Here we see how Jacobi fields arise as variations of the exponential map. This should be of no surprise since Jacobi fields are variations of geodesics and the exponential map is defined in terms of geodesics.

Throughout we will assume that \((M, g)\) is complete so that \((M, d_g)\) is a complete metric space, or equivalently by the Hopf-Rinow theorem, \((M, g)\) is geodesically complete so that \(\exp_x\) is smoothly defined on all of \(T_x M\). In this case, for any \(x, y \in M\) there exists a shortest geodesic joining \(x\) to \(y\). That is, there exists a geodesic \(\gamma : [0, L] \to M\) such that \[ \gamma(0) = x, \quad \gamma(L) = y, \quad d(x, y) = L(\gamma). \] No assumption about uniqueness of \(\gamma\) is made and in general this is false as in the examples below.

Now, let \(x \in M\). We have \[ \exp_x : T_x M \to M \] is a diffeomorphism on an open neighbourhood of the origin. Given \(X, Y \in T_x M\) we think of \(Y \in T_X(T_x M)\) via \[ \rho_X: Y \in T_x M \mapsto \partial_t|_{t=0} (X + t Y) \] which defines a linear isomorphism \(T_x M \to T_X(T_x M)\).

Then we define \[ d(\exp_x)_X (Y) = \partial_s|_{s=0} \exp_x(X + s Y). \] Note that since \(M\) is assumed complete, such an expression is defined.

As to Jacobi fields:

Let \(\gamma: [0, L] \to M\) be a geodesic. If there exists a Jacobi field along \(\gamma\) such that \(J(0) = 0\) and \(J(t_0) = 0\) for some \(t_0 \in (0, L]\), then \(y = \gamma(t_0)\) is called a conjugate point of \(x = \gamma(0)\).

Let \(X, Y \in T_x M\) and let \(\gamma_X\) be the unique geodesic with \(\gamma_X(0) = x\) and \(\gamma_X'(0) = X\). Let \(J(t)\) be the Jacobi field along \(\gamma_X\) with \(J(0) = 0\) and \(J'(0) = Y\). Then \[ d(\exp_x)_{tX} \cdot tY = J(t). \] In particular, \(d(\exp_x)_{tX}\) is singular if and only if the point \(\exp_x(tX)\) is conjugate to \(x\).

We have \[ d(\exp_x)_{tX} \cdot tY = \partial_s|_{s=0} \exp_x (tX + s t Y) = \partial_s|_{s=0} \gamma_{X + s Y} (t). \] On the other hand, \[ \gamma_s(t) = \exp_x t(X + sY) \] is a geodesic for each hence, hence \[ J(t) = \partial_s|_{s=0} \gamma_{X + s Y} (t) \] is a Jacobi field.

Then \[ J(0) = \partial_s|_{s=0} \gamma_{X + s Y} (0) = \partial_s|_{s=0} x = 0 \] and \[ J'(t) = \partial_t \partial_s|_{s=0} \gamma_{X + sY} (t) = \partial_s|_{s=0} \partial_t \gamma_{X + s Y} = \partial_s|_{s=0} (X + s Y) = Y. \]

For the last part, \(d(\exp_x)_{tX}\) is singular precisely when it has a non-zero kernel. That is when there exists a non-zero \(Y\) such that \[ d(\exp_x)_{tX} Y = 0. \] Since we know that \(d(\exp_x)_0\) is non-singular we must have \(t > 0\) and taking \(J(0) = 0\), \(J'(0) = \tfrac{1}{t} Y\) we get \[ J(t) = d(\exp_x)_{tX} (t t^{-1} Y) = 0. \]

The open (metric) ball \(B_r(x)\) of radius \(r>0\) and centre \(x \in M\) is the set \[ B_r(x) = \{y : d(x, y) < r\}. \] The closed ball has the inequality replaced by equality.

Let \(X \in T_x M\) with \(|X| < r\). That is \(X \in B_r(0) \subseteq T_x M\). Then \[ d(x, \exp_x(X)) \leq L[\gamma_X(1)] = \int_0^1 |\gamma'(t)| dt = \int_0^1 |X| dt = |X| < r. \] Thus \[ \lvert X\rvert < r \Rightarrow \exp_x(X) \in B_r(x). \] Conversely, by the Hopf-Rinow theorem, if \(y \in B_r(x)\), there exists a length minimising geodesic \(\gamma\) joining \(x\) to \(y\). Let \[ X = \frac{d(x, y)}{|\gamma'(0)|} \gamma'(0) \] so that \[ \lvert X\rvert = d(x, y) < r \] and hence \(X \in B_r(0)\) such that \[ \exp_x(X) = \gamma_X(1) = y. \] That is \[ y \in \exp_x(B_r(0)). \]

Therefore, \[ B_r(x) = \exp_x(B_r(0)). \]

In fact, as just observed if \(y \in B_r(x)\), then there exists an \(X \in B_r(0) \subseteq T_x M\) such that \(d(x, y) = L[\gamma_X]\) and \(y = \exp_x(X)\). In other words, \(\gamma_X\) minimises length between \(x\) and \(\gamma_x(t)\) for each \(t \in [0, 1]\): /if there were a shorter geodesic \(\mu\) joining \(x\) to \(\gamma_X(t)\) for some \(t \in (0, 1)\), then following first \(\mu\) from \(x\) to \(\gamma_X(t)\), and then \(\gamma_X\) to \(y\) would give a shorter geodesic joining \(x\) to \(y\).

Let, \[ C_x = \{X \in T_x M : \gamma_X \text{ minimises length for all } t \in [0, 1]\}. \] The boundary \(\partial C_x\) is called the cut locus of \(x \in M\). The interior, \(\overset{\circ}{C_x}\) is called the injectivity domain.

Notice that homogeneity of geodesics, \(\gamma_{tX}(s) = \gamma_X (ts)\), if \(X \in C_x\), then so is \(\lambda X\)for all \(\lambda \in [0, 1]\). Thus \(C_x\) is star shaped. Moreover, by the Hopf-Rinow theorem, \(M = \exp_x(C_x\).

On the other hand, for \(X \in \overset{\circ}{C_x}\), there does not exist any other \(Y \in \overset{\circ}{C_x}\) such that \(\exp_x(X) = \exp_x(Y)\). This follows since if there is another geodesic \(\mu\) minimising length between \(x\) and \(y = \exp_x(X) = \gamma_X(1)\), then \(\gamma_X\) does not minimise length from \(x\) to \(\gamma_X(1+\epsilon)\) for any \(\epsilon\). The reason for this is that one can prove minimising geodesics are smooth and hence \(mu\) must be parallel or anti-parallel with \(\gamma_X\) at \(y\) so either \(\mu = \gamma_X\), or \(\mu\) meets \(\gamma_X\) coming from the other direction and hence minimises length to \(\gamma_X(1 + \epsilon) = \mu(1-\epsilon)\).

In other words, \(\exp_x\) is injective on \(\overset{\circ}{C_x}\).

Now, as in Eschenburg chapter 5 and the references therein, a geodesic \(\gamma\) fails to minimise length from \(x = \gamma(0)\) to \(y = \gamma(1)\) only if either there exists another minimising geodesic joining \(x\) to \(y\) or if \(y\) is a conjugate point to \(x\) along \(\gamma\). We just observed the first possibility and so what remains to prove is that a geodesic does not minimise past a conjugate point (see Eschenburg for the reference to the proof). Thus since \(X \in \overset{\circ}{C_x}\) implies \(\gamma_X\) minimises up to \(1 + \epsilon\) for some \(\epsilon > 0\), neither possibility occurs. Then not only is \(\exp_x\) injective on \(\overset{\circ}{C_x}\) it is also an immersion since \(d(\exp_x)_X Y = J(t)\) for a Jacobi field along \(\gamma_X\) and since \(\gamma_X\) has no conjugate points \(J(t) \ne 0\) unless \(Y = 0\).

In other words, \(\exp_x\) is an immersion on \(\overset{\circ}{C_x}\).

Therefore in fact \(\exp_x\) is an injective immersion from \(\overset{\circ}{C_x} \to M\). In fact, it is an embedding and hence \(\exp_x\) is diffeomorphic with it's image. Moreover since \(M = \exp_x C_x\) and \(C_x = \overline{\overset{\circ}{C_x}}\) is the closure of \(\overset{\circ}{C_x}\).

Therefore \(\exp_x(\overset{\circ}{C_x})\) is a dense, open, contractible subset of \(M\). Since contractible sets have trivial topology, all the topology of \(M\) is contained in the boundary \(\partial C_x\)! Another way to put this is that the map \(\exp_x : C_x \to M\) is a surjective map from a topologically trivial set onto \(M\) and thus all the topological information is contained in the map \(\exp_x\) and what's more that information is encoded precisely on the \(\exp_x\) restricted to the boundary. Since the topology of a manifold can be arbitrarily complex, the behaviour of \(\exp_x\) can be arbitrarily bad along the boundary.

We have \[ B_r(x) = \exp_x(B_r(0) \cap C_x). \] This just follows since as discussed above, for any \(y \in M\), there always exists \(X \in C_x\) such that \(\gamma_X\) is length minimising; that is since \(\exp_x : C_x \to M\) is a surjection.

The Bonnet-Myers theorem says that if a complete Riemannian manifold has a positive lower bound, then there is a corresponding upper bound on diameter and hence the manifold is compact. Myers gave the first proof. Prior to that Bonnet proved the statement when the Ricci curvature bound is replaced by the stronger hypothesis that the sectional curvature is bounded. Cheng's contribution was to prove that the equality case is attained precisely on the sphere with it's standard "round" metric.

Let \(k > 0\) and let \(\mathbb{S}^n (k)\) denote the sphere of constant sectional curvature \(k\).

Let \((M, g)\) be a Riemannian manifold such that \[ \operatorname{Ric} \geq (n-1) k = \operatorname{Ric} (\mathbb{S}^n(k)) \] for some \(k > 0\).

Then \[ \operatorname{diam} (M, g) := \sup\{d(x, y) : x, y \in M\} \leq \frac{\pi}{\sqrt{k}}. \] Moreover, equality is attained if and only if \((M, g)\) is isometric to \(\mathbb{S}^n(k)\).

Let \(\gamma\) be a geodesic with unit length tangent vector \(V = \gamma'\). The assumptions of the theorem imply that \[ \operatorname{Tr} \operatorname{R}_V = \operatorname{Ric}(V) \geq (n-1) k. \] The average comparison theorem for the Riccati equation then implies that for any solution of the Riccati equation, the function \(a = \tfrac{1}{n-1} \operatorname{Tr} A\) satisfies \[ a(t) \leq \bar{a}(t) \] where \(\bar{a}(t)\) satisfies \(a' + a^2 + k = 0\) and \(\bar{a}(0) = a(0)\).

Here we need to take the extended version with \(A(t) \sim 1/t\) as \(t \to 0\) so that our Jacobi fields satisfying \(J(0) = 0\) solve \(J' = A J\).

Then as in the discussion before the average Jacobi field comparison theorem, we take a basis \(\{J_1, \cdots, J_{n-1}\}\) of Jacobi fields solving \(J' = A J\) (equivalently a basis of Jacobi fields such that \(J_i(0) = 0\)). Then by the average Jacobi field comparison theorem, the function \(j = \det (J_1, \cdots, J_{n-1})\) satisfies \[

j' \bar{j}

\] where \(\bar{j}\) is formed from the corresponding fields for \(R = k \operatorname{Id}\) and satisfying \(\bar{j}' = \bar{a} \bar{j}\).

Direct computation verifies that \[ \bar{a}(t) = \sqrt{k} \cot(\sqrt{k} t) \] and \[ \bar{J}_i(t) = \sin(\sqrt{k} t) e_i \] is a linearly independent set of solutions. Therefore, since each \(\bar{J}_i\) has a zero (in fact the first zero) at \(\pi/\sqrt{k}\) we have \[ \bar{j}(\pi/\sqrt{k}) = 0. \] Since \(|j| \leq |\bar{j}|\), there exists a \(t_0 \leq \pi/\sqrt{k}\) with \[ j(t_0) = 0. \] But \(j = \det(J_1, \cdots, J_{n-1})\) and \(J_i\) are a linearly independent set of solutions so that at least one of \(J_i\) must satisfy \(J_i(t_0) = 0\).

Now let \(x, y \in M\) be arbitrary and let \(\gamma\) be a unit length, length minimising geodesic joining \(x\) to \(y\). We have just proven that there is a \(t_0 \leq \pi/\sqrt{k}\) such that \(\gamma(\pi/\sqrt{k})\) is conjugate to \(x\). Since geodesics do not minimise distance past their first conjugate point, we must have \[ y = \gamma(t_1) \text{ for } t_1 \leq \pi/\sqrt{k} \] and \[ d(x, y) = t_1 \leq \pi/\sqrt{k}. \] That is for all \(x, y \in M\), \[ d(x, y) \leq \pi/\sqrt{k} \] and hence \[ \operatorname{diam} (M, g) \leq \frac{\pi}{\sqrt{k}}. \]

Note that if we have equality, then there is a geodesic realising the diameter and we must have equality in the average Riccati comparison theorem along that geodesic. This gives \(\operatorname{R}_V = (n-1)k \operatorname{Id}\) along the geodesic. One one needs to show that it in fact, the diameter equality on holds on all geodesics. For this see Eschenburg by applying the maximum principle to certain sub-harmonic functions.

With the same assumptions as in the theorem, \(M\) is compact and its universal cover is compact hence \(M\) has finite fundamental group.

The theorem implies that both \(M\) and its universal cover \(\hat{M}\) have finite diameter and hence are closed and bounded subsets of themselves. The Hopf-Rinow theorem then implies that both \(M\) and \(\hat{M}\) are compact. The fundamental group is finite, because \(\hat{M} \to M\) is a compact covering hence finite sheeted.

Next we a theorem on the growth of the volume of metric balls.

Let \((M, g)\) be a complete manifold with \[ \operatorname{Ric} \geq k(n-1) \] where \(k \in \mathbb{R}\). Let \(\bar{B}_r\) be a ball of radius \(r > 0\) in \((\bar{M}, g_k)\) the complete, simply connected manifold of constant sectional curvature \(k\) (sphere, Euclidean space or hyperbolic space). Then for any \(x \in M\) writing \(B_r(x)\) for the ball of radius \(r\), the function \[ r \mapsto \frac{\operatorname{Vol} B_r(x)}{\operatorname{Vol} \bar{B}_r} \] is monotonically decreasing down to \(1\). In particular, for \(r_1 < r_2\), \[ \frac{\operatorname{Vol} B_{r_2}(x)}{\operatorname{Vol} B_{r_1}(x)} \leq \frac{\operatorname{Vol} \bar{B}_{r_2}}{\operatorname{Vol} \bar{B}_{r_1}} \] If equality holds at some \(r_1 < r_2\) then \(B_{r_2}(x)\) is isometric to \(\bar{B}_{r_1}\).

Another way to phrase the last inequality is that for say \(r > 1\), \[ \operatorname{Vol} B_{r}(x) \leq \frac{\operatorname{Vol} B_1(x)}{\operatorname{Vol} \bar{B}_1} \operatorname{Vol} \bar{B}_r = C(x) \operatorname{Vol} \bar{B}_r \] and we get an estimate for the growth rate of the volume of metric balls.

We have \[ \exp_x : B_r(0) \cap \overset{\circ}{C_x} \to B_r(x) \] is a diffeomorphism onto a dense subset of \(B_r(x)\) so that \[ \operatorname{Vol} (B_r(x)) = \int_{B_r(0) \cap \overset{\circ}{C_x}} \det(d(\exp_x)_X) dX. \] For \(X \in T_x M\) with \(\|X\| = 1\) (i.e. \(X \in \mathbb{S}^{n-1}\), let \(\operatorname{cut}(X) = \sup\{t : \gamma_X \text{ is minimising} \}\) and \(r(X) = \min\{r, \operatorname{cut}(X)\}\). Then in polar coordinates, \[ \operatorname{Vol} (B_r(x)) = \int_{\mathbb{S}^{n-1}} \int_0^{r(v)} \det(d(\exp_x)_{tX}) t^{n-1} dt dX. \] Letting \(e_1, \cdots, e_{n-1}\) be a basis for \(X^{\perp}\) gives \[ d(\exp_x)_{tX} e_i = \frac{1}{t} d(\exp_x)_{tX} t e_i = \frac{1}{t} J_i(t) \] where \(J_i\) is the Jacobi field with \(J_i(0) = 0\), \(J_i'(0) = e_i\). Therefore \[ \det d(\exp_x)_{tX} = \frac{1}{t^{n-1}} j_X \] and \[ \operatorname{Vol} (B_r(x)) = \int_{\mathbb{S}^{n-1}} \int_0^{r(v)} j(t) dt dX. \]

Then \(j/\bar{j}\) is monotonically decreasing where \(\bar{j}\) is the corresponding quantity in constant curvature. Therefore the function \[ q(t) = \frac{1}{\operatorname{Vol} \mathbb{S}^{n-1} } \int_{\mathbb{S}^{n-1}} \frac{j_X(t)}{\bar{j}(t)} dX \] is also monotone decreasing. Then \[ \frac{\operatorname{Vol} B_r(x)}{\operatorname{Vol} \bar{B}_r} = \frac{\int_0^r \int_{\mathbb{S}^{n-1}} j_X(t) dt dX}{\operatorname{Vol} \mathbb{S}^{n-1} \int_0^r \bar{j}(t) dt} = \frac{\int_0^r q(t) \bar{j}(t) dt}{\int_0^r \bar{j}(t) dt} \] is the mean of \(\bar{q}\) with respect to the non-negative, weighted measure \(\bar{j}(t) dt\). But means of decreasing functions are decreasing in the upper limit of integration \(r\).

Prove that if \(q\) is a decreasing function and \(\bar{j} \geq 0\), then \[ r \mapsto \frac{\int_0^r q(t) \bar{j}(t) dt}{\int_0^r \bar{j}(t) dt} \] is decreasing. Hint: Differentiate with respect to \(r\).