{dE}{ds}(0)=0$ then $c$ is a geodesic which intercepts the submanifold $N$ orthogonally (is free boundary).

I managed to get to,

$$\frac{1}{2}E'(0)=-\int^a_{0}\left\langle V(t),\frac{D}{dt}\cdot\frac{dc}{dt} \right\rangle dt-\sum^{k}_{i=1}\left\langle V(t_i),\frac{dc}{dt}(t^{+}_{i})-\frac{dc}{dt}(t^{-}_{i})\right\rangle- \left\langle V(0),\frac{dc}{dt}(0)\right\rangle+\left\langle V(1),\frac{dc}{dt}(1)\right\rangle$$

where $V$ is the variational field defined by $V(t)=g(t)\frac{D}{dt}\frac{dc}{dt}$, where $g:[0,1] \to \mathbb{R}$ is a piecewise differentiable function with $g(t)>0$ if $g(t_i)\neq 0$, remembering that $t_i$ are the partition terms for which $c$ is differentiable.

We know that the variational field is non-zero, and $E'(0)=0$, to finish the proof I need to conclude that

$$\frac{D}{dt}\frac{dc}{dt}=0\quad\text{and}\quad \frac{dc}{dt}(0)=\frac{dc}{dt}(1)=0$$

But I'm not able to do that. I don't know if I took the correct path, I used the main ideas of the proof of the previous proposition. I can go into more detail if needed, but does anyone have any suggestions?

Let $M$ be a Riemannian manifold and $N$ be a closed submanifold. We say that $c:I\rightarrow M$ is a free-boundary geodesic if $c(a),c(b)\in N \text{ and } c'(a),c'(b ) \perp N$.

The first result mentioned can be found in Manfredo P. do Carmo book Riemannian Geometry Proposition 2.5.

Edit: A small change in the statement is necessary to make the question make sense. It was not assumed that $c(0),c(1)\in N$.

Additional Information: This question was probably inspired by exercise 1 in chapter 9. You can see the exercise in this image.

  • 1
    $\begingroup$ You’re on a first name basis with DoCarmo?!! $\endgroup$ May 14 at 3:00
  • 2
    $\begingroup$ @TedShifrin Here some famous Brazilian authors are addressed by their first name, it's not a lack of respect or familiarity. An example is Manfredo do Carmo, "Manfredo's book on differential geometry" or Elon Larges Lima, "Elon's book of analysis". Obviously this does not extend to academic articles. Look here. Anyway... $\endgroup$
    – Mrcrg
    May 14 at 19:59
  • $\begingroup$ Interesting cultural patterns we don’t all know about. Thanks. Quite unusual. I’m still puzzled whether it stays appropriate in an international setting like this. $\endgroup$ May 14 at 20:01


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