I'm reading the John Lee's book 'Riemannian Manifolds-An Introduction to Curvature', proof of the Theorem 11.7. and stuck at some statements. The theorem states as follows :
Theorem 11.7. ( Bonnet's Theorem ) Let $M$ be a complete, connected, Riemannian manifold all of whose sectional curvatures are bounded below by a positive constant $1/R^2$. Then $M$ is compact, with a finite fundamental group, and with diameter less than or equal to $\pi R$.
Proof. The first step is to show that the diameter of $M$ is no greater than $\pi R$. Suppose the contrary : then there are points $p,q \in M$, and (by the Hopf-Rinow theorem : his book Cor.6.15) a minimizing unit speed geodesic segment $\gamma$ from $p$ to $q$ of length $L > \pi R$. Since $\gamma$ is minimizing, its index form is nonnegative (his book Cor. 10.13), We will derive a contradiction by constructing a proper (c.f. his book p.98) normal vector field $V$ along $\gamma$ such taht $I(V,V) <0$. By direct computation,
Let $E$ be any parallel normal unit vector field along $\gamma$, and let $$ V(t) := ( \sin\frac{\pi t}{L})E(t).$$
Q.1. My first question is, why can we choose parallel normal unit vector field $E$ along $\gamma$? Here, 'parallel' means $D_t E =0$ ( the covariant derivative along $\gamma$ ). We have a theorem (his book Theorem 4.11) : Given a curve $\gamma : I \to M$, $t_0 \in I$, and a vector $V_0 \in T_{\gamma(t_0)}M$, there exists a unique parallel vector field $V$ along $\gamma$ such that $V(t_0) = V_0$.
And the 'normal' means $E(t) \perp \dot{\gamma}(t)$ for all $t$. Anyway, can we construct parallel normal unit vector field along $\gamma$? How?
And, he continue to argue as follows :
Observe that $V$ vanishes at $t=0$ and $t=L$, so $V$ is a proper normal vector field along $\gamma$. ( Note the similarity between $V$ and the formulas $(10.5)$, $(10.6)$ for Jacobi fields on the sphere of radius $L/\pi$ ). By direct computation,
$$ D_tV(t) = \frac{\pi}{L}(\cos \frac{\pi t}{L})E(t)$$ $$ D_t^2V(t) = -\frac{\pi^2}{L^2}(\sin \frac{\pi t}{L})E(t),$$
and so
$$ I(V,V) = - \int_{0}^{L} \langle D_t^2V + R(V,\dot{\gamma})\dot{\gamma},V \rangle dt$$
..( Omitted )
Q.2. I don't know why this final equality is true. It seems that he used next theorem ( his book, Proposition 10.14 )
Proposition 10.14. For any pair of proper normal vector fields $V,W$ along a geodesic segment $\gamma$, $ I(V,W) = -\int_{a}^{b} \langle D_t^2 V + R(V,\dot{\gamma})\dot{\gamma},W \rangle dt -\Sigma_{i=1}^{k}\langle \Delta_iD_t V , W (a_i) \rangle $, where $\{ a_i \}$ are the points where $V$ is not smooth, and $\Delta_{i}D_t V$ is the jump in $D_tV$ at $t=a_i$.
My question is, in our case, why $\Sigma_{i=1}^{k}\langle \Delta_iD_t V , V (a_i) \rangle = 0$ ? What the term 'jump' exactly means?
Now, accept next statement : Let $M$ be a complete, connected Riemannian manifold all of whose sectional curvatures are bounded below by a positive constant $1/ R^2$, then the diameter $\operatorname{diam}(M):= \sup \{ d(p,q) : p, q \in M\} \le \pi R$.
In the third paragraph in the proof, he argues as follows :
To show that $M$ is compact, we just choose a base point $p$ and note that every point in $M$ can be connected ( by the Hopg-Rinow theorem ) to $p$ by a geodesic segment of length at most $\pi R$. Therefore, $\exp_p : \bar{B}_{\pi R}(0) \to M $ is surjective, so $M$ is the continuous image of a compact set.
Q.3. I don't understand why the $\exp_p : \bar{B}_{\pi R}(0) \to M $ is surjective, although it seems possible. Let $\mathcal{E} := \{ V \in TM := \uplus_{p\in M}T_p M : \gamma_V \operatorname{is defined on an interval containing} [0,1] \}$, where $\gamma_V$ is the maximal geodesic. Note that since $M$ is complete, each $\gamma_V$ is defined for all $t\in \mathbb{R}$. Then the exponential map $\exp$ is defined by $\exp(V) := \gamma_V(1)$. And for each $p \in M$, the restricted exponential map $\exp_p$ is the restriction of $\exp$ to the set $\mathcal{E}_p := \mathcal{E} \cap T_pM$.
To show that $\exp_p : \bar{B}_{\pi R}(0) \to M $ is surjective, let $q \in M$ be an another element. As he argued above, also by the Hopf-Rinow theorem ( his book Cor. 6.15 ) there exists a geodesic segment $\gamma : [0,L] \to M$ ( up to unit speed reparametrization ( his book p.93, Exercise 6.2. ) connecting $p$ and $q$ of length $L \le \pi R$. From these, how can we find $V \in \bar{B}_{\pi R}(0)$ such that $\exp_p(V) = \gamma_V(1) = q$ ( through some reparametrization )?
And, in the final paragraph, he argues as follows :
Finally, let $\pi : \tilde{M} \to M$ denote the universal covering space of $M$, with the metric $\tilde{g} := \pi^{*}g$. Note that in this case, $\pi$ is local isometry. By the result of Problem 6-11, $\tilde{M}$ is complete. Note that $\tilde{g}$ also has sectional curvatures bounded below by $1/R^2$, so $\tilde{M}$ is compact by the argument above. By the theory of covering spaces.. (omitted ).
Q.4. Finally I want to understand the bold statement. Can we try to use next Lemma (his book p.119, Lemma 7.2.) ?
Lemma 7.2. The Riemann curvature endomorphism and curvature tensor are local isometry invariants. More precisely, if $\varphi : (M,g) \to (\tilde{M},\tilde{g})$ is a local isometry, then $\varphi^{*}(\tilde{Rm}) = Rm$.
It seems that there is possibility that this trial works but I can't make formal proof until now.
Understanding these issues will enhance understanding theory of Riemannian Geometry. Thanks for reading. Can anyone helps?
Let $V_0\perp \gamma '$ in Thm. 4.11.
By "jump" it means the following for broken curve segment (FIGURE 6.6. in the book):
To prove that $\exp$ is surjective here, we need a direction $V$ and a $t$. (Proposition 5.7. - b). Let $\gamma$ be a geodesic connection $p$ to $q$ of length $\leq \pi R$. now let $V$ to be initial velocity of this geodesic and we are sure that there is a $t$ that make this equation valid because we defined $\gamma$ be an geodesic connection $p$ to $q$.
It follows from local isometry property of Riemannian universal covering space (that follows from Lemma 7.2.) that $K_M(\pi_*\Pi) = K_{\widetilde{M}}(\Pi)$. So both have same sectional curvature lower bound.