I am looking for a proof of the fundamental inequality of the index form, which I have seen as references or statements in a lot of sources, but without a proof. This is the statement:
Let $M$ be a riemannian manifold and $\gamma : [a,b] \to M$ be a geodesic. Let $E_1, ... , E_n$ be an orthonormal frame along $\gamma$ and let $J_1, ... J_n$ be the unique Jacobi fields along $\gamma$ with $J_i(a)=0$ and $J_i(b)=E_i$ for all i. Then: $I(J_i) \leq I(\frac{t}{d(a,b)}E_i)$
As a reminder:
The index form is defined as
$I:\mathfrak{X}(M)\times\mathfrak{X}(M)\to\mathbb{R}$, $I(X,Y) := \int_a^b (\langle \nabla_{\dot{\gamma}} X, \nabla_{\dot{\gamma}} Y\rangle - R(\dot{\gamma},X,\dot{\gamma},Y))dt.$
and with just one argument:
$I:\mathfrak{X}(M)\to\mathbb{R}$, $I(X) := I(X,X).$
Thanks a lot!
(please correct me, if I have misstated anything)
A proof can be found in this book:
Jurij Dmitrievic Burago and Viktor Abramovic Zalgaller. Geometric Inequalities. Springer, 1988.
on page 237.
One can take a look at the proof here:
https://books.google.de/books?id=Gpz6CAAAQBAJ&printsec=235&hl=de&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false
(Luckily: When I was looking, pages 235 - 237 were available)