A rigidity theorem for manifolds without conjugate points

Christopher B. Croke^{∗}  Bruce Kleiner^{†}

August, 1996

Abstract

In this paper we show that some non-simply connected manifolds without conjugate points exhibit rigidity phenomena similar to that studied in [B-G-S] section I.5. This is a companion paper to [Cr-Kl1] that deals with the simply connected case. In particular we show that one cannot make a nontrivial, compactly supported, change to a complete flat metric without introducing conjugate points.

1 Introduction

In a companion paper [Cr-Kl1] we showed that if (X, g_{X}) is a complete simply connected manifold without conjugate points then the product manifold (M, g_{0}) = (X × R, g_{X} × dt^{2}) has the property that any metric g_{1} on M without conjugate points that agrees with g_{0} on the complement of a compact set K must in fact be isometric to g_{0}. We refer to this situation by the phrase ‘there are no nontrivial NCP perturbations of (M, g_{0}) supported on K’. The purpose of this paper is to extend this result to certain non-simply connected manifolds.

Our first theorem is that all noncompact flat manifolds have a similar rigidity property:

∗Supported by NSF grants #DMS-9001707,#DMS-9505175, and #DMS-9626232 .

†Supported by an NSF Postdoctoral Fellowship and NSF grants #DMS-9505175 and #DMS-9626911.

Theorem 1.1 Let (M, g_{0}) be a complete, noncompact flat manifold. Then any metric without conjugate points on M which agrees with g_{0} outside a compact set K ⊂ M is isometric to g_{0}.

The case where (M, g_{0}) is compact in contained in Burago and Ivanov’s recent proof of the Hopf conjecture that an n-torus without conjugate points must be flat [Bu-Iv] (proved after the first draft of this paper was written), but the general noncompact case does not follow from their method^{1}. The case where M is isometric to E^{n} was done for n = 2 in [Gr-Gu] and for all n in [Cr1] (also see [Cr2]).

Many of our results involve a growth condition on the fundamental group. So we need the following definition.

Definition 1.2 Let Γ be a finitely generated group. Then we will say that Γ grows more slowly than r^{n} if for any choice of finite generating set Σ ⊆ Γ, we have

lim_{r→∞} \frac{Card(\{g ∈ Γ||g|_{Σ} ≤ r\})}{r^{n}} → 0

where | · |_{Σ} : Γ ⟶ R denotes the word norm on Γ defined by the generating set Σ (see [Grm]).

We call a Riemannian manifold (M, g_{0}) an NCP manifold if it is complete and has no conjugate points. In particular the universal cover M̃ is diffeomorphic to R^{n}. For K ⊂ M, an NCP perturbation of M supported on K, or an NCP perturbation of (M, K, g_{0}), we mean a Riemannian metric g such that (M, g) is an NCP manifold and g = g_{0} on M − K. Such a perturbation is called trivial if g is isometric to g_{0}.

Our second theorem states a special case of what we can show in a more general setting.

Theorem 1.3 Assume that (X, g_{X}) is a noncompact manifold without conjugate points where π_{1}(X) has polynomial growth. If K ⊂ X ×R is a compact totally convex subset (i.e. any geodesic segment with endpoints in K lies in K), then there are no nontrivial NCP perturbations of (X × R, g_{X} × dt^{2}) supported on K.

1Their approach makes essential use of the finiteness of volume.

The condition of total convexity on K above is quite strong and there are many other cases where we can prove the nonexistence of NCP perturbations. However, every case that we can handle does involve a growth condition on the fundamental group.

To consider compact subsets K of a Riemannian manifold (M, g) that are not totally convex, we will need to consider the union, C_{1}(K, g), of the geodesic segments in M with endpoints in K. We can now state a general theorem.

Theorem 1.4 Let (X, g_{X}) be a noncompact n-1 dimensional NCP manifold such that π_{1}(X) grows slower than r^{n} and g an NCP perturbation of (X × R, K, g_{0}) where K is compact. Then if C_{1}(K) is bounded and if at least one of the following hold:

1. (X, g_{X}) has no focal points.

2. π_{1}(X) grows slower than r^{n−1}.

3. (X, g_{X}) = (Y × R, g_{Y} + dt^{2}).

4. K is totally convex.

then g is isometric to g_{0}.

One can see some of the added complications when π_{1}(X) is not trivial by considering compact X and the metric g_{1} = f^{2}(t)g_{X} × dt^{2} where f(t) = 1 for t ∉ [−1, 1] and f(t) > 1 for t ∈ (−1, 1). Although g_{1} will have conjugate points, it is not hard to see that geodesics that do not stay in the perturbed region have no conjugate points (in fact they are lines when lifted to the universal cover). The existence of geodesics that stay in the perturbed region (or more generally return to it for arbitrarily large parameter values) will be a new complication. In particular it will become important to show that their unit tangent vectors form a set of measure 0 in the unit tangent bundle.

Hence a crucial ingredient in the proof of theorems 1.3 and 1.4 is lemma 4.4. Below we state a version holding for compact smooth domains (i.e. compact sets with smooth boundary that are the closure of an open set), which may be of independent interest.

Proposition 1.5 Let (M^{n}, g) be a manifold without conjugate points, K ⊂ M a connected compact smooth domain, and Γ = i_{∗}(π_{1}(K)) ⊂ π_{1}(M) where i : K → M is the inclusion. Assume that Γ grows more slowly than r^{n}. If W ⊂ π^{−1}(K) ⊂ SM is a measurable subset of the unit sphere bundle of M which is invariant under the geodesic flow, then W has Liouville measure 0.

2 Notation and preliminaries

We will always assume that geodesics are parametrized by arclength unless otherwise stated. We will often consider two NCP metrics g_{0} and g on a given manifold M. S(M, g_{0}) and S(M, g) will represent the respective unit sphere bundles. If v is a g-unit vector of M (i.e. v ∈ S(M, g)) then γ_{v} will represent the g-geodesic with γ′(0) = v. We similarly define the g_{0}-geodesic, γ_{0v}, if v ∈ S(M, g_{0}). We will let g^{t} (resp. g_{0}^{t}) represent the geodesic flow for time t, i.e g^{t}(v) = γ′_{v}(t).

If K is a subset of M we let Ω_{g}^{+}(K) = {v ∈ S(X, g)| there are t_{i} with t_{i} → +∞ and γ_{v}(t_{i}) ∈ K}. We let Ω_{g}^{−}(K) = −Ω_{g}^{+}(K), Ω_{g}^{±}(K) = Ω_{g}^{+}(K) ∪ Ω_{g}^{−}(K), and Ω_{g}(K) = Ω_{g}^{+}(K) ∩ Ω_{g}^{−}(K). We note that Ω_{g}^{+}(K), and hence Ω_{g}^{−}(K), Ω_{g}^{±}(K), and Ω_{g}(K) are invariant under the geodesic flow on S(X, g). Further, if v ∈ Ω_{g}(K) then the geodesic γ_{v} lies in the set C_{1}(K) defined in the introduction. We can similarly define Ω_{g_{0}}^{+}(K), Ω_{g_{0}}^{−}(K), Ω_{g_{0}}^{±}(K), and Ω_{g_{0}}(K).

Let (M, g_{0}) be an NCP manifold, K ⊂ M a closed subset, and g an NCP perturbation of (M, K, g_{0}). If γ_{0} is a g_{0}-geodesic such that γ′_{0}(t) ∉ Ω_{g_{0}}^{±}(K), i.e. there is number N such that γ_{0}(t) ∉ K for |t| > N, then a g-geodesic γ is said to be a compact perturbation of γ_{0} if γ(t) = γ_{0}(t) for all |t| > N. The point here being that γ_{0} has a compact perturbation if the g-geodesic that coincides with γ_{0}(t) when t ∈ [−∞, −N] also coincides (including parameter) when t ∈ [N, ∞].

We will call the perturbation a lens equivalent perturbation if for v ∈ S(M − K, g_{0}) ∩ Ω^{+}(K, g_{0}) then v ∈ S(M − K, g) ∩ Ω^{+}(K, g) and for v ∈ S(M − K, g_{0}) − Ω^{±}(K, g_{0}) then γ_{0v} has a compact perturbation in g.

We will often be considering products M = X × R^{l} where X (which is not necessarily compact or simply connected) has an NCP metric g_{X} and M is given the product metric g_{0} = g_{X} × g_{E} where g_{E} is the euclidean metric.

An important special case for us is when l = 1. B(a) will represent the ball of radius a in R^{l}. We will use π_{X} and π_{R} to represent projection onto the factors (supressing the exponent l), and will call a vector v which is tangent to a euclidean factor (i.e. π_{X∗}(v) = 0) a vertical vector, and one tangent to an X factor (i.e. π_{R∗}(v) = 0) a horizontal vector. We will let S^{nh}(M) = {v ∈ S(M)|v is not horizontal}.

Let K be a subset of M such that π_{R}(K) is compact. We note that Ω^{±}(K, g_{0}) consists of only horizontal vectors. An NCP perturbation g of (M, K, g_{0}) will be called a weak lens equivalence if for each nonhorizontal v ∈ S(M, g_{0}), γ_{0v} has a compact perturbation in g. It will be called a local lens equivalence if there is an ϵ > 0 such that for every v ∈ S(M) such that |π_{X∗}(v)| < ϵ, γ_{0v} has a compact perturbation in g. A lens equivalent perturbation is of course a weak lens equivalence, which is in turn a local lens equivalence.

Lemma 2.1 Let (M, g_{0}) be an NCP manifold whose universal cover (M̃, g_{0}) is (X̃ × R^{l}, g_{0}), where (X̃, g_{X}) is a simply connected NCP manifold and g_{0} = g_{X} × g_{E}. Let K be a compact subset of M whose lift K̃ ⊂ M̃ is a subset of X̃ × B(a). Let g be an NCP perturbation of (M, K, g_{0}) such that the induced metric (also called g) on M̃ is a weak lens equivalent perturbation of (M̃, K̃, g_{0}) . If V ol(C_{1}(K, g_{0})) is finite and V ol(Ω_{g}(K)) = 0 (i.e.Ω_{g}(K) has Liouville measure 0) then V ol(K, g_{0}) = V ol(K, g).

Proof. Since V ol(C_{1}(K, g_{0})) is finite and K ⊂ C_{1}(K), it is sufficient to show that V ol(C_{1}(K, g_{0})) = V ol(C_{1}(K, g)). We define a smooth map F : S^{nh}(M̃, g_{0}) → S(M̃, g) − Ω_{g}(K̃) defined by F (v) = γ′(0), where γ is the ggeodesic which is the compact perturbation of the g_{0}-geodesic γ_{0v}. It is easy to see that F is the identity on S^{nh}(M̃ −C_{1}(K̃), g_{0}) and maps S^{nh}(C_{1}(K̃), g_{0}) injectively to S(C_{1}(K̃), g). It is also clear that F is a conjugacy of the geodesic flows, i.e. F ∘g_{0}^{t} = g^{t} ∘F . From this and the invariance of the Liouville measure under the geodesic flow one sees that F preserves the Liouville measures. Below we will show that W = S(C_{1}(K̃, g)) − F (S^{nh}(C_{1}(K̃, g_{0}))) has Liouville measure 0. This combined with the invariance of F under the deck group, and the fact that the horizontal vectors form a set of Liouville measure 0, tell us that V ol(C_{1}(K, g_{0})) = V ol(C_{1}(K, g)) and hence yields the result.

To see that W has Liouville measure 0, we note that W can be discribed as {v ∈ S(C_{1}(K̃), g)| π_{R}(γ_{v}(t)) ∈ B(a) for all times t} = Ω_{g}(K) ∪ ⋃_{i=0}^{∞} W_{i}^{+} ∪ ⋃_{i=0}^{∞} W_{i}^{−} where W_{i}^{+} = {v ∈ W |γ_{v}(t) ∉ K for all t ≥ i}, and W_{i}^{−} = −W_{i}^{+} . Thus we need only show that for every integer i > 0 the set W_{i}^{+} (and hence W_{i}^{−}) has g-Liouville measure 0. For v ∈ W_{i}^{+} and t > i, γ_{v}(t) is a g_{0}-geodesic such that π_{R}(γ_{v}(t)) ∈ B(a). Hence π_{R}(γ_{v}(t)) for t > i is in fact a single point. Thus for t > i, π_{R∗}(γ′_{v}(t)) = 0. Now by the invariance of the Liouville measure under the geodesic flow g^{t} we see V ol(W_{i}^{+}) = V ol(g^{i}(W_{i}^{+})). Now g^{i}(W_{i}^{+}) ⊂ {v ∈ S(M̃ −K̃, g_{0})|π_{R∗}(v) = 0} which has g_{0} and hence g-Liouville measure 0. The lemma follows.

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Proposition 2.2 Let (M, g_{0}) be an NCP manifold whose universal cover (M̃, g_{0}) = (X̃ × R^{l}, g_{0}), where (X̃, g_{X}) is a simply connected NCP manifold and g_{0} = g_{X} × g_{E}. Let K be a compact subset of M whose lift K̃ ⊂ M̃ is a subset of X̃ × B(a). Let g be an NCP perturbation of (M, K, g_{0}) such that the induced metric (also called g) on M̃ is a local lens equivalent perturbation of (M̃, K̃, g_{0}) . Then V ol_{g}(K) ≥ V ol_{g_{0}}(K) with equality if and only if g is isometric to g_{0}.

A proof of proposition 2.2 may be found in section 7.

Combining Proposition 2.2 with lemma 2.1 we get :

Corollary 2.3 Let (M, g_{0}) be an NCP manifold whose universal cover (M̃, g_{0}) is (X̃ × R^{l}, g_{0}), where (X̃, g_{X}) is a simply connected NCP manifold and g_{0} = g_{X} × g_{E}. Let K be a compact subset of M such that V ol(C_{1}(K, g_{0})) is finite and whose lift K̃ ⊂ M̃ is a subset of X̃ × B(a). Let g be an NCP perturbation of (M, K, g_{0}) such that the induced metric (also called g) on M̃ is a weak lens equivalent perturbation of (M̃, K̃, g_{0}). Then g is isometric to g_{0} if and only if V ol_{g}(Ω_{g}(K)) = 0.

Remark. The proof of Proposition 2.2 and Corollary 2.3 hold more generally than for compact subsets K. In particular, they hold if K has finite volume and π_{R}(K̃) is compact. However, Proposition 2.2 does not hold for arbitrary finite volume sets since there are closed (and open) finite volume subsets, K, of R^{2} such that every line intersects K in a noncompact set and hence any NCP perturbation is a lens equivalence. Further there are K ⊂ R^{2} of finite volume that admit NCP perturbations. On the other hand we have no counterexample to Corollary 2.3 when it is only assumed that C_{1}(K, g_{0}) has finite volume.

3 Busemann functions and lens equivalence

We will say v ∈ S(M, g) is a ray if the geodesic γ_{v} is a ray in the sense that d(γ(0), γ(t)) = t for all t > 0. We can then define the busemann function b_{v} : M → R by b_{v}(x) = lim_{t→∞} d(x, γ_{v}(t)) − t. We note that b_{v}(γ_{v}(t)) = −t for t > 0 (or all t if γ_{v} is a line, i.e. minimizes between any two points on it). In the case when M is a simply connected NCP manifold then b_{v} is defined for all v ∈ S(M, g). For v ∈ S(M, g_{0}) we use the notation b_{0v} for the busemann function. If v ∈ S(M, g) is a ray and x ∈ M we can define the horoball B_{v}(x) = {y ∈ M|b_{v}(y) ≤ b_{v}(x)} and the horosphere ∂B_{v}(x) = {y ∈ M|b_{v}(y) = b_{v}(x)}. Similarly, for v ∈ S(M, g_{0}) a ray we can define B_{0v}(x) and ∂B_{0v}(x).

The reader is referred to [E] for results on busemann functions on simply connected NCP manifolds. In particular busemann functions are C^{1,1}, and the vector field V (x) = ∇b_{v}(x) is a unit vector field whose integral curves are precisely those geodesics γ such that \frac{d}{dt}b_{v}(γ(t)) = 1.

If M is a simply connected NCP manifold, C ⊂ M is closed, and v ∈ S(M−C, g) is a ray then the shadow S(C, v) of C is the set {x ∈ M|γ_{∇b_{v}(x)}(t) ∈ C for some t ≤ 0}. Now if g is a perturbation of (M, C, g_{0}) then we also have that v ∈ S(M − C, g_{0}) and it is easy to check that for y ∈ M − S(C, v) we have b_{v}(y) = b_{0v}(y).

On simply connected NCP manifolds, if Nu is the outward normal at x to B_{v}(x) (i.e. ν = ∇b_{v}(x)), then, by an easy triangle inequality argument, B_{v}(x) ∩ B_{ν}(x) ⊂ ∂B_{v}(x) ∩ ∂B_{ν}(x). We will use this fact extensively. We will also use the fact that on product manifolds (X_{1} × X_{2}, g_{1} × g_{2}) we have b_{v}(x_{1}, x_{2}) = a_{1}b_{v_{1}}(x_{1}) + a_{2}b_{v_{2}}(x_{2}) where π_{i∗}(v) = a_{i}v_{i} and v_{i} ∈ S(X_{i}, g_{i}).

If (M̃, g_{0}) = (X̃ ×R^{l}, g_{X} ×g_{E}), where (X̃, g_{X}) is a simply connected NCP manifold , we will let S_{a}^{+}(M̃) = {v ∈ S(M̃)|π_{R}(γ_{0v}(t)) ∉ B(a) for any t ≥ 0}.

The point is that if g is any NCP perturbation of (M̃, X̃ × B(a), g_{0}) then for v ∈ S_{a}^{+}(M) we have that γ_{v}(t) = γ_{0v}(t) for all t > 0.

Lemma 3.1 Let (M, g_{0}) be an NCP manifold and K ⊂ M compact. Assume that (M̃, g_{0}) = (X̃ × R^{l}, g_{X} × g_{E}), where (X̃, g_{X}) is a simply connected NCP manifold, and further assume that K̃ ⊂ X̃ × B(a). Let g be an NCP perturbation of (M, K, g_{0}). Then there is a constant C(g_{0}, g) such that for every x, y ∈ M̃ we have |d(x, y) − d_{0}(x, y)| < C(g, g_{0}).

In particular for each nonhorizontal vector v ∈ S(M̃ − X̃ × B(a)) and for every x ∈ M̃ we have |b_{v}(x) − b_{0v}(x)| < C(g, g_{0}). Further for every v ∈ S_{a}^{+}(M) and every x ∈ M̃ −X̃×B(a), we have π_{R∗}(∇b_{v}(x)) = π_{R∗}(∇b_{0v}(x)) = π_{R∗}(−v). In the special case that (X̃, g_{X}) is also euclidean we have ∇b_{v}(x) is a parallel field on M̃ − X̃ × B(a).

Proof: Since π_{1}(M) acts cocompactly on K̃, we have c_{1}^{−2}g̃_{0} ≤ g̃ ≤ c_{1}^{2}g̃_{0} for a suitable constant c_{1}. Let x_{1}, x_{2} be such that π_{R}(x_{i}) ∈ ∂B(a). Let (γ_{0}(t), τ_{0}(t)) ∈ X̃ × R^{l} be the geodesic from x_{1} to x_{2}. The length of τ_{0} is less than 2a. Let γ̄(t) = (γ(t), π_{R}(x_{1})) and τ̄_{0}(t) = (π_{X}(x_{2}), τ_{0}(t)) be curves in M̃. γ̄ is both a g and a g_{0} geodesic (and hence mimimizing) of length L, while the g_{0} length of τ̄ is ≤ 2a, and the g length of τ̄ is ≤ 2ac_{1}. The path γ̄ ∪ τ̄ goes from x_{1} to x_{2} so we see by the triangle inequality that L − 2a ≤ d_{0}(x_{1}, x_{2}) ≤ L + 2a and L − 2ac_{1} ≤ d(x_{1}, x_{2}) ≤ L + 2ac_{1} hence |d(x_{1}, x_{2}) − d_{0}(x_{1}, x_{2})| ≤ 2a + 2ac_{1} = c_{2}.

Let x and y be in M such that that π_{R}(x) and π_{R}(y) lie outside B(a) − ∂B(a). Let γ : [0, d_{0}(x, y)] → M̃ be the g_{0}-geodesic from x to y. By the convexity of X̃ × B(a) either γ misses X̃ × B(a) or there is an interval [t_{1}, t_{2}] such that γ ∩ X × B(a) = γ([t_{1}, t_{2}]). Now

d_{0}(x, y) = d_{0}(x, γ(t_{1})) + d_{0}(γ(t_{1}), γ(t_{2})) + d_{0}(γ(t_{2}), y) ≥
≥ d(x, γ(t_{1})) + d(γ(t_{1}), γ(t_{2})) − c_{2} + d(γ(t_{2}), y) ≥ d(x, y) − c_{2}.

Similarly starting with the g-geodesic from x to y we conclude d(x, y) ≥ d_{0}(x, y) − c_{2}.

If π_{R}(x) ∈ B(a) − ∂B(a) there is a point x̄ such that π_{R}(x̄) ∈ ∂B(a), d_{0}(x, x̄) ≤ a, and d(x, x̄) ≤ ac_{1}. Similarly for y. Hence the lemma will hold for all x and y if we take C(g, g_{0}) = c_{2} + 2a + 2ac_{1}.

The busemann functions b_{v} and b_{0v} are limits as t goes to ∞ of functions b_{t}(x) = d(γ_{v}(t), x) − t and b_{0t} = d(γ_{v}(t), x) − t respectively. Since for t > 0, γ_{v}(t) = γ_{0v}(t), we see that |b_{t}(x) − b_{0t}(x)| < C(g, g_{0}) and hence |b_{v}(x) − b_{0v}(x)| < C(g, g_{0}).

We now compare the gradients of b_{v} and b_{0v} at a point x for v ∈ S_{a}^{+}(M̃). Let w = ∇b_{v}((x)) and assume that w ∈ S_{a}^{+}(M̃) (otherwise −w ∈ S_{a}^{+}(M̃) and the proof goes through). Since b_{v}(γ_{w}(s)) = s+b_{v}(x) we see that |b_{0v}(γ_{w}(s))− s| < 2C(g, g_{0}) and hence γ′_{w}(s) is asymptotic to ∇b_{0v} thus π_{R∗}(w) = π_{R∗}(−v). In the case (X̃, g_{X}) is euclidean we see that w is parallel to −v.

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Lemma 3.2 Let (X, g_{X}) be an NCP manifold, K ⊂ X ×B(a) ⊂ X ×R^{l} be a (not necessarily compact) subset , and g an NCP perturbation of (M, K, g_{0}). Assume further that for each v ∈ S_{a}^{+}(M̃) and for every x ∈ M̃ − X̃ × B(a) we have b_{v}(x) = b_{0v}(x). Then g is a weak lens equivalence.

Proof: It is sufficient to prove weak lens equivalence for the induced metrics on M̃. Let γ_{0} be a non-horizontal g_{0}-geodesic. π_{R}(γ_{0}(t)) = L(t) for some line L(t) ∈ R^{l}. We may assume that L(t) ∉ B(a) whenever t ∉ [−d, d]. In particular v = γ′_{0}(d) ∈ S_{a}^{+}(M̃). Let x = γ_{0}(−d), y = γ_{0}(d), γ the ggeodesic from x to y with γ(d) = y, γ(−d′) = x , and w = γ′(d).

We first show that w ∈ S_{a}^{+}(M̃). If π_{R}(γ([−d′, d])) ∩ B(a) = ∅ then γ(t) = γ_{0}(t) (and d = d′) for all t since then γ is a g_{0}-geodesic from x to y. If π_{R}(γ([−d′, d])) ∩ B(a) ≠ ∅ then, since B(a) is convex, π_{R}(γ(t)) ∉ B(a) for t ∉ [−d′, d] so w ∈ S_{a}^{+}(M̃).

We see 2d = d_{0}(x, y) = b_{0v}(x) = b_{v}(x) ≤ d(x, y), with equality holding if and only if x = γ_{v}(−2d). On the other hand, d(x, y) = b_{w}(x) = b_{0w}(x) ≤ d_{0}(x, y). Thus the equality holds and we see that γ(t) = γ_{v}(t) is a compact perturbation of γ_{0}(t) and thus that g is a weak lens equivalence.

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4 Liouville measure of Ω(K)

When considering NCP perturbations, g, of (X̃ × R^{l}, K̃, g_{X} × g_{E}) in the case where M is not simply connected, one of the major problems is to show that V ol(Ω_{g}(K)) = 0. When (X̃, g_{X}) has negative curvature we can show that Ω_{g}(K) contains no open sets, but this gives us no information about the Liouville measure. In general, this seems to be a very hard problem and it is the principal obstacle to extending theorem 1.3 to a much broader class of products X × R. In this section we show that V ol(Ω_{g}(K)) = 0 under auxiliary assumptions concerning volume or π_{1} growth.

Let C ⊂ M, and C̃ ⊂ M̃ be the preimage of C in universal cover M̃ of M. We will say that C has volume growth less than r^{n} if there is a function f(r) > 0 with lim_{r→∞} f(r) = 0 such that for every x̃ ∈ C̃, V ol(C̃ ∩B(x̃, r)) < f(r)r^{n}.

For (M, g) an NCP manifold, W ⊂ SM a subset of the unit tangent bundle, x ∈ M, and r > 0 we will let B^{W} (x, r) = {exp_{x}(tV)|V ∈ W ∩ T_{x}M and 0 ≤ t ≤ r} be the W − ball at x of radius r. We similarly define the W − sphere S^{W} (x, r). Let π(W) represent the projection of W to M, and B̃^{W} (x, r) represent a W − ball in the universal cover centered at a point that projects to x (hence only defined up to the deck group).

The proof of Theorem 1 of [Cr3] involves integration over the unit sphere bundle S(M) a of a compact NCP manifold M. Restricting these arguments to a subset W of S(M) yields:

Theorem 4.1 Let (M, g) be an NCP manifold and W ⊂ SM be a measurable subset such that

1. W = −W

2. W is geodesic flow invariant

3. π(W) has finite volume and bounded curvature

Then for each fixed r > 0

Ave(V ol(B̃^{W} (x, r))) ≥ \frac{V ol(W)}{V ol(π(W))} \frac{r^{n}}{n}

where the average is taken over all x ∈ π(W) with respect to the Riemannian measure and V ol(W) represents the Liouville measure of W.

The proof is given in section 7 and involves the volumes of W − spheres.

Remark: Both Theorems 1 and 3 of [Cr3], including the equality discussions, have analogues in this setting however they will not be needed in this paper.

We will apply this theorem when W = Ω(K). We let C_{2}(K) denote the projection of Ω(K) to M and note that C_{2}(K) ⊂ C_{1}(K).

Corollary 4.2 Let K be a compact subset of an n-dimensional complete NCP manifold, M. If C_{1}(K) (or C_{2}(K)) has finite volume, bounded curvature, and volume growth less than r^{n} then Ω(K) has Liouville measure 0.

Proof: Since C_{2}(K) ⊂ C_{1}(K), we need only prove the C_{2}(K) version. Let W = Ω(K), C = C_{2}(K), M̃ be the universal cover of M with covering projection p, C̃ = p^{−1}(C), and W̃ = (p_{∗})^{−1}(W). Now for x ∈ π(W), and x̃ ∈ M̃ such that p(x̃) = x we see that BW̃ (x̃, r) ⊂ C̃ ∩ B(x̃, r) and hence V ol(BW̃ (x̃, r) ≤ V ol(C̃ ∩ B(x̃, r)) < f(r)r^{n}, where f(r) comes from the definition of volume growth less than r^{n}. Hence Theorem 4.1 along with the assumption that C_{2}(K) has volume growth less than r^{n} implies that

V ol(W) = 0. □

If g is an NCP perturbation of (M, K, g_{0}), then C_{1}(K, g) = C_{1}(K, g_{0}), however it is not in general the case that C_{2}(K, g) = C_{2}(K, g_{0}) even though both are subsets of the set C_{1}(K, g) = C_{1}(K, g_{0}).

Corollary 4.3 Let g be an NCP perturbation of (M, K, g_{0}), where K a compact subset of an n-dimensional complete NCP manifold M. If C_{1}(K, g_{0}) has finite volume, bounded curvature, and volume growth less than r^{n} with respect to g_{0}, then Ω_{g}(K) has Liouville measure 0.

Proof: We first note that C_{1}(K, g) has finite volume, and bounded curvature since C_{1}(K, g_{0}) and C_{1}(K, g) are the same subset C and disagree metrically only on the compact set K. Thus by corollary 4.2 we need only show that C has volume growth less than r^{n}. Since g and g_{0} only disagree on a compact set there is a constant c_{0} such that for every vector v, \frac{1}{c_{0}} g_{0}(v, v) ≤ g(v, v) ≤ c_{0}g_{0}(v, v). Hence for any x̃ ∈ M̃, B_{g}(x, r) ⊂ B_{g_{0}}(x̃, c_{0}r), and for any subset S ⊂ M̃, V ol_{g}(S) ≤ c_{0}^{n}V ol_{g_{0}}(S). Hence V ol_{g}(C̃ ∩ B_{g}(x̃, r)) ≤ V ol_{g}(C̃ ∩ B_{g_{0}}(x̃, c_{0}r)) ≤ c_{0}^{n}V ol_{g_{0}}(C̃ ∩ B_{g_{0}}(x̃, c_{0}r)) ≤ c_{0}^{n}f_{0}(c_{0}r)(c_{0}r)^{n}, where f_{0} comes from the definition of C having growth rate less than r^{n} in g_{0}. So we can take f(r) = c_{0}^{2n}f_{0}(c_{0}r) and the corollary follows.

These results are related to π_{1} growth via the following lemma.

Lemma 4.4 Let M be an n-dimensional NCP manifold such that π_{1}(M) grows more slowly than r^{n}, and B a bounded subset. Then the set W = {v ∈ SM|γ_{v}(t) ∈ B for all t} has Liouville measure 0.

proof: Fix x_{0} ∈ B and let D be such that B ⊂ B(x_{0}, D). Let Σ be any generating set of π_{1}(M) which includes all elements of π_{1} which can be represented by loops at x_{0} of length less than 3D. We will let p : M̃ → M be the projection from the universal cover, and x̃_{0} ∈ p^{−1}(x_{0}). For r > 0 let N(r) = {x̃_{i} ∈ p^{−1}(x_{0})|x̃_{i} can be connected to x̃_{0} via a path in p^{−1}(B(x_{0}, D)) of length less than r+2D}. Let x ∈ B and x̃ ∈ p^{−1}(x)∩B(x̃_{0}, D). Then each ỹ ∈ B^{W} (x̃, r) can be connected to x̃ by a path of length less than r in p^{−1}(B) hence in p^{−1}B(x_{0}, D). Since there is some x̃_{i} ∈ p^{−1}(x_{0}) with ỹ connected to x̃_{i} in B(x̃_{i}, D) we see that x̃_{i} ∈ N(r). Thus

B^{W} (x̃, r) ⊂ ⋃_{x̃_{i}∈N(r)} B(x̃_{i}, D).

Hence V ol(B^{W} (x̃, r)) ≤ #N(r)V ol(B(x̃_{0}, D)).

By Theorem 4.1 , we need only show lim_{r→∞}#N(r)/r^{n} = 0 to complete the proof. To do this we will show that each x̃_{i} ∈ N(r) is of the form α(x̃_{0}) where α ∈ π_{1}(M) and |α|_{Σ} ≤ r/D + 2 and then the lemma follows from the growth rate of π_{1}(M).

Fix x̃_{i} ∈ N(r) and let τ(t), 0 ≤ t ≤ L ≤ r+2D, be a path with arc length parameter from x̃_{0} to x̃_{i} in p^{−1}(B(x_{0}, D)). For each integer 1 ≤ j ≤ [L/D] let σ_{j} be a path of length less than D from τ(Dj) to an x̃_{i_{j}} ∈ p^{−1}(x_{0}). Let x̃_{i_{0}} = x̃_{0} and x̃_{i_{[L/D]+1}} = x̃_{i}. Thus it is easy to see that for each 0 ≤ j ≤ [L/D] there is a path of length less than 3D from x_{i_{j}} to x_{i_{j+1}} and hence there is an α_{j} ∈ Σ such that x_{i_{j+1}} = α_{j}(x_{i_{j}}). So we see that for α = α_{[L/D]}...α_{1}α_{0}, x̃_{i} = α(x̃_{0}), and |α|_{Σ} ≤ r/D + 2. The lemma follows.

Applying the above lemma with B = C_{2}(K) we get:

Corollary 4.5 Let M be an n-dimensional NCP manifold such that π_{1}(M) grows more slowly than r^{n}, and K a compact subset. If C_{2}(K) is bounded then Ω(K) has Liouville measure 0.

Of course the same holds if C_{1}(K) is bounded, since then C_{2}(K) is also bounded.

proof of proposition 1.5: The proof is the same as that of lemma 4.4 except that we take D to be the intrinsic diameter of K (i.e. for every two points in K there is a path between them in K of length ≤ D), take Σ to be a generating set of Γ including all elements representable by loops lying in K of length less than 3D, and make sure that all the paths we choose lie in K̃ instead of p^{−1}(B(x_{0}, D)).

5 Flat case

In this section we will prove theorem1.1.

We begin by recalling some facts about flat manifolds (see [Cha]). If (M, g_{0}) is a complete flat manifold, then we may identify its universal cover M̃ with Euclidean space E^{n}, and there is a π_{1}(M)-invariant k-dimensional affine subspace A^{k} ⊆ E^{n} with A^{k}/π_{1}(M) compact. We will write E^{n} = A^{k} × R^{l}. Consequently, for every a > 0 the a-tubular neighborhood of A^{k}/π_{1}(M) ⊂ M is compact, and so we may assume without loss of generality that (M, g) is an NCP perturbation of (M, C, g_{0}), where C is such an a-tubular neighborhood. Passing to universal covers, we find that (M̃, g̃) is an NCP perturbation of (E^{n}, C̃, g̃_{0}) where C̃ = {x ∈ E^{n}|d(x, A^{k}) ≤ a} = A^{k} × B(a). In particular, since C̃ is convex we see that C_{1}(K) ⊂ C and hence Corollary 4.5 tells us that Ω_{g}(K) has Liouville measure 0.

By lemma 3.1 we see that if v ∈ S(M̃ − C̃), then ∇b_{v} is a constant vector field on E^{n} − C̃.

In the case codimension(A^{k}) ≥ 2, E^{n} − C̃ is connected, so ∇(b_{v} − b_{v}^{0}) = 0 forces b_{v} = b_{v}^{0}. By lemma 3.2 (M̃, g̃) is thus a weak lens equivalent perturbation of (M̃, C̃, g̃_{0}). Corollary 2.3 now gives the result.

When A^{k} has codimension 1, E^{n} − C̃ has two components, each isometric to a half-space. We know that a g-geodesic starting in one half-space which is not parallel to the hyperplanes bounding C̃ will intersect C̃ in a finite segment, and reemerge in the other half-space with the same direction (measured with respect to E^{n}).

After passing to a 2-fold cover if necessary, we may assume that M has two ends, and that (M, g_{0}) is isometric to a Riemannian product R × N where N is a compact flat manifold. Our set C is isometric to I × N for some interval I = [a, b]. If the perturbation of g_{0} is supported away from ∂C, then we may form a smooth closed Riemannian manifold X by gluing together the two (isometric) boundary components of (C, g) by the obvious map.

We now claim that X has no conjugate points. To see this, note that if v ∈ S(M̃, g̃) with π(v) ∈ ∂C̃, then γ_{v} carries a global solution of the Riccati equation which is zero on the portion of γ_{v} lying outside C̃. Hence any geodesic in X̃ which hits a copy of ∂C̃ will carry a global solution of the Riccati equation which is zero near each copy of ∂C̃ in X̃; in particular, such a geodesic has no conjugate points. On the other hand, if a geodesic in X̃ never hits a copy of ∂C̃, then it lies in a copy of (C̃, g̃), so it has no conjugate points.

Note that for each choice of v ∈ S(C, g)|_{∂C}, we get a vector field on X which looks like the gradient of a Busemann function. The union of the images of these (smoothly varying) vector fields contains everything but Ω(C) ∪ V ert, which has measure zero in SX. Hence the results of [Cr-Kl2] imply that X – and consequently (M̃, g̃) – is flat.

Alternatively, one could apply the recent result of Burago and Ivanov [Bu-Iv] that a torus without conjugate points must be flat. □

6 X × R

In this section we consider perturbations g of (X ×R, K, g_{0}) where K is compact and g_{0} = g_{X} ×dt^{2} for more general (X, g_{X}). When X is compact and not flat there is very little more we can say. So we will be most interested in the case where (X, g_{X}) is not simply connected, not flat, and not compact. The proof of such rigidity theorems breaks up into showing that V ol(Ω_{g}(K)) = 0 , and that g is a weak lens equivalence.

Proposition 6.1 Let (X, g_{X}) be a noncompact n-1 dimensional NCP manifold and g an NCP perturbation of (X ×R, K, g_{0}) where K is compact. Then g will be a weak lens equivalence if at least one of the following hold:

1. (X, g_{X}) has no focal points.

2. π_{1}(X) grows slower than r^{n−1}.

3. (X, g_{X}) = (Y × R, g_{Y} × dt^{2}).

4. K is totally convex and π_{1}(X) has polynomial growth.

Before proving the proposition we need a preliminary lemma.

Lemma 6.2 Let (X, g_{X}) be a simply connected manifold with no focal points, and v ∈ S(X). If f : X → R is a C^{1,1} function on X such that |∇f(x)| = 1 and |b_{v}(x) − f(x)| < C for all x ∈ X and some constant C, then f(x) = b(x) + c for some other constant c.

Proof. We first note that the integral curves γ(t) of ∇f are geodesics. To see this, let τ(s), 0 ≤ s ≤ d(γ(t_{1}), γ(t_{2})) be the minimizing geodesic from γ(t_{1}) to γ(t_{2}). Then integrating g_{X}(∇f(τ(s)), τ′(s)) we see that t_{2} − t_{1} = f(γ(t_{2})) − f(γ(t_{1})) ≤ d(γ(t_{1}), γ(t_{2})), and hence γ is a geodesic.

Fix x ∈ X and let w = −∇f(x) and u = −∇b_{v}(x). Let g(t) = b_{v}(γ_{w}(t))− f(γ_{w}(t)) = b_{v}(γ_{w}(t))+t−f(x). Now if w ≠ u we see that g′(0) > 0. Further, since there are no focal points, b_{v} is convex, and hence g′′(t) ≥ 0, and thus g is unbounded. This contradicts the assumption of the lemma and hence we conclude that ∇f = ∇b_{v} and the lemma follows.

□

Proof of proposition 6.1.

We will let M = X × R, K ⊂ X × [−a, a], and M̃ = X̃ × R represent the universal covering space. Let v ∈ S_{a}^{+}(M̃) and fix t_{0} ∉ [−a, a]. By lemma 3.1, the gradients of the restrictions of b_{v} and b_{0v} to X̃ × {t_{0}} have the same constant norm α. They thus, after multiplying by 1/α, induce functions b_{v}^{t_{0}} and b_{0v}^{t_{0}} on X̃ with gradients of norm 1 and of a bounded distance from each other. Further since g_{0} is the product metric b_{0v}^{t_{0}} is a busemann function of X̃.

Proof of 1. By lemma 6.2 we see that b_{v}^{t_{0}}(x) = b_{0v}^{t_{0}}(x) + c for some constant c. On the other hand, since X is noncompact and K is compact, there is an x_{0} ∈ X̃ such that (x_{0}, t_{0}) is not in the shadow S(K̃, v) hence b_{v}(x_{0}) = b_{0v}(x_{0}) and hence c = 0. Thus lemma 3.2 gives the result.

Proof of 2. For D > 0 and C ⊂ X̃ let T_{D}(C) represent the D tubular neighborhood of C. Let W^{D} = {w ∈ S(X)|γ_{w}(t) ∈ T_{D}(π_{X}(K)) for all t} where π_{X} is the projection from M to X. Assumption 2. along with lemma 4.4 tells us that W^{D} (and hence its lift W̃^{D} to S(X̃)) has Liouville measure 0 for all D > 0.

Let S^{t_{0}} = πX̃(S(K̃, v) ∩ X̃ × {t_{0}}), i.e. the shadow at t_{0}. Now if γ(s) is a gradient geodesic of b_{0v}^{t_{0}}(x) on X̃ and for some s_{0}, γ(s_{0}) ∉ S^{t_{0}}, then b_{0v}^{t_{0}}(γ(s_{0})) = b_{v}^{t_{0}}(γ(s_{0})) and ∇b_{0v}^{t_{0}}(γ(s_{0})) = ∇b_{v}^{t_{0}}(γ(s_{0})). Thus γ(s) is also a gradient geodesic of b_{v}^{t_{0}}, and we see b_{0v}^{t_{0}}(γ(s)) = b_{v}^{t_{0}}(γ(s)) for all s . On the other hand, if γ(s_{0}) ∈ S^{t_{0}} then for some t ∈ [−a, a] we have (γ(s_{0} + \frac{t_{0}+t}{\sqrt{1−α^{2}}} ), t) ∈ K̃ so γ(s_{0} + \frac{t_{0}}{\sqrt{1−α^{2}}} ) ∈ T_{D}πX̃(K̃) where D = \frac{a}{\sqrt{1−α^{2}}} . Thus if γ(s) ∈ S^{t_{0}} for all s, then γ(s) ∈ T_{D}πX̃(K̃) for all s, i.e. γ′(s) ∈ W̃^{D}.

Let γ be a g-geodesic with v = γ′(0) ∈ S_{a}^{+}(M̃) and π_{R}(γ(t)) < −a for all t < T. If \frac{π_{X∗}(γ′(T))}{|π_{X∗}(γ′(T))|} ∉ W̃^{D} the above tells us that b_{v}(γ(t)) = b_{0v}(γ(t)) for all t < T. Also for γ_{0} a g_{0}-geodesic with v = γ′_{0}(0) ∈ S_{a}^{+}(M̃) and π_{R}(γ_{0}(t)) < −a for all t < T. If \frac{π_{X∗}(γ′_{0}(T))}{|π_{X∗}(γ′_{0}(T))|} ∉ W̃^{D} the above tells us that b_{v}(γ_{0}(t)) = b_{0v}(γ_{0}(t)) for all t < T.

We finish the proof by modifying the proof of lemma 3.2. It is sufficient to prove weak lens equivalence for the induced metrics on M̃. Let γ_{0} be a nonhorizontal g_{0}-geodesic in M̃. We can assume (by changing the orientation and shifting the parameter of γ_{0} if necessary) that π_{R}(γ(t)) = bt for some 0 < b ≤ 1. Let d > a/b, x = γ_{0}(−d), y = γ_{0}(d), and v = γ′_{0}(d). Since W̃^{D} has measure 0 we can choose v_{i} ∈ S(M̃) converging to v such that \frac{π_{X∗}(γ′_{v_{i}}(−2d))}{|π_{X∗}(γ′_{v_{i}}(−2d))|} ∉

W̃^{D}. Let x_{i} = γ_{0v_{i}}(−2d) and y_{i} = γ_{0v_{i}}(0). We see 2d = d_{0}(x_{i}, y_{i}) = b_{0v_{i}}(x_{i}) = b_{v_{i}}(x_{i}) ≤ d(x_{i}, y_{i}), and hence taking limits d_{0}(x, y) ≤ d(x, y). On the other hand, if γ is the g-geodesic from x to y (with γ(0) = y), let w = γ′(0) and as above choose w_{i} converging to w such that \frac{π_{X∗}(γ′_{w_{i}}(−2d))}{|π_{X∗}(γ′_{w_{i}}(−2d))|} ∉ W̃^{D}. Let x′_{i} = γ_{0w_{i}}(−2d) and y′_{i} = γ_{0w_{i}}(0). Then d(x′_{i}, y′_{i}) = b_{w_{i}}(x′_{i}) = b_{0w_{i}}(x′_{i}) ≤ d_{0}(x′_{i}, y′_{i}), and taking limits d(x, y) ≤ d_{0}(x, y). Thus we get for every large d 2d = d_{0}(γ_{0}(d), γ_{0}(−d)) = d(γ_{0}(d), γ_{0}(−d)) and we see that γ_{0} has a compact perturbation and thus that g is a weak lens equivalence.

Proof of 3. As usual we will let M = X × R, K ⊂ Y × B(a) ⊂ X × [−a, a], and M̃ = X̃ × R = Ỹ × R^{2} represent the universal covering space. We will let C = Y × B(a) so K ⊂ C.

We will prove the result using lemma 3.2. To that end let v ∈ S_{a}^{+}(M̃). We first study the product metric g_{0}. For each y ∈ Y let P_{y} = {y}×R^{2} be

the totally geodesic two plane through (y, 0, 0). Since g_{0} = g_{Y} × dz^{2} where dz^{2} is the flat metric on R^{2}, we see that for every y ∈ Y, b_{0v}|_{P_{y}} is linear, and the level set, LS_{0v}(r, s) = ∂B_{0v}(r, s) ∩ P_{y}, of b_{0v}|_{P_{y}} through (r, s) is the straight line perpendicular to the projection of v to R^{2}. It is also easy to see that the shadow S(C, v) ∩ P_{y} is the a-tubular neighborhood of the ray τ through the origin parallel and opposite to the projection of v onto R^{2}.

For ease of discussion we will put coordinates (r, s) on the plane P_{y} such that τ is the positive r-axis. So b_{0v}((r, s)) = L(r) for some linear function L : R → R with L′(r) > 0.

By applying lemma 3.2, we need only show that b_{0v} and b_{v} agree on P_{y} − B(a). Since b_{0v} and b_{v} agree on M −S(C, v), we have for (r, s) ∈ P_{y} −S(C, v), b_{v}(r, s) = L(r). For (r_{0}, s_{0}) ∈ P_{y} with r_{0} > a let ψ = ∇b_{v}(r_{0}, s_{0}) ∈ T_{(r_{0},s_{0})}M be the outward normal to the level set of b_{v}. Then ∂B_{ψ}(r_{0}, s_{0}) intersects P_{y} in a line, LS_{ψ}(r_{0}, s_{0}), through (r_{0}, s_{0}) (since it coincides with ∂B_{0ψ}(r_{0}, s_{0}) ∩ P_{y}) which must lie “to the right” of the level set LS_{v}(r_{0}, s_{0}) of b_{v} (i.e. if (r_{1}, s_{1}) ∈ LS_{ψ}(r_{0}, s_{0}) then b_{v}(r_{1}, s_{1}) ≥ b_{v}(r_{0}, s_{0}) ).

For r_{0} > a and |s_{0}| > a we have LS_{v}(r_{0}, s_{0})∩{(r, s)||s| > a} = {(r_{0}, s)||s| > a} and hence LS_{ψ}(r_{0}, s_{0}) = {(r_{0}, s)}. Hence it is easy to see that for any (r_{1}, s_{1}) ∈ LS_{v}(r_{0}, s_{0}) we have r_{1} ≤ r_{0}. On the other hand if r_{0} is chosen large enough such that LS_{v}(r_{0}, 2a) ⊂ {(r, s)|r > a} and (r_{1}, s_{1}) ∈ LS_{v}(r_{0}, 2a), then the above argument forces the set {(r_{0}, s)||s| > a} to lie to the left of the line LS_{ψ}(r_{1}, s_{1}) (ψ = ∇b_{v}(r_{1}, s_{1})) through (r_{1}, s_{1}) and hence r_{1} ≥ r_{0}. Thus for r_{0} sufficiently large the lines r = r_{0} are level sets of b_{v}|_{P_{y}} and hence b_{v} and b_{0v} agree on all of P_{y} − B(a). The lemma follows.

Proof of 4. First note that π_{1}(K) ≡ π_{1}(X × R) ≡ π_{1}(X) (since X is connected) since loops in X based at some point p ∈ K are homotopic to geodesic loops based at p, which lie entirely in K since K is totally convex. π_{1}(K) has polynomial growth, so it is a virtually nilpotent group by [Grm], and hence it’s virtually abelian by [Cr-S]. The homological dimension of π_{1}(K) is < n − 1 since π_{1}(K) ≡ π_{1}(X) and X is noncompact; hence π_{1}(K) is virtually Z^{k} with k < n − 1, and we are reduced to case 2 of the proposition.

□

Proof of theorem 1.3. By the proof of part 4 of proposition 6.1, we know that π_{1}(K) has growth slower than dim(X). Hence we may apply corollary 4.4, proposition 6.1, and finally corollary 2.3 to conclude that g is isometric to g_{0}. □

We can now combine the results of sections 4 and 6 with Corollary 2.3 to prove Theorem 1.4.

7 Jacobi tensors and proofs

We will be considering Lagrange tensors along a geodesic γ(t).(For a detailed discussion see [Es-OS] and [E].) 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 ∈ SM there are two such Lagrange tensors along γ_{v} (the geodesic with γ′(0) = v) of interest to us. J_{v} will be the Lagrange tensor with J_{v}(0) = 0 and J′_{v}(0) = Id. A_{v} will be the Lagrange tensor with A_{v}(0) = Id and A′_{v}(0) = 0. J_{v} and A_{v} are Lagangian since Jacobi tensors (i.e. tensors satisfying (1) above) which satisfy (2) (resp. (3)) for some t must satisfy (2) (resp. (3)) for all t. If we fix an orthonormal basis for v^{⊥}, then with respect to parallel bases along γ_{v} we can think of J_{v}(t) and A_{v}(t) as matrices where the columns represent jacobi fields with respect to these bases.

It is not hard to check, since A_{v} is Lagrange, (see [Es-OS]) that if A_{v}(t) is nonsingular for all t ∈ [0, s] we have:

J_{v}(s) = A_{v}(s) ∫_{0}^{s} A_{v}^{−1}(t)A_{v}^{−1∗}(t)dt.  (1)

When γ_{v} is a geodesic in an NCP manifolds we define a tensor U(v) = lim_{t→∞} J′_{g^{t}v}(−t)J_{g^{t}v}^{−1}(−t). Note that J′_{g^{t}v}(−t)J_{g^{t}v}^{−1}(−t) is a tensor at γ_{v}(0) for all t that represents the second fundamental form of the sphere of radius t centered at γ_{v}(−t). See [Es-OS] for a proof that U(v) exists. We will let u(v) = tr(U(v)). Then it is easy to see, by looking at u(v) as the limit of mean curvatures of spheres, that u(v) + u(−v) ≥ 0.

If M is an NCP manifold and W ⊂ SM is a measurable set, invariant under the geodesic flow, such that π(W) has finite volume and bounded sectional curvatures then (as was shown in [Es-OS] and [G]) u(v) is bounded and varies measurably in v for v ∈ W. Hence we can define h_{μ}^{W} = ∫_{W} u(v)\overline{dv} where \overline{dv} is the normalized Liouville measure i.e. \overline{dv} = \frac{1}{V ol(W)}dv. We note in particular that when −W = W we have h_{μ}^{W} ≥ 0. We remark in passing that when W = SM then h_{μ}^{W} is the measure entropy of the geodesic flow (see [Fr-Ma]).

The following lemma is lemma 1 of [Cr3].

Lemma 7.1 Let J_{v}, and u be as above. Then

det(J_{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}.

Proof of Theorem 4.1:

Ave(V ol(S̃^{W} (x, l))) = \frac{1}{V ol(π(W))} ∫_{W} det(J_{v}(l))dv.

≥ \frac{V ol(W)}{V ol(π(W))}[∫_{W} ∫_{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 7.1, 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^{W} (x, l)))V ol(π(W))}{V ol(W)}]^{\frac{1}{n−1}} ≥

= ∫_{0}^{l} ∫_{W} 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\{∫_{W} \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} and W are invariant under the geodesic flow g^{z} yields:

∫_{W} ∫_{0}^{t} u(g^{z}v)dz\overline{dv} = ∫_{0}^{t} ∫_{W} u(g^{z}v)\overline{dv}dz = ∫_{0}^{t} ∫_{W} u(v)\overline{dv}dz = th_{μ}^{W}

Applying this to the above inequality yields:

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

where the last inequality follows (after performing the integral) since h_{μ}^{W} ≥ 0. Thus we see that Ave(V ol(S̃(x, l))) ≥ \frac{V ol(W)}{V ol(π(W))}l^{n−1}. Integrating both sides as l runs from 0 to r yields the theorem.

□

The next lemma is the primary ingredient in the proof of Proposition 2.2. We will let B^{n−1}(ϵ) represent the ϵ-ball in E^{n−1}, dx represent the standard volume form, and π : B^{n−1}(ϵ) × R → B^{n−1}(ϵ) represent the projection.

Lemma 7.2 Let (M, g_{0}) be an NCP manifold and C ⊂ M compact. Let F_{0} : B^{n−1}(ϵ) × R → M be a smooth injective map such that

1. F_{0}^{∗}(g_{0}) is a product metric.

2. For each fixed x, γ_{0x}(t) = F_{0}(x, t) is a geodesic parametrized by arclength.

3. There is a T such that F_{0}^{−1}(C) ⊂ B^{n−1}(ϵ) × [−T, T].

Let g be an NCP perturbation of g_{0} supported on C, which is a local lens equivalence in the sense that all “nearly vertical” g_{0}-geodesics (i.e. those with initial tangents suficiently close to γ′_{0x}(t) for some x and t), have ggeodesics that are compact perturbations. Let F : B^{n−1}(ϵ) × R → M be the corresponding map to (M, g) using the compact perturbations γ_{x}(t). Let μ_{0} (resp. μ) be the pullback under F_{0} (resp. F) of the g_{0}-Riemannian measure (resp. g-Riemannian measure). Define f_{0} : B^{n−1}(ϵ) → R^{+} (resp. f) by f_{0}(x)dx = π_{∗}(μ_{0}|_{F_{0}^{−1}(C)}) (resp. f).

Then f_{0}(x) ≥ f(x) for all x with equality holding if and only if F^{∗}(g) is also a product metric (hence isometric to F_{0}^{∗}(g_{0})).

proof Since g and g_{0} agree outside C we can assume that F^{−1}(C) = F_{0}^{−1}(C) = B^{n−1}(ϵ)×[−T, T]. For x ∈ B^{n−1}(ϵ) we will use the notation A(x, t) (resp. J(x, t)) for the matrix representation of A_{v}(t + T) (resp., J_{v}(t + T)), where v = γ′_{0x}(−T)) = γ′_{x}(−T), along the g-geodesic γ_{x}(t) with respect to a parallel orthonormal basis . We similarly define A_{0}(x, t) and J_{0}(x, t) along the g_{0}-geodesic γ_{0v} we choose the orthonormal basis to agree at (x, −T). Since g_{0} is the product metric A_{0}(x, t) = Id and J_{0}(x, t) = (t + T)Id.

If for some t_{0}, A(x, t_{0}) is singular (say A(x, t_{0})r⃗ = 0 for some r⃗ ∈ R^{n−1}) then there would be a jacobi field, j (i.e. j(t) = A(x, t)r⃗), along γ_{x} such that j(t_{0}) = 0 and j(t) is parallel and hence bounded for t ≤ −T. But since the curvatures are bounded along γ_{x}, Theorem 1 of [Es-OS] says that there are no such jacobi fields . Hence A(x, t) is always nonsingular.

With respect to our chosen orthonormal bases, A(x, t) represents dF (x, t) restricted to vectors perpendicular to γ′_{x}(t) . So we see that F is an immersion and

f_{0}(x) = ∫_{−T}^{T} Det(A_{0}(x, t))dtdx = 2T and f(x) = ∫_{−T}^{T} Det(A(x, t))dtdx.

By the local lens equivalence, nearly vertical g_{0}-geodesics have g-geodesics as compact perturbations . Hence, by taking a variations of such geodesics, we see that jacobi fields j_{0}(t) and j(t) along γ_{x} and γ_{0x} respectively, with j_{0}(−T) = j(−T) and j′_{0}(−T) = j′(T) have j_{0}(t) = j(t) for all t ∉ [−T, T] (note γ_{x}(t) = γ_{0x}(t) for t ∉ [−T, T]). Hence we see that jacobi tensors along γ_{x} and γ_{0x} respectively with the same initial conditions at −T must agree for t ∉ [−T, T]. (Note that this does not necessarily mean that they have the same matrix representation since parallel bases along γ_{x} and γ_{0x} that are the same for t ≤ −T may differ by a fixed orthogonal transformation for t ≥ 2r. However the determinants will be the same.) Thus we see from equation 1

Det[\frac{1}{2T} ∫_{−T}^{T} A^{−1}(x, t)A^{−1∗}(x, t)dt] = Det[\frac{1}{2T} A^{−1}(x, T)J(x, T)] =

= Det[\frac{1}{2T} A_{0}^{−1}(x, T)J_{0}(x, T)] = 1.

Using the above equation and the convexity of Det^{\frac{−1}{2}} on the space of self adjoint positive definite matrices [B] we see that:

f(x) = 2T \frac{1}{2T} ∫_{−T}^{T} Det(A(x, t)dt ≥

≥ 2T Det^{\frac{−1}{2}} [\frac{1}{2T} ∫_{−T}^{T} A^{−1}(x, t)A^{−1∗}(x, t)dt] = 2T = f_{0}(x).

Equality holds above if and only if A^{−1}(x, t)A^{−1∗}(x, t) is constant in t hence is the identity. But this is the case if and only if A(x, t) is orthogonal, hence A′(x, t)A^{−1}(x, t) is skew symmetric. On the other hand we know that A′(x, t)A^{−1}(x, t) is symmetric, thus A′(x, t) = 0 and A(x, t) = Id. Hence equality holds if and only if A(x, t) = Id. If equality holds for all x we see that , F^{∗}(g) is isometric to F^{∗}(g_{0}). □

Proof of Proposition 2.2: If M is itself a product with a line then lemma 7.2 gives the result rather easily. In our more general setting we need to be somewhat more careful. We will assume that l is the maximum such that M̃ = X̃ × R^{l} and K̃ ⊂ X̃ × B(a) for some choice of X̃ (if M̃ is euclidean for example there are a lot of choices for splittings). This maximality implies that the deck group, π_{1}(M), acts so as to preserve the splitting.

We let H̃_{0} be the collection of vertical v ∈ S(M̃, g_{0}). H̃_{0} carries the natural n-dimensional foliation F̃_{0} with leaves consisting of parallel vectors. We define H̃ to be the collection of v ∈ S(M̃, g) such that there is a t ∈ R with γ_{v}(t) ∈ M̃ − X̃ × B(a) and γ′_{v}(t) ∈ H̃_{0}. H̃ ⊂ S(M̃, g) is invariant under the geodesic flow of g, and since the amount of time that γ_{v} spends in X̃ × B(a) is uniformly bounded for v ∈ H̃ (see lemma 3.1), it is clear that H̃ is a closed submanifold of S(M̃, g). We may define a foliation F̃ of H̃ by declaring the leaves to be the sets of v ∈ H̃ such that γ′_{v}(t) ∈ S(M̃ −X̃×B(a)) is parallel to a fixed w ∈ S(M̃ − X̃ × B(a), g_{0}). H̃ is invariant under the natural action of π_{1}(M) on S(M̃, g), and we set H = H̃/π_{1}(M) ⊂ S(M, g).

It is obvious that π_{0} maps each leaf of F̃_{0} diffeomorphically onto M̃, where π_{0} : S(M̃, g_{o}) → M̃ is the projection. It is also true that π is a diffeomorphism on each leaf of F̃. This follows since for each leaf of F̃ there is a natural diffeomorphism, d , to the corresponding leaf of F̃_{0}, and the composition π ∘ d^{−1} ∘ π_{0}^{−1} is a proper (by lemma 3.1) immersion (by the argument that A(x, t) is nonsingular in the proof of lemma 7.2) and hence is a diffeomorphism.

We now put a measure on each leaf of F̃_{0} (resp. F̃) by pulling back the Riemannian volume form from (M̃, g_{0}) (resp. (M̃, g)), and define a transverse measure on F̃_{0} (resp. F̃) by identifying the leaf space of F̃_{0} (resp. F̃) with the set of directions S^{l−1} in R^{l}, and using normalized Lebesgue measure on S^{l−1}. Combining these two measures, we get a measure μ̃_{0} on H̃_{0} (resp. μ̃ on H̃) such that π_{0∗}μ̃_{0} (resp. π_{∗}μ̃) is the Riemannian volume on (M̃, g_{0}) (resp. (M̃, g̃)). if we let G̃_{0} (resp. G̃) be the orbit space of H̃_{0} (resp. H̃) under the geodesic flow, then G̃_{0} and G̃ are naturally diffeomorphic; moreover using the local product structures of the measures and applying lemma 7.2 locally on the leaves of F̃_{0} and F̃ we see that the pushforward of μ|_{π^{−1}(X̃×B(a))} to G̃ ≃ G̃_{0} under the quotient map H̃ ⟶ G̃ is ≥ the pushforward of μ_{0}|_{π^{−1}(X̃×B(a))} to G̃_{0} under the projection H̃_{0} ⟶ G̃_{0}, with equality only if the leaves of F̃ correspond to parallel vector fields on (M̃, g).

If we denote by μ_{0} (resp. μ) the corresponding measures in H_{0} (resp. H), then π_{∗}μ_{0} (resp. π_{∗}μ) will be the Riemannian measure for (M, g_{0}) (resp. (M, g)). In particular, we have ∫_{H∩π^{−1}(K)} μ = V ol_{g}(K) and ∫_{H_{0}∩π_{0}^{−1}(K)} μ = V ol_{g_{0}}(K). If we now let G_{0} (resp. G) be the orbit space of H_{0} (resp. H) under the geodesic flow, then again G_{0} and G are naturally diffeomorphic; and the pushforward of μ|_{π^{−1}(K)} to G ≃ G_{0} under the quotient map H ⟶ G

is ≥ the pushforward of μ_{0}|_{π^{−1}(K)} to G_{0} under the projection H_{0} ⟶ G_{0}, with equality only if the leaves of F̃ correspond to parallel vector fields on (M̃, g̃).

Thus we see that V ol_{g}(K) ≥ V ol_{g_{0}}(K) with equality holding only if g is isometric to g_{0}.

□

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