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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!

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    $\begingroup$ "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. $\endgroup$
    – Didier
    Commented Nov 30, 2021 at 9:36

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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.

BTW, in case anyone wants to look this up in the book, this is Problem 10-12, not Exercise 10.12. The problems are at the ends of the chapters, while exercises are incorporated into the text of the chapters.

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