VOLUMES OF BALLS IN MANIFOLDS WITHOUT CONJUGATE POINTS

Christopher B. Croke ^{∗}

October 1991

Introduction

In this paper we consider volumes of balls in the universal cover M̃ of a compact Riemannian manifold M without conjugate points. We will let V ol(B̃(x, r)) (resp. V ol(S̃(x, r))) be the volume of a metric ball (resp. sphere) of radius r in M̃ centered at a point x̃ that projects to x. For f any function on M we will let Ave(f(x)) be the average of f with respect to the Riemannian measure dx on M. We will let β(n) be the volume of the unit n ball and α(n) be the volume of the unit n sphere.

We first show:

Theorem 1 Let M be a compact manifold without conjugate points. Then for every r > 0:

Ave(V ol(B̃(x, r))) ≥ β(n)r^{n} with equality if and only if M is flat.

Further for each x ∈ M:

lim inf_{r→∞} \frac{V ol(B̃(x, r))}{β(n)r^{n}} ≥ 1

with equality if and only if M is flat.

∗supported in part by NSF grant #DMS85-05550 while at MSRI and NSF grant #DMS90-01707

In 1958 L. Green [G], extending the two dimensional result of E. Hopf [H], showed that on a compact manifold without conjugate points the total scalar curvature is nonpositive and equal to zero if and only if M is flat. If we let f(r) = Ave(V ol(B̃(x, r))), then one can interpret this theorem as saying that f(r) ≥ β(n)r^{n} for sufficiently small r.

Since in M̃ the injectivity radius, inj(M̃), is infinite theorem 1 is related to results about the volume of balls whose radius is less than the injectivity radius. In [C4] we showed that there is a constant c_{1}(n), depending only on n, such that for any n dimensional Riemannian manifold M, any x ∈ M, and any r ≤ inj(M) we have V ol(B(x, r)) ≥ c_{1}(n)r^{n}. The constant c_{1}(n) is not sharp. Berger’s isoembolic inequality (see [B2] and [B-K]) says that for every compact Riemannian manifold M, V ol(M) ≥ c_{2}(n)inj(M)^{n} with equality holding if and only if M is a round sphere. This was generalized in [C1] in a number of directions. In particular, it was shown that for r ≤ inj(M), Ave(V ol(B(x, r))) ≥ c_{3}(n)r^{n}, with equality holding if and only if M is isometric to a round sphere of injectivity radius r.

Two Riemannian manifolds ,M and N, are said to have conjugate geodesic flows if there is a homeomorphism F : S^{1}M → S^{1}N between the unit tangent bundles which commutes with the geodesic flows, i.e. F ∘ g_{M}^{t} = g_{N}^{t} ∘ F where g_{M}^{t} (resp. g_{N}^{t}) is the geodesic flow on M (resp. N). We use theorem 1 to prove the following:

Theorem 2 Let M^{n} be a compact manifold whose geodesic flow is conjugate to the geodesic flow of a flat manifold N^{n} which has the same volume as M. Then M and N are isometric.

We prove this theorem by showing that M must have no conjugate points, and that volumes of balls in M̃ grow asymptotically like euclidean balls and hence by Theorem 1, M must be flat.

One could ask if the volume condition is necessary in theorem 2. This problem will be taken up in a future paper with Bruce Kleiner where in particular it will be shown that no such assumption is needed if F is assumed to be C^{1}.

By the results in [M] and [F-M], f(r) will grow exponentially unless the topological entropy, h, of the geodesic flow is 0. We will let h_{μ} be the measure entropy of the geodesic flow with respect to the normalized Liouville measure μ = \overline{dv} = \frac{1}{V ol(S^{1}M)}dv where dv is the Liouville measure. It was shown in [F-M] that this is the average of the mean curvature, u(v)(the trace of the second fundamental form – see section 1 for a definition), of the horospheres in M̃. For manifolds without conjugate points it is clear that h_{μ} ≥ 0. Let δ = δ(h_{μ}) = −(\frac{h_{μ}}{n−1})^{2} be such that the simply connected space M_{δ}^{n} of constant curvature δ has horospheres with mean curvature h_{μ}. Let V ol_{δ}B(r) be the volume of a ball (and V ol_{δ}S(r) be the volume of a sphere) of radius r in M_{δ}^{n}. We show:

Theorem 3 Let M^{n} be a compact manifold without conjugate points and let δ = δ(h_{μ}), then for every r > 0:

Ave(V ol(B̃(x, r))) ≥ V ol_{δ}B(r) with equality holding if and only if M has constant curvature δ.

Further

lim inf_{r→∞} \frac{Ave(V ol(B̃(x, r)))}{V ol_{δ}B(r)} ≥ 1

with equality if and only if M has constant curvature δ.

Remark: Similar statements are true for V ol(S̃(x, r)). In fact in the proof of the theorem we prove the results for V ol(S̃(x, r)) and get the theorem by integration.

For a manifold of negative curvature, Margulis [Ma] considered the function

C(x) = lim_{r→∞} \frac{V ol(B̃(x, r))}{V ol_{δ(h)}B(r)}

where h is the topological entropy. Margulis showed that C(x) exists and is continuous. Recent work of Yue [Y] shows, among other things, that C(x) is in fact smooth and generically not constant. In general h ≥ h_{μ}. The asymptotic estimate in theorem 3 is interesting only in the case where h = h_{μ} (for otherwise the left hand side is infinite) where it can be phrased as Ave(C(x)) ≥ 1 with equality if and only if M has constant curvature.(One can pass the limit through the Ave since, as is not hard to show, the functions V ol(B̃(x, r))/V ol_{δ}(r) converge uniformly to C(x).)

We note that theorem 1 is a consequence of theorem 3 since h_{μ} ≥ 0 and since for fixed x_{0}

lim inf_{r→∞} \frac{V ol(B̃(x_{0}, r))}{β(n)r^{n}} = lim inf_{r→∞} \frac{Ave(V ol(B̃(x, r)))}{β(n)r^{n}}.

You can see this by letting D be the diameter of a fundamental domain and noticing that

Ave(V ol(B̃(x, r + D))) ≥ V ol(B̃(x_{0}, r)) ≥ Ave(V ol(B̃(x, r − D))).

Remark: The conjugacy problem in theorem 2 is closely related to the boundary rigidity problem discussed in [C2]. A Riemannian manifold (M, ∂M, g_{0}) with boundary ∂M is called boundary rigid if every (N, ∂M, g_{1}) (with diffeomorphic boundary) with d_{g_{0}}(p, q) = d_{g_{1}}(p, q) for every p, q ∈ ∂M, must be isometric to M (d_{g} represents the distance in M between boundary points). It has been shown (see [Gr] sec5.5B, [Mi], and [C2]) that if M^{n} can be isometrically immersed in R^{n} then it is boundary rigid. Theorem 2 along with the arguments in sections 5 and 7 of [C2] yield the fact that SGM subdomains of compact flat manifolds are boundary rigid. The condition SGM is a condition on d_{g_{0}} (see [C2] sec. 1) which loosely speaking says that every geodesic segment whose interior lies in the interior of M is the unique geodesic between its endpoints. While all M that isometrically immerse in R^{n} are SGM there are SGM subdomains of compact flat manifolds that do not isometrically immerse in R^{n}, so we get some new examples of boundary rigid manifolds. Other examples of boundary rigid manifolds are: subdomains of an open hemisphere of a round sphere (see [Mi] and [Gr] sec. 5.5B) and any SGM surface (two dimensional) of nonpositive curvature [C3].

Remark:One can ask similar questions for non-compact complete manifolds. Innami ([I1] and [I2]) and Guimarães ([Gu]) proved extensions of Greens theorem [G] to the complete case. The argument given here for fixed r in theorem 3 extends in a standard way to the complete finite volume case whenever δ = −(\frac{1}{n−1} ∫_{S^{1}M} u(v)\overline{dv})^{2} is defined, i.e. |u(v)| is integrable. In particular, this holds if the curvature is bounded. The key point is that u(v) is always defined, measurable, and bounded on compact sets (see for example [I1]).

The author would like to thank Bruce Kleiner for helpful conversations and useful comments and to thank Jianguo Cao for a careful reading of the first draft.

1 The Proofs for Finite r

In this section we prove the inequality in theorem 3 for fixed finite r.

We will be considering Lagrange tensors along a geodesic γ(t).(For a detailed discussion see [E-OS].) These are tensors, A, which satisfy:

1. the Jacobi equation:

A(t)′′ + R(t)A(t) = 0

where R is the curvature tensor R(., γ′(t))γ′(t).

2. A′(t)A^{−1}(t) is symmetric whenever A^{−1} exists.

3. A(t) is nondegenerate in the sense that the kerA(t) ∩ kerA′(t) = 0.

For v ∈ S^{1}M there are two such Lagrange tensors along γ_{v} (the geodesic with γ′(0) = v) of interest to us. A_{v} will be the Lagrange tensor with A_{v}(0) = 0 and A′_{v}(0) = Id. J_{v} will be the unstable Lagrange tensor with J_{v}(0) = Id (i.e. J_{v} is the limit as t_{0} → ∞ of the Lagrange tensors J with J(0) = Id and J(−t_{0}) = 0 see ([E-OS],[F-M] and [G]). U(v) = J′_{v}(0)J_{v}^{−1}(0) = J′_{v}(0) will be the second fundamental form of the horosphere determined by v. It is a bounded symmetric solution to the Riccati equation U′ + U^{2} + R = 0.

We will let u(v) = tr(U(v)) be the mean curvature of the stable horosphere through v. It was shown in [E-OS],[F-M] and [G] that u(v) is bounded and varies measurably in v. In [F-M] it was shown that h_{μ} is the average of u(v) with respect to the Liouville measure dv on S^{1}M (i.e. h_{μ} = ∫_{S^{1}M} u(v)\overline{dv}).

Since J_{g^{t}(v)}(s) = J_{v}(t + s)J^{−1}(t) it follows that J′_{v}(t)J_{v}^{−1}(t) = U(g^{t}v). Hence d/dt log(det(J_{v}(t))) = tr(J′_{v}(t)J_{v}^{−1}(t)) = u(g^{t}v). Thus:

(1.1)  det(J_{v}(t)) = exp{∫_{0}^{t} u(g^{z}v)dz}.

Lemma 1 Let A_{v}, J_{v}, and u be as above. Then

det(A_{v}(l)) ≥ [∫_{0}^{l} exp\{\frac{1}{n − 1}[∫_{0}^{l} u(g^{z}v)dz − 2 ∫_{0}^{s} u(g^{z}v)dz]\}ds]^{n−1}

with equality holding if and only if J_{v}(t)J_{v}^{∗}(t) = λ(t)J for a function λ and a constant endomorphism J.

Proof of Lemma 1: Since J_{v}(t) is a non singular Lagrange tensor and J_{v}(0) = Id following [Be] appendix D we can write:

(1.2)  A_{v}(l) = J_{v}(l) ∫_{0}^{l} J_{v}^{−1}(s)J_{v}^{−1∗}(s)ds.

From Corollary 1 section 2 (or see [Be] appendix D):

det[∫_{0}^{l} J_{v}^{−1}(s)J_{v}^{−1∗}(s)ds] ≥ [∫_{0}^{l}(det(J_{v}(s)))^{\frac{−2}{n−1}} ds]^{n−1}

with equality holding if and only if J_{v}^{∗}(t)J_{v}(t) = λ(t)J. Applying this to equations 1.1 and 1.2 yields:

det(A_{v}(l)) ≥ exp\{∫_{0}^{l} u(g^{z}v)dz\}[∫_{0}^{l} exp\{\frac{−2}{n − 1} ∫_{0}^{s} u(g^{z}v)dz\}ds]^{n−1}

= [∫_{0}^{l} exp\{\frac{1}{n − 1} ∫_{0}^{l} u(g^{z}v)dz\} exp\{\frac{−2}{n − 1} ∫_{0}^{s} u(g^{z}v)dz\}ds]^{n−1}

The lemma now follows. □

Next we prove a linear algebra lemma that will be useful in the study of the equality cases.

Lemma 2 If A(t) is a differentiable curve in Gl(n) such that A(0) = Id, A′(t)A^{−1}(t) is symmetric, and A^{∗}(t)A(t) = λ(t)Id then A(t) = \sqrt{λ(t)}Id.

Proof of Lemma 2: We can and will assume that λ(t) = 1 by considering B(t) = 1/\sqrt{λ(t)}A(t) which satisfies the conditions of the lemma with λ = 1. But then A^{∗}(t) = A^{−1}(t). Differentiating A(t)A^{∗}(t) = Id tells us that A′(t)A^{∗}(t) is skew symmetric. Thus A′(t)A^{−1}(t) is both symmetric and skew symmetric and so is 0. Thus A′(t) = 0 and so A(t) = Id. □

We are now in a position to prove the first part of Theorem 3 (and hence the first part of Theorem 1). In order to prove the second part of the theorem we will need to be more careful about the inequalities we use (including Lemma 1 ), but for the sake of clarity we will give the straight forward proof of the first part and then modify it as necessary for the second part.

Proof of first part of Theorems 1 and 3:

Ave(V ol(S̃(x, l))) = \frac{1}{V ol(M)} ∫_{S^{1}M} det(A_{v}(l))dv.

≥ α(n − 1)[∫_{S^{1}M} ∫_{0}^{l} exp\{\frac{1}{n − 1}[∫_{0}^{l} u(g^{z}v)dz − 2 ∫_{0}^{s} u(g^{z}v)dz]\}ds\overline{dv}]^{n−1}

by Lemma 1, the definition of \overline{dv}, and a Höelder inequality. Hence by changing order of integration, and then applying Jensen’s inequality to the convex function exp we see:

[\frac{Ave(V ol(S̃(x, l)))}{α(n − 1)}]^{\frac{1}{n−1}} ≥

= ∫_{0}^{l} ∫_{S^{1}M} exp\{\frac{1}{n − 1}[∫_{0}^{l} u(g^{z}v)dz − 2 ∫_{0}^{s} u(g^{z}v)dz]\}\overline{dv}ds

≥ ∫_{0}^{l} exp\{∫_{S^{1}M} \frac{1}{n − 1}[∫_{0}^{l} u(g^{z}v)dz − 2 ∫_{0}^{s} u(g^{z}v)dz]\overline{dv}\}ds.

Now the fact that \overline{dv} is invariant under the geodesic flow g^{z} yields:

∫_{S^{1}M} ∫_{0}^{t} u(g^{z}v)dz\overline{dv} = ∫_{0}^{t} ∫_{S^{1}M} u(g^{z}v)\overline{dv}dz = ∫_{0}^{t} ∫_{S^{1}M} u(v)\overline{dv}dz = th_{μ} Applying this to the above inequality yields:

[\frac{Ave(V ol(S̃(x, l)))}{α(n − 1)}]^{\frac{1}{n−1}} ≥ ∫_{0}^{l} exp\{\frac{1}{n − 1}(l − 2s)h_{μ}\}ds

= ∫_{0}^{l} e^{(l−2s)\sqrt{−δ}}ds ≡ C_{δ}(l) = \{ \begin{matrix}l, & if δ = 0; \\ 1/\sqrt{−δ} sinh(\sqrt{−δ}l), & if δ < 0. \end{matrix}

Thus we see that Ave(V ol(S̃(x, l))) ≥ V ol_{δ}S(l). Integrating both sides as l runs from 0 to r yields the inequalities in the theorems.

If equality holds then we must have equality in the Jensen inequality for a.e. s and in lemma 1 for a.e. v. Hence for a.e. s and a.e. v:

(l − 2s)\sqrt{−δ} = \frac{1}{n − 1}[∫_{0}^{l} u(g^{z}v)dz − 2 ∫_{0}^{s} u(g^{z}v)dz]

so u(v) = (n − 1)\sqrt{−δ} for a.e. v.

Equality in Lemma 1 says J_{v}^{∗}(t)J_{v}(t) = λ(t)J for a.e. v. We can take J = Id since J_{v}(0) = Id. By equation 1.1 and the above we see λ(t) = e^{2t\sqrt{−δ}}Id. Lemma 2 then says that J_{v}(t) = e^{t\sqrt{−δ}}Id for a.e. v and hence by the Jacobi equation (or the Riccati equation) we see that M has constant curvature δ.

□

2 The Function Det

We still need to prove that lim inf_{r→∞} \frac{Ave(V ol(B̃(x,r)))}{V ol_{δ}(r)} = 1 implies that M has constant curvature δ. To do this we need to study more closely the errors introduced in the two convexity inequalities we used in the proof.

We now consider the strictly convex function G = det^{−1/2} on the space, SA, of positive definite self adjoint endomorphisms (see [B1] 11.8.9.5). For fixed S_{0} we define the linear part , L_{S_{0}}, of G by L_{S_{0}}(S) = DG_{S_{0}}(S − S_{0}) + G(S_{0}) and the remainder, R_{S_{0}}, of G by G(S) = L_{S_{0}}(S) + R_{S_{0}}(S). The strict convexity of G implies that R_{S_{0}} is non-negative and strictly convex.

Let U be such that det(U) = 1 then S ↦ USU^{∗} is a linear isomorphism on SA which preserves G and hence:

(2.0)  R_{US_{0}U^{∗}}(USU^{∗}) = R_{S_{0}}(S).

Let S be a measurable map from a space P, with probability measure dp, to SA. Integrating G = L_{S_{0}} + R_{S_{0}} with S_{0} = ∫_{P} S(p)dp we see:

(2.1)  ∫_{P} G(S(p))dp = G(∫_{P} (S(p))dp) + ∫_{P} R_{∫_{P} S(p)dp}(S(p))dp.

Similarly if we define ep(x) = exp(x) − 1 − x (i.e. ep is the remainder at 0 of exp) and if a real valued function f has average 0 on P then:

(2.2)  ∫_{P} exp(f(p))dp = 1 + ∫_{P} ep(f(p))dp.

Lemma 3 If J(t) is continuous in GL(n − 1) and det(J(t)) > 0 then:

det[∫_{0}^{l} J^{−1}(t)J^{−1∗}(t)dt] =

[1 − ∫_{0}^{l} R_{I_{l}}(det(J(t))^{\frac{2}{n−1}} J^{−1}(t)J^{−1∗}(t))dμ]^{−2}[∫_{0}^{l} det(J(z))^{\frac{−2}{n−1}} dz]^{n−1}

where:

dμ = \frac{det(J(t))^{\frac{−2}{n−1}} dt}{∫_{0}^{l} det(J(z))^{\frac{−2}{n−1}} dz} and I_{l} = ∫_{0}^{l} det(J(t))^{\frac{2}{n−1}} J^{−1}(t)J^{−1∗}(t)dμ.

Proof of lemma 3: Apply equation 2.1 to the space [0, l] with measure dμ to get:

∫_{0}^{l} det ^{−1/2}(det(J(t))^{\frac{2}{n−1}} J^{−1}(t)J^{−1∗}(t))dμ =

= det ^{−1/2}[∫_{0}^{l} det(J(t))^{\frac{2}{n−1}} J^{−1}(t)J^{−1∗}(t)dμ]+∫_{0}^{l} R_{I_{l}}(det(J(t))^{\frac{2}{n−1}} J^{−1}(t)J^{−1∗}(t))dμ.

An easy computation shows that the left hand side is 1. Hence:

[1 − ∫_{0}^{l} R_{I_{l}}(det(J(t))^{\frac{2}{n−1}} J^{−1}(t)J^{−1∗}(t))dμ]^{−2} = det(\frac{∫_{0}^{l} J^{−1}(t)J^{−1∗}(t)dt}{∫_{0}^{l} det(J(z))^{\frac{−2}{n−1}} dz}) =

[∫_{0}^{l} det(J(z))^{\frac{−2}{n−1}} dz]^{−(n−1)} det[∫_{0}^{l} J^{−1}(t)J^{−1∗}(t)dt].

Rearranging terms yields the lemma. □

Remark: By the above 0 ≤ ∫_{0}^{l} R_{I_{l}}(det(J(t))^{\frac{2}{n−1}} J^{−1}(t)J^{−1∗}(t))dμ < 1.

Corollary 1 With J as above we have:

det[∫_{0}^{l} J^{−1}(s)J^{−1∗}(s)ds] ≥ [∫_{0}^{l}(det(J(s)))^{\frac{−2}{n−1}} ds]^{n−1}

with equality holding if and only if J_{v}^{∗}(t)J_{v}(t) = λ(t)J for a function λ(t) and a constant endomorphism J.

Proof: The inequality follows directly from the above. If equality holds then det(J(t))^{\frac{2}{n−1}} J^{−1}(t)J^{−1∗}(t) = I_{l} for all t. Hence I_{l} is invertible and symmetric hence J(t)J^{∗}(t) = det(J(t))^{\frac{2}{n−1}} I_{l}^{−1}. The other way is clear. □

3 The Proof of the Asymptotic Case

For this section we assume lim inf_{r→∞} \frac{Ave(V ol(B̃(x,r)))}{V ol_{δ}B(r)} = 1 and hence that

lim inf_{r→∞} \frac{Ave(V ol(S̃(x,r)))}{V ol_{δ}S(r)} = 1.

Our first goal is to show for a.e v ∈ S^{1}Mthat:

(3.1)  u(v) = (n − 1)\sqrt{−δ}

We define \overline{u}(v) = u(v) − (n − 1)\sqrt{−δ} and let

F (v, s, l) = (2s − l)\sqrt{−δ} + \frac{1}{n − 1}[∫_{0}^{l} u(g^{z}v)dz − 2 ∫_{0}^{s} u(g^{z}v)dz].

= \frac{1}{n − 1}[∫_{0}^{l} \overline{u}(g^{z}v)dz − 2 ∫_{0}^{s} \overline{u}(g^{z}v)dz]

We saw in the proof of the first part of the theorems that ∫_{S^{1}M} F (v, s, l)\overline{dv} = 0 for any choice of s and l, and that

[\frac{Ave(V ol(S̃(x, l)))}{α(n − 1)}]^{\frac{1}{n−1}} ≥ ∫_{0}^{l} ∫_{S^{1}M} exp(F (v, s, l))\overline{dv}e^{(l−2s)\sqrt{−δ}}ds

which by equation 2.2 (and the definition of C_{δ} in sec. 1) is

= C_{δ}(l) + ∫_{0}^{l} ∫_{S^{1}M} ep(F (v, s, l))\overline{dv}e^{(l−2s)\sqrt{−δ}}ds.

Dividing both sides by C_{δ}(l) and taking lim inf_{l→∞} we see:

0 = lim inf_{l→∞} \frac{1}{C_{δ}(l)} ∫_{0}^{l} ∫_{S^{1}M} ep(F (v, s, l))\overline{dv}e^{(l−2s)\sqrt{−δ}}ds

and hence:

(3.2)  0 = lim inf_{l→∞} \frac{e^{l\sqrt{−δ}}}{C_{δ}(l)} ∫_{0}^{l} ∫_{S^{1}M} ep(F (v, s, l))\overline{dv}e^{−2s\sqrt{−δ}}ds.

To study this we introduce the function

H(v, a, c) = ∫_{a}^{a+1} ep(\frac{1}{n − 1}(c − 2 ∫_{0}^{s} \overline{u}(g^{z}v)dz))e^{−2s\sqrt{−δ}}ds.

We claim that

H(v, a, c) = e^{−2a\sqrt{−δ}}H(g^{a}v, 0, c − 2 ∫_{0}^{a} \overline{u}(g^{z}v)dz).

This follows from:

H(v, a, c) = ∫_{a}^{a+1} ep(\frac{1}{n − 1}([c − 2 ∫_{0}^{a} \overline{u}(g^{z}v)dz]−2 ∫_{a}^{s} \overline{u}(g^{z}v)dz))e^{−2s\sqrt{−δ}}ds

by making first the substitution \overline{z} = z − a in the second internal integral and then the substitution \overline{s} = s − a we get:

H(v, a, c) = e^{−2a\sqrt{−δ}} ∫_{0}^{1} ep(\frac{1}{n − 1}([c−2 ∫_{0}^{a} \overline{u}(g^{z}v)dz]−2 ∫ \overline{s} ; 0 \overline{u}(g^{\overline{z}+a}v)d\overline{z}))e^{−2\overline{s}\sqrt{−δ}}d\overline{s}

= e^{−2a\sqrt{−δ}}H(g^{a}v, 0, c − 2 ∫_{0}^{a} \overline{u}(g^{z}v)dz).

Let H(v, a) = inf{H(v, a, c)|c ∈ R} ≥ 0. By the above, H(v, a) = e^{−2a\sqrt{−δ}}H(g^{a}v, 0). We now have for N the greatest integer less than l:

∫_{S^{1}M} ∫_{0}^{l} ep(F (s, l))e^{−2s\sqrt{−δ}}ds\overline{dv} ≥ ∑_{a=0}^{N−1} ∫_{S^{1}M} ∫_{a}^{a+1} ep(F (s, l))e^{−2s\sqrt{−δ}}ds\overline{dv}

≥ ∑_{a=0}^{N−1} e^{−2a\sqrt{−δ}} ∫_{S^{1}M} H(g^{a}v, 0)\overline{dv} = ∫_{S^{1}M} H(v, 0)\overline{dv}[ ∑_{a=0}^{N−1} e^{−2a\sqrt{−δ}}].

Since lim inf_{l→∞} \frac{e^{l\sqrt{−δ}}}{C_{δ}(l)} ∑_{a=0}^{N−1} e^{−2a\sqrt{−δ}} > 0, equation 3.2 implies that H(v, 0) = 0 for a.e. v. Since ep is strictly convex it is easy to see that this can happen only if ∫_{0}^{s} \overline{u}(g^{z}v)dz is constant a.e., so \overline{u}(v) = 0 a.e., and thus we shown equation 3.1.

Our next goal is to show that J_{v}(t) = e^{\sqrt{−δ}t}Id. Using equation 3.1 we already know that det J_{v}(t) = e^{(n−1)\sqrt{−δ}t}. Hence by lemma 3 for a.e. v:

det[∫_{0}^{l} J_{v}^{−1}(s)J_{v}^{−1∗}(s)ds] = [1−∫_{0}^{l} R_{I_{l}^{v}}(e^{2\sqrt{−δ}t}J_{v}^{−1}(t)J_{v}^{−1∗}(t))dμ]^{−2}[∫_{0}^{l} e^{−2\sqrt{−δ}z}dz]^{n−1}

where dμ = \frac{e^{−2\sqrt{−δ}t}dt}{C_{δ}^{1}(l)} where C_{δ}^{1}(l) = ∫_{0}^{l} e^{−2\sqrt{−δ}t}dt.

Using this along with equation 1.2 we see:

det(A_{v}(l)) ≥ C_{δ}(l)[1 − ∫_{0}^{l} R_{I_{l}^{v}}(e^{2\sqrt{−δ}t}J_{v}^{−1}(t)J_{v}^{−1∗}(t))dμ]^{−2}

Hence:

1 = lim inf_{l→∞} \frac{Ave(V ol(S̃(x, l)))}{V ol_{δ}S(r)} ≥ lim inf_{l→∞} ∫_{S^{1}M} [1−∫_{0}^{l} R_{I_{l}^{v}}(e^{2\sqrt{−δ}t}J_{v}^{−1}(t)J_{v}^{−1∗}(t))dμ]^{−2}\overline{dv}.

By the convexity of x^{−2} we conclude that:

(3.3) 0 = lim inf_{l→∞} ∫_{S^{1}M} \frac{1}{C_{δ}^{1}(l)} ∫_{0}^{l} R_{I_{l}^{v}}(e^{2\sqrt{−δ}t}J_{v}^{−1}(t)J_{v}^{−1∗}(t))e^{−2\sqrt{−δ}t}dt\overline{dv}.

For Q in SA let

\overline{H}(v, a, Q) = ∫_{a}^{a+1} R_{Q}(e^{2\sqrt{−δ}t}J_{v}^{−1}(t)J_{v}^{−1∗}(t))e^{−2\sqrt{−δ}t}dt.

Since J_{v}(u+a) = J_{g^{a}v}(u)J_{v}(a), using the substitution u = t−a and equation 2.0 we see:

\overline{H}(v, a, Q) = ∫_{0}^{1} R_{Q}(e^{2\sqrt{−δ}u}e^{2\sqrt{−δ}a}J_{v}^{−1}(u + a)J_{v}^{−1∗}(u + a))e^{−2\sqrt{−δ}(u+a)}du =

= e^{−2\sqrt{−δ}a} ∫_{0}^{1} R_{Q}(\{e^{\sqrt{−δ}a}J_{v}^{−1}(a)\}e^{2\sqrt{−δ}u}J_{g^{a}v}^{−1}(u)J_{g^{a}v}^{−1∗}(u)\{e^{\sqrt{−δ}a}J_{v}^{−1}(a)\}^{∗})e^{−2\sqrt{−δ}u}du =

= e^{−2\sqrt{−δ}a}\overline{H}(g^{a}v, 0, \{e^{\sqrt{−δ}a}J_{v}^{−1}(a)\}^{−1}Q\{e^{\sqrt{−δ}a}J_{v}^{−1}(a)\}^{−1∗})

As before, letting \overline{H}(v, a) = inf\{\overline{H}(v, a, Q)| Q ∈ SA\} we see that \overline{H}(v, a) = \overline{H}(g^{a}v, 0). Hence, as in the previous argument, equation 3.3 implies that:

0 = ∫_{S^{1}M} \overline{H}(v, o)\overline{dv} lim inf_{l→∞} \frac{1}{C_{δ}^{1}(l)} ∑_{a=0}^{N−1} e^{−2\sqrt{−δ}a}

and hence that \overline{H}(v, 0) = 0 for a.e. v. But this implies that e^{2\sqrt{−δ}t}J_{v}^{−1}(t)J_{v}^{−1∗}(t) is constant for t ∈ [0, 1]. Since J_{v}(0) = Id, lemma 2 yields J_{v}(t) = e^{\sqrt{−δ}t}Id for a.e. v and t ∈ [0, 1] which shows that M has constant curvature δ. □

4 Proof of Theorem 2

We now consider the proof of Theorem 2. This theorem was proved for n = 2 in [C3] so we will assume that n ≥ 3. In particular we see that the conjugacy F : S^{1}M → S^{1}N lifts to a homeomorphism F̃ : S^{1}M̃ → S^{1}Ñ and that the map on closed geodesics induced by F induces an isomorphism from π_{1}(M) to π_{1}(N). We will show that M is flat. It then follows from Bieberbach’s second theorem that M is isometric to N. To see this we note that Bieberbach’s second theorem ([Ch] page 19) says that the actions (as deck transformations) of π_{1}(M) and π_{1}(N) on R^{n} are conjugate by an affine isomorphism, A. Further the linear part of A is built out of the translations in the groups. Since F preserves the lengths of closed geodesics and hence the lengths of the corresponding translations it is easy to see that the linear part of A is orthogonal and hence that M is isometric to N. (Note that the angle between translation vectors is preserved since the length of each of the translations as well as the length of the sum is preserved.)

Lemma 4 If the geodesic flow of M^{n} is conjugate to the geodesic flow of a compact riemannian manifold N^{n} of nonpositive curvature then M has no conjugate points.

Proof: The case n = 2 was proved in [C3]. The general case follows the same argument. In N every closed geodesic is the shortest curve in its free homotopy class. Hence the same is true for M. Since this applies to all iterates of a given closed geodesic, any lift of a closed geodesic to M̃ must be minimizing. Thus M has no conjugate points as the closed geodesics are dense in S^{1}M since they are dense in S^{1}N. □

Proof of Theorem 2: By the above arguments and Theorem 1 it is sufficient to show that for x ∈ M, lim_{r→∞} \frac{V ol(B̃(x,r))}{β(n)r^{n}} ≤ 1.

We will let πÑ represent the projection from S^{1}Ñ to Ñ (define πM̃ similarly) . Let D = max_{x̃∈M̃} Diam(πÑ(F̃(Sx̃))), where Sx̃ is the sphere of unit vectors at x̃. Of course D exists since we could restrict x̃ to a compact fundamental domain.

Fix x̃ ∈ M̃ and V ∈ Sx̃. Let p̃ ∈ Ñ be πÑ F̃(V). We claim that F̃(S^{1}B(x̃, r)) ⊂ S^{1}B(p̃, r + 2D). Let W ∈ S^{1}B(x̃, r) and ỹ = πM̃(W). Let γ̃ be the geodesic from x̃ to ỹ, V_{1} = γ̃′(x̃) and W_{1} = γ̃′(ỹ). Since F commutes with the geodesic flow we see d(πÑ(F̃(V_{1})), πÑ(F̃(W_{1}))) = Length(γ̃) ≤ r. Hence

d(p, πÑ(F̃(W)) ≤ d(πÑ(F̃(V)), πÑ(F̃(V_{1}))) + d(πÑ(F̃(V_{1})), πÑ(F̃(W_{1})))+

+d(πÑ(F̃(W_{1})), πÑ(F̃(W)))+ ≤ D + r + D.

This proves the claim.

Let us now fix a compact fundamental domain K ⊂ M̃ and let d be its diameter. Since V ol(M)) = V ol(N), V ol(F̃(S^{1}K))) = V ol(S^{1}K). The same is true for any translate gK for g ∈ π^{1}(M) and hence for a union of such translates. Now let BK(x̃, r)) = ∪{gK| gK ∩ B(x̃, r) ≠ ∅}. Then BK(x̃, r)) ⊂ B(x̃, r + d) and hence:

α(n − 1)V ol(B(x̃, r)) = V ol(S^{1}B(x̃, r)) ≤ V ol(S^{1}KB(x̃, r)) =

= V ol(F̃(S^{1}KB(x̃, r))) ≤ V ol(S^{1}B(ỹ, r+d+D))) = α(n−1)β(n)(r+d+D)^{n}.

Hence

lim_{r→∞} \frac{V ol(B(x̃, r))}{β(n)r^{n}} ≤ lim_{r→∞} \frac{(r + d + D)^{n}}{r^{n}} = 1.

□

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