# If $\gamma(b)$ is not conjugate to $\gamma(a)$ along $\gamma$. Show that $I(V, V)=I(J, J)$ if and only if $V=J .$

Exercice 10-12 (Introduction to Riemannian Geometry, Lee):

Let $$(M, g)$$ be a Riemannian manifold. Suppose $$\gamma:[a, b] \rightarrow M$$ is a unitspeed geodesic segment with no interior conjugate points, $$J$$ is a normal Jacobi field along $$\gamma$$, and $$V$$ is any other piecewise smooth normal vector field along $$\gamma$$ such that $$V(a)=J(a)$$ and $$V(b)=J(b)$$.

(a) Show that $$I(V, V) \geq I(J, J)$$.

(b) Now assume in addition that $$\gamma(b)$$ is not conjugate to $$\gamma(a)$$ along $$\gamma$$. Show that $$I(V, V)=I(J, J)$$ if and only if $$V=J .$$

I think that if $$\gamma$$ has no interior conjugate points then it minimize the distance(!). Let $$V$$ be a proper and normal field along $$\gamma$$, then $$I(V,V) \geqslant 0$$, or $$I(J,J)=0$$(since it's a Jacobi field), then $$I(V,V) \geqslant I(J,J)$$ For the second question, suppose that $$I(V,V)=I(J,J)$$ and $$V\not =J$$ Then if V is proper, $$V(a)=J(a)=0$$ and $$V(b)=J(b)=0$$. Since $$J(a)=J(b)=0$$ then $$J=0$$ everywhere then $$I(V,V)=0$$ and $$V\not =0$$, so $$V$$ is a Jacobi field and hence $$\gamma(a)$$ and $$\gamma(b)$$ are two conjugate points which contradicts our hypothesis ! I don't really know is this is true or not, so any help is appreciated!

• "I think that if γ has no interior conjugate points then it minimize the distance": this is false. The flat torus is a counter-example. The fact there is no conjugate points along $\gamma$ says that there cannot exist a smooth family of geodesics joining $p$ and $q$ in a neighbourhood of $\gamma$, unless the constant family. Commented Nov 30, 2021 at 9:36

There are a number of problems with your argument. As @Didier commented, it does not follow from the absence of conjugate points that $$\gamma$$ is length-minimizing. More importantly, most of your claims about $$I(V,V)$$ and $$I(J,J)$$ would be justified if $$V$$ and $$J$$ were proper (i.e., vanishing at the endpoints), but there's no such assumption in the problem statement.