A rigidity theorem for simply connected manifolds without conjugate points

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

January, 1994

Abstract

In this paper we show that manifolds without conjugate points exhibit rigidity phenomena similar to that studied in [B-G-S] section I.5. The main theorem is that if X is a complete, simply connected Riemannian manifold without conjugate points, and M = X × R is given the Riemannian product metric g, then any metric without conjugate points on M which agrees with g outside a compact set is isometric to g.

1 Introduction

If (M, g_{0}) is a complete Riemannian manifold satisfying some property P, then we will say that (M, g_{0}) is rigid under compactly supported perturbations (respecting the property P) if any metric g_{1} on M which satisfies property P and coincides with g_{0} outside a compact set K ⊂ M is isometric to g_{0}. Euclidean space was shown to be rigid under compactly supported perturbations with nonpositive (or nonnegative) sectional curvature in [Gr-Wu]. In [B-G-S, pp.69](see also [S-Z]), certain classes of simply connected manifolds with nonpositive curvature were shown to be rigid under compactly supported perturbation; for instance, if (M, g) is a symmetric space of noncompact type of rank ≥ 2, or if (M, g) is a Riemannian product of manifolds with nonpositive sectional curvature, then (M, g) is rigid under compactly supported perturbations with nonpositive curvature.

∗Supported by NSF grant #DMS90-01707 .

†Supported by an NSF Postdoctoral Fellowship.

Manifolds without conjugate points enjoy many of the global properties of manifolds of nonpositive curvature. [Gr-Gu, Cr2, Cr1] established that metrics without conjugate points on Euclidean space are rigid under compactly supported perturbations having no conjugate points. The purpose of this paper is to extend this last rigidity statement to a much larger class of manifolds without conjugate points:

Theorem 1.1 Let X be a 1-connected Riemannian manifold without conjugate points, and let g_{0} be the Riemannian product metric on X × R. Then g_{0} is rigid under compactly supported deformations having no conjugate points.

Given the similarities between manifolds without conjugate points and manifolds with nonpositive curvature, theorem 1.1 leads us to:

Conjecture 1.2 If (M, g_{0}) is a complete Riemannian manifold without conjugate points, must any metric without conjugate points g on M agreeing with g_{0} outside a compact set satisfy rank(g_{1}) = rank(g_{0}) (Here the rank of (M, g_{i}) is the largest integer k such that every geodesic γ : R ⟶ M in (M, g_{i}) factors through a totally geodesic isometric immersion i : E^{k} ⟶ M).

When M is simply connected, the methods of this paper reduce conjecture 1.2 to the problem of “lens rigidity” for manifolds with rank ≥ 2 (see remark 2.2). We should point out that not all such theorems about nonpositive curvature extend to the no conjugate points setting. In particular, the example given in [Kl] shows that the main theorem in [A-S] does not extend to manifolds without conjugate points, while the example in [Bu] shows that the flat strip theorem does not hold in manifolds without conjugate points. In fact, almost none of the techniques used to prove rigidity results for manifolds of nonpositive curvature are applicable in our case, and hence the proofs in this paper are completely different.

Outline of proof. The proof of theorem 1.1 breaks up into three parts. The first part (proposition 2.1) uses the properties of Busemann functions to show that the two metrics are “lens equivalent”, i.e. if γ^{0} is a g_{0}-geodesic such that γ^{0}(t) ∉ K for |t| > T then there is a g_{1}-geodesic γ^{1} such that γ^{0}(t) = γ^{1}(t) for all |t| > T. The second part uses this lens equivalence, along with the invariance of the Liouville measures under the geodesic flows, to conclude (corollary 2.3) that for any domain D with K ⊂ D we have V ol_{g_{0}}(D) = V ol_{g_{1}}(D). In the third part we use lens equivalence along with an extremal property of the Jacobi equation in product manifolds to show that for such D we have V ol_{g_{0}}(D) ≥ V ol_{g_{1}}(D) with equality holding if and only if the metrics are isometric.

In a companion paper [Cr-Kl], we generalize theorem 1.1 by allowing certain manifolds X with nontrivial fundamental group; in particular we show that any flat manifold is rigid under compactly supported perturbations having no conjugate points. Although the proof in these cases involves analogues of the three steps mentioned in the outline above, a number of new ideas are required, leading to considerably longer proofs. The main new complication is that some geodesics remain in a compact region K forever; in particular, it becomes important to control the measure of the set of such geodesics.

2

2.1 Notation

We will always assume that geodesics are parametrized by arclength unless otherwise stated. We will be considering two metrics g_{i}, i = 0, 1 on a given manifold M. S(M, g_{i}) will denote the unit sphere bundle of (M, g_{i}), S_{p}(M, g_{i}) the fiber of S(M, g_{i}) at p ∈ M. If v ∈ S(M, g_{i}), then γ_{v}^{i} will denote the g_{i}-geodesic with initial velocity v, exp^{i} : T M ⟶ M is the exponential map for g_{i}, Φ_{t}^{i} : S(M, g_{i}) ⟶ S(M, g_{i}) is the geodesic flow after time t. d^{i} : M × M ⟶ R is the distance function associated with g_{i}, and if x ∈ M then d_{x}^{i} : M ⟶ R is the distance from x.

When π_{1}(M) = {e} and g_{i} has no conjugate points, then γ_{v}^{i}|_{[0,∞]} is a ray for every v ∈ S(M, g_{i}). The Busemann function b_{v}^{i} associated with v ∈ S(M, g_{i}) is given by b_{v}^{i}(x) = lim_{t→∞}(d^{i}(x, γ_{v}^{i}(t))−t). The convergence in this limit is in the C^{1} topology while the functions b_{v}^{i} are C^{1,1} (see [E]). In particular we see that (∇b_{v}^{0})(p) = lim_{t→∞}(∇d_{γ_{v}^{i}(t)}^{i})(x). We define horoballs by B_{v}^{i}(x) = (b_{v}^{i})^{−1}((−∞, x]). The g_{i}-horoball (respectively g_{i}-horosphere) generated by v ∈ S(M, g_{i}) is B_{v}^{i} ≡ B_{v}^{i}(0), (respectively ∂B_{v}^{i} = (b_{v}^{i})^{−1}(0)). If B is a g_{i}-horoball and w is an outward normal at some p ∈ ∂B, then by the triangle inequality B_{w}^{i} is disjoint from Interior(B). See [E] for a discussion of horospheres in manifolds without conjugate points.

If K ⊂ M is a closed set, we let Chord^{i}(K) be the union of the g_{i}-geodesic segments with endpoints in K; if g_{1}|_{M−K} = g_{0}|_{M−K} then Chord^{0}(K) = Chord^{1}(K) ≡ Chord(K).

2.2 Lens rigidity

The first step in the proof of theorem 1.1 is to show that if a manifold without conjugate points, M, has Rank(M) = k ≥ 2, then the geodesic flow of M is “rigid” under compactly supported perturbations without conjugate points:

Proposition 2.1 Let (M, g_{0}) be a 1-connected manifold without conjugate points, and assume that rank(M, g_{0}) = k ≥ 2 in the sense that every v ∈ S(M, g_{0}) is tangent to a k-flat (i.e a totally geodesic copy of E^{k}). Let g_{1} be a compactly supported perturbation of g_{0} having no conjugate points, and let K ⊂ M be a compact subset such that g_{1}|_{M−K} = g_{0}|_{M−K}. If x, y ∈ M − Chord(K), then

1. d^{1}(x, y) = d^{0}(x, y),

2. (exp_{x}^{1})^{−1}(y) = (exp_{x}^{0})^{−1}(y).

Proof. Our first goal is to show that b_{v}^{1}(y) = b_{v}^{0}(y) for all y ∉ Chord(K) provided γ_{v}^{i}([0, ∞)) ∩ K = ∅.

If v ∈ S(M, g_{i}), then we define Shadow^{i}(K, v) to be the set of x ∈ M such that the g_{i}-gradient line of −b_{v}^{i} starting at x hits K; alternately, y ∉ Shadow^{i}(K, v) if for all sufficiently large t the g_{i}-geodesic segment from y to γ_{v}^{i}(t) is disjoint from K. If γ_{v}^{i}([0, ∞)) ∩ K = ∅ then Shadow^{0}(K, v) = Shadow^{1}(K, v) ≡ Shadow(K, v), and for all y ∉ Shadow(K, v), b^{0}(y) = b^{1}(y).

By definition, a (k − 1)-flat F in (M, g_{0}) is a v-admissible (k − 1)-flat if v is in the unit normal bundle of F, and there is a k-flat F̄ ⊂ M containing F which is tangent to v.

Pick v ∈ S(M, g_{0}), and let F̄ be a k-flat tangent to v. Then γ_{v}^{0} ⊂ F̄, and b_{v}^{0}|F̄ is a Busemann function on F̄ ≃ E^{k}. Hence F = (b_{v}^{0})^{−1}(0) ∩ F̄ is a hyperplane in F̄, and a v-admissible (k −1)-flat in (M, g_{0}). If w = −(∇b_{v}^{0})(p) for some p ∈ F̄, then b_{w}^{0} = b_{v}^{0} + c. This follows since clearly b_{w}^{0} = b_{v}^{0} + c on F̄, hence on the geodesics γ_{v} and γ_{w}, and an easy triangle inequality argument (see for example the proof of proposition 2.5 of [C-S]) then shows b_{w}^{0} = b_{v}^{0} + c everywhere.

We now consider v ∈ S(M, g_{0}) such that γ_{v}^{0}([0, ∞)) ∩ K = ∅. Pick r_{1}, r_{2} such that for the horoballs B_{v}^{0}(r_{1}) and B_{v}^{1}(r_{2}) we have K ⊂ B_{v}^{0}(r_{1}) ⊂ B_{v}^{1}(r_{2}); this is possible because K and B_{v}^{0}(r) ∩ Shadow(K, v) is compact for every r. We claim that B_{v}^{1}(r_{2}) = B_{v}^{0}(r_{2}). To see this note that if w is an outward normal to ∂(B_{v}^{1}(r_{2})) then B_{w}^{1} is disjoint from Interior(B_{v}^{1}(r_{2})), and since K ⊂ B_{v}^{1}(r_{2}) we have B_{w}^{1} = B_{w}^{0} and B_{v}^{1}(r_{2}) is supported at π(w) by any w-admissible (k − 1)-flat F. If we let w run over all outward normals to B_{v}^{1}(r_{2}) based at points in (M − Shadow(K, v)) then the union of the wadmissible (k − 1)-flats is disjoint from Interior(B_{v}^{1}(r_{2})) and it coincides with ∂B_{v}^{0}(r_{2}). This implies that B_{v}^{1}(r_{2}) ⊆ B_{v}^{0}(r_{2}). Let r_{3} = Max{r|B_{v}^{0}(r) ⊆ B_{v}^{1}(r_{2})} = Min(b_{v}^{0}|_{∂(B_{v}^{1}(r_{2}))}). Then B_{v}^{0}(r_{3}) ⊆ B_{v}^{1}(r_{2}) is tangent to ∂B_{v}^{1}(r_{2}) at some point p, and if w is normal to ∂B_{v}^{1}(r_{2}) at p then any w-admissible (k − 1)-flat F supports ∂B_{v}^{1}(r_{2}) from both the inside and outside, forcing F ⊂ ∂B_{v}^{1}(r_{2}). But since b_{v}^{0} and b_{v}^{1} are both constant on F and b_{v}^{0}(x) = b_{v}^{1}(x) for x ∈ F −Shadow(K, v) ≠ ∅, we conclude that r_{3} = r_{2} and B_{v}^{1}(r_{2}) = B_{v}^{0}(r_{2}).

We now know that b_{v}^{0}(x) = b_{v}^{1}(x) when x ∉ Shadow(K, v) or when b_{v}^{0}(x) ≥ Max(b_{v}^{0}|_{K}). Pick x ∉ Chord(K), and let w = (∇b_{v}^{0})(x). If γ_{w}^{0}([0, ∞)) ∩ K = ∅, then for large enough t, γ_{w}^{0}(t) will lie in the region where b_{v}^{1} = b_{v}^{0}, so (∇b_{v}^{0})(γ_{w}^{0}(t)) = (∇b_{v}^{1})(γ_{w}^{0}(t)), which implies that γ_{w}^{0}|_{[0,∞)} is a gradient line for both b_{v}^{0} and b_{v}^{1}, which implies that b_{v}^{1}(x) = b_{v}^{0}(x). Similar reasoning applies if γ_{w}^{0}((−∞, 0]) ∩ K = ∅ so we have b_{v}^{0}(x) = b_{v}^{1}(x) as desired.

If x, y ∉ Chord(K), consider the g_{0}-geodesic segment η from x to y. If η lies in M − K, then d^{0}(x, y) = length_{g_{0}}(η) = length_{g_{1}}(η) = d^{1}(x, y). Otherwise, letting v = −η′(0) ∈ S_{x}(M, g_{0}), we have γ_{v}^{0}([0, ∞)) ∩ K = ∅, so b_{v}^{1}(y) = b_{v}^{0}(y) by the arguments of the previous paragraphs. Hence

d^{0}(x, y) = b_{v}^{0}(y) − b_{v}^{0}(x) = b_{v}^{1}(y) − b_{v}^{1}(x)
≤ d^{1}(x, y)

since b_{v}^{1} has Lipschitz constant 1. Symmetric reasoning gives d^{1}(x, y) ≤ d^{0}(x, y), so we have part 1 of proposition 2.1. The second part of proposition 2.1 follows immediately from the first, since (exp_{x}^{i})^{−1}(y) = −d^{i}(x, y)(∇d_{y}^{i})(x).

□

Remark 2.2 Proposition 2.1 reduces conjecture 1.2 – in the simply connected case – to the “lens rigidity” question posed in [Cr2].

By the proposition, we may define a diffeomorphism F : S(M, g_{0}) ⟶ S(M, g_{1}) by F (v) = Φ_{t}^{1}(Φ_{−t}^{0}(v)) where t ∈ R is chosen so that Φ_{s}^{0}(v) ∈ M − K for all s ≤ −t. Using proposition 2.1 and the fact that the geodesic flow preserves the canonical measure on S(M, g_{i}), it is easy to verify that F satisfies:

1. F is smooth,

2. F (−v) = −F (v),

3. If π(v) ∉ Chord(K), then F (v) = v.

4. F is a measure preserving map.

Hence we have:

Corollary 2.3 If D ⊂ M is a domain containing K, then V ol_{g_{0}}(D) = V ol_{g_{1}}(D).

Proof. The triangle inequality implies that Chord(K) is compact, so we first assume that Chord(K) ⊂ D.

F (S(M − D, g_{0})) = S(M − D, g_{1}) by property 3, so F (S(D, g_{0})) = S(D, g_{1}). Hence

V ol_{g_{0}}(D) = \frac{1}{c_{n}} V ol_{g_{0}}(S(D, g_{0})) = \frac{1}{c_{n}} V ol_{g_{1}}(S(D, g_{1})) = V ol_{g_{1}}(D),

where c_{n} is the volume of the standard unit sphere in E^{n}.

We now know that V ol_{g_{0}}(K) = V ol_{g_{1}}(K), so the corollary follows immediately. □

2.3 The proof of theorem 1.1

Let M = X × R, and let g_{1} be a metric without conjugate points on M such that g_{1}|_{M−K} = g_{0}|_{M−K}, where K = K_{1} × [a, b], and K_{1} ⊂ X is a compact domain with smooth boundary. Since g_{0} has rank ≥ 2, proposition 2.1 applies, and we conclude that g_{1}-geodesics coincide with g_{0}-geodesics outside the slab X × [a, b]. For each x ∈ K_{1} let γ_{x}^{i} be the g_{i}-geodesic such that γ_{x}^{i}(t) = (x, t) for all t ∉ [a, b]. Thus we have a well-defined map G : K ⟶ K given by G(x, t) = γ_{x}^{1}(t).

For fixed x ∈ K and parallel orthonormal frames along the γ^{i} we see that the differential D(G)(x, t) represents a jacobi tensor along γ_{x}^{1}(t) that is orthogonal for t ∉ [a, b]. Hence by the argument in [Cr2] page 459, ∫_{a}^{b} Jac(G)(x, t)dt ≥ b − a ∀x ∈ K_{1}, with equality holding if and only if with respect to parallel frames along the γ_{x}^{i}, D(G)(x, t) is a fixed orthogonal matrix. (A complete argument, in a slightly more general context then we have here, can be found in the last section of [Cr-Kl]).

Hence

V ol(K, g_{1}) = ∫_{K} Jac(G)vol_{g_{0}}
= ∫_{K_{1}} (∫_{a}^{b} Jac(G)(x, t)dt) dx.
V ol_{g_{1}}(K) ≥ ∫_{K_{1}}(b − a)dx = V ol_{g_{0}}(K).  (1)

By corollary 2.3 equality holds in inequality 1, so ∫_{a}^{b} Jac(G)(x, t)dt = b − a ∀x ∈ K_{1}, which forces G to be an isometry. □

References

[A-S] M. Anderson, V. Schroeder, Existence of flats in manifolds of nonpositive curvature, Invent. Math., 85(1986), no. 2, 303-315.

[B-G-S] W. Ballmann, M. Gromov, V. Schroeder, Manifolds of nonpositive curvature, Progress in Mathematics vol. 61, Birkhauser, 1985.

[B] M. Berger, Géométrie (5 volumes), Paris, CEDIC-Nathan, 1976.

[Bu] K. Burns, The flat strip theorem fails for surfaces with no conjugate points, Proc. Amer. Math. Soc., 115(1992),199-206.

[Cr1] C. Croke, Simply connected manifolds without conjugate points which are flat outside a compact set, Proc. Amer. Math. Soc., 111, (1991), no. 1, 297-298.

[Cr2] C. Croke, Rigidity and the distance between boundary points, J. Diff. Geom.,33 (1991),445-464.

[Cr-Kl] C. Croke, B. Kleiner A rigidity theorem for non-simply connected manifolds without conjugate points (in preparation).

[C-S] C. Croke and V. Schroeder, The fundamental group of compact manifolds without conjugate points, Comm. Math. Helv., 61(1986), 161-175.

[E] J. Eschenburg, Horospheres and the stable part of the geodesic flow, Math. Zeit, 153, (1977), 237-251.

[Es-OS] J. Eschenburg, J. O’Sullivan, Growth of Jacobi fields and divergence of geodesics, Math. Z., 150, (1976), no. 3, 221-237.

[Gr-Gu] L. Green, R. Gulliver, Planes without conjugate points, J. Diff. Geom., 22, (1985), no. 1, 43-47.

[Gr-Wu] R. Greene, H. H. Wu, On a new gap phenomenon in Riemannian Geometry, Proc. Nat. Acad. Sci., 79, (1982), no. 2, part 1, 714-715.

[Kl] B. Kleiner (in preparation)

[S-Z] V. Schroeder, W. Ziller, Local rigidity of symmetric spaces, Trans. AMS, 320, (1990), no. 1, 145-160.