Lorentz and semi-Riemannian spaces with Alexandrov curvature bounds

# Lorentz and semi-Riemannian spaces with Alexandrov curvature bounds

Stephanie B. Alexander 1409 W. Green St., Urbana, Illinois 61801  and  Richard L. Bishop 1409 W. Green St., Urbana, Illinois 61801
###### Abstract.

A semi-Riemannian manifold is said to satisfy (or ) if spacelike sectional curvatures are and timelike ones are (or the reverse). Such spaces are abundant, as warped product constructions show; they include, in particular, big bang Robertson-Walker spaces. By stability, there are many non-warped product examples. We prove the equivalence of this type of curvature bound with local triangle comparisons on the signed lengths of geodesics. Specifically, if and only if locally the signed length of the geodesic between two points on any geodesic triangle is at least that for the corresponding points of its model triangle in the Riemannian, Lorentz or anti-Riemannian plane of curvature (and the reverse for ). The proof is by comparison of solutions of matrix Riccati equations for a modified shape operator that is smoothly defined along reparametrized geodesics (including null geodesics) radiating from a point. Also proved are semi-Riemannian analogues to the three basic Alexandrov triangle lemmas, namely, the realizability, hinge and straightening lemmas. These analogues are intuitively surprising, both in one of the quantities considered, and also in the fact that monotonicity statements persist even though the model space may change. Finally, the algebraic meaning of these curvature bounds is elucidated, for example by relating them to a curvature function on null sections.

###### 1991 Mathematics Subject Classification:
53B30,53C21,53B70
The research described in this paper was made possible in part by Award No. RMI-2381-ST-02 of the U.S. Civilian Research & Development Foundation for the Independent States of the Former Soviet Union (CRDF). Also, important parts were developed while the first author visited IHES in Bures-sur-Yvette, France.

## 1. Introduction

### 1.1. Main theorem

Alexandrov spaces are geodesic metric spaces with curvature bounds in the sense of local triangle comparisons. Specifically, let denote the simply connected 2-dimensional Riemannian space form of constant curvature . For curvature bounded below (CBB) by , the distance between any two points of a geodesic triangle is required to be the distance between the corresponding points on the “model” triangle with the same sidelengths in . For curvature bounded above (CBA), substitute “”. Examples of Alexandrov spaces include Riemannian manifolds with sectional curvature or . A crucial property of Alexandrov spaces is their preservation by Gromov-Hausdorff convergence (assuming uniform injectivity radius bounds in the CBA case). Moreover, CBB spaces are topologically stable in the limit [P], a fact at the root of landmark Riemannian finiteness and recognition theorems. (See Grove’s essay [Ge].) CBA spaces are also important in geometric group theory (see [Gv, BH]) and harmonic map theory (see, for example, [GvS, J, EF]).

In Lorentzian geometry, timelike comparison and rigidity theory is well developed. Early advances in timelike comparison geometry were made by Flaherty [F], Beem and Ehrlich [BE], and Harris [H1, H2]. In particular, a purely timelike, global triangle comparison theorem was proved by Harris [H1]. A major advance in rigidity theory was the Lorentzian splitting theorem, to which a number of researchers contributed; see the survey in [BEE], and also the subsequent warped product splitting theorem in [AGH]. The comparison theorems mentioned assume a bound on sectional curvatures of timelike 2-planes . Note that a bound over all nonsingular 2-planes forces the sectional curvature to be constant [Ki], and so such bounds are uninteresting.

This project began with the realization that certain Lorentzian warped products, which may be called Minkowski, de Sitter or anti-de Sitter cones, possess a global triangle comparison property that is not just timelike, but is fully analogous to the Alexandrov one. The comparisons we mean are on signed lengths of geodesics, where the timelike sign is taken to be negative. In this paper, length of either geodesics or vectors is always signed, and we will not talk about the length of nongeodesic curves. The model spaces are , or , where is the simply connected -dimensional Lorentz space form of constant curvature , and is with the sign of the metric switched, a space of constant curvature .

The cones mentioned above turn out to have sectional curvature bounds of the following type. For any semi-Riemannian manifold, call a tangent section spacelike if the metric is definite there, and timelike if it is nondegenerate and indefinite. Write if spacelike sectional curvatures are and timelike ones are ; for , reverse “timelike” and “spacelike”. Equivalently, if the curvature tensor satisfies

 (1.1) R(v,w,v,w)≥K(⟨v,v⟩⟨w,w⟩−⟨v,w⟩2),

and similarly with inequalities reversed.

The meaning of this type of curvature bound is clarified by noting that if one has merely a bound above on timelike sectional curvatures, or merely a bound below on spacelike ones, then the restriction of the sectional curvature function to any nondegenerate -plane has a curvature bound below in our sense: (as follows from [BP]; see §6 below). Then means that may be chosen independently of .

Spaces satisfying (or ) are abundant, as warped product constructions show. They include, for example, the big bang cosmological models discussed by Hawking and Ellis [HE, p. 134-138] (see §7 below). Since there are many warped product examples satisfying for all in a nontrivial finite interval, then by stability, there are many non-warped product examples.

Searching the literature for this type of curvature bound, we found it had been studied earlier by Andersson and Howard [AH]. Their paper contains a Riccati equation analysis and gap rigidity theorems. For example: A geodesically complete semi-Riemannian manifold of dimension and index , having either or and an end with finite fundamental group on which , is [AH]. Their method uses parallel hypersurfaces, and does not concern triangle comparisons or the methods of Alexandrov geometry. Subsequently, Díaz-Ramos, García-Río, and Hervella obtained a volume comparison theorem for “celestial spheres” (exponential images of spheres in spacelike hyperplanes) in a Lorentz manifold with or [DGH].

Does this type of curvature bound always imply local triangle comparisons, or do triangle comparisons only arise in special cones? In this paper we prove that curvature bounds or are actually equivalent to local triangle comparisons. The existence of model triangles is described in the Realizability Lemma of §2. It states that any point in represents the sidelengths of a unique triangle in a model space of curvature , and the same holds for under appropriate size bounds for .

We say is a normal neighborhood if it is a normal coordinate neighborhood (the diffeomeorphic exponential image of some open domain in the tangent space) of each of its points. There is a corresponding distinguished geodesic between any two points of , and the following theorem refers to these geodesics and the triangles they form. If in addition the triangles satisfy size bounds for , we say is normal for . All geodesics are assumed parametrized by , and by corresponding points on two geodesics, we mean points having the same affine parameter.

###### Theorem 1.1.

If a semi-Riemannian manifold satisfies , and is a normal neighborhood for , then the signed length of the geodesic between two points on any geodesic triangle of is at least  (at most)  that for the corresponding points on the model triangle in , or .

Conversely, if triangle comparisons hold in some normal neighborhood of each point of a semi-Riemannian manifold, then .

In this paper, we restrict our attention to local triangle comparisons (i.e., to normal neighborhoods) in smooth spaces. In the Riemannian/Alexandrov theory, local triangle comparisons have features of potential interest to semi-Riemannian and Lorentz geometers: they incorporate singularities, imply global comparison theorems, and are consistent with a theory of limit spaces. Our longer-term goal is to see what the extension of the theory presented here can contribute to similar questions in semi-Riemannian and Lorentz geometry.

### 1.2. Approach

We begin by mentioning some intuitive barriers to approaching Theorem 1.1. In resolving them, we are going to draw on papers by Karcher [Kr] and Andersson and Howard [AH], putting them to different uses than were originally envisioned.

First, a fundamental object in Riemannian theory is the locally isometrically embedded interval, that is, the unitspeed geodesic. These are the paths studied in [Kr] and [AH]. However, in the semi-Riemannian case this choice constrains consideration to fields of geodesics all having the same causal character. By contrast, our construction, which uses affine parameters on , applies uniformly to all the geodesics radiating from a point (or orthogonally from a nondegenerate submanifold).

Secondly, a common paradigm in Riemannian and Alexandrov comparison theory is the construction of a curve that is shorter than some original one, so that the minimizing geodesic between the endpoints is even shorter. In the Lorentz setting, this argument still works for timelike curves, under a causality assumption. However, spacelike geodesics are unstable critical points of the length functional, and so this argument is forbidden.

Thirdly, while the comparisons we seek can be reduced in the Riemannian setting to -dimensional Riccati equations (as in [Kr]), the semi-Riemannian case seem to require matrix Riccati equations (as in [AH]). Such increased complexity is to be expected, since semi-Riemannian curvature bounds below (say) have some of the qualities of Riemannian curvature bounds both below and above.

Let us start by outlining Karcher’s approach to Riemannian curvature bounds. It included a new proof of local triangle comparisons, one that integrated infinitesimal Rauch comparisons to get distance comparisons without using the “forbidden argument” mentioned above. Such an approach, motivated by simplicity rather than necessity in the Riemannian case, is what the semi-Riemannian case requires.

In this approach, Alexandrov curvature bounds are characterized by a differential inequality. Namely, has CBB by in the triangle comparison sense if and only if for every and unit-speed geodesic , the differential inequality

 (1.2) (f∘γ)′′+Kf∘γ≤1

is satisfied (in the barrier sense) by the following function :

 (1.3) mdKdq=⎧⎪⎨⎪⎩(1/K)(1−cosh√−Kdq),K<0(1/K)(1−cos√Kdq),K>0d2q/2,K=0.

The reason for this equivalence is that the inequalities (1.2) reduce to equations in the model spaces ; since solutions of the differential inequalities may be compared to those of the equations, distances in may be compared to those in . The functions then provide a convenient connection between triangle comparisons and curvature bounds, since they lead via their Hessians to a Riccati equation along radial geodesics from .

We wish to view this program as a special case of a procedure on semi-Riemannian manifolds. For a geodesic parametrized by , let

 (1.4) E(γ)=⟨γ′(0),γ′(0)⟩.

Thus In this paper, we work with normal neighborhoods, and set where is the geodesic from to that is distinguished by the normal neighborhood.

 (1.5) E(p,q)=Eq(p)=inf{E(γ): γ is a geodesic joining p \ and q},

under hypotheses that ensure the two definitions agree locally. In (1.5), if and are not connected by a geodesic.)

Now define the modified distance function at by

 (1.6) hK,q=⎧⎨⎩(1−cos√KEq)/K=∑∞n=1(−K)n−1(Eq)n(2n)!,K≠0Eq/2,K=0.

Here, the formula remains valid when the argument of cosine is imaginary, converting to . In the Riemannian case, . The CBB triangle comparisons we seek will be characterized by the differential inequality

 (1.7) (hK,q∘γ)′′+KE(γ)hK,q∘γ≤E(γ),

on any geodesic parametrized by .

The self-adjoint operator associated with the Hessian of may be regarded as a modified shape operator. It has the following properties: in the model spaces, it is a scalar multiple of the identity on the tangent space to at each point; along a nonnull geodesic from , its restriction to normal vectors is a scalar multiple of the second fundamental form of the equidistant hypersurfaces from ; it is smoothly defined on the regular set of , hence along null geodesics from (as the second fundamental forms are not); and finally, it satisfies a matrix Riccati equation along every geodesic from , after reparametrization as an integral curve of .

We shall also need semi-Riemannian analogues to the three basic triangle lemmas on which Alexandrov geometry builds, namely, the Realizability, Hinge and Straightening Lemmas. The analogues are intuitively surprising, both in one of the quantities considered, and also in the fact that monotonicity statements persist even though the model space may change. The Straightening Lemma is an indicator that, as in the standard Riemannian/Alexandrov case, there is a singular counterpart to the smooth theory developed in this paper.

### 1.3. Outline of paper

We begin in §2 with the triangle lemmas just mentioned. In §3, it is shown that the differential inequalities (1.7) become equations in the model spaces, and hence characterize our triangle comparisons.

Comparisons for the modified shape operators under semi-Riemannian curvature bounds are proved in §4, and Theorem 1.1 is proved in §5.

In §6, semi-Riemannian curvature bounds are related to the analysis by Beem and Parker of the pointwise ranges of sectional curvature [BP], and to the “null” curvature bounds considered by Uhlenbeck [U] and Harris [H1].

Finally, §7 considers examples of semi-Riemannian spaces with curvature bounds, including Robertson-Walker “big bang” spacetimes.

## 2. Triangle lemmas in model spaces

Say three numbers satisfy the strict triangle inequality if they are positive and the largest is less than the sum of the other two. Denote the points of whose coordinates satisfy the strict triangle inequality by , and their negatives by . A triple, one of whose entries is the sum of the other two, will be called degenerate. Denote the points of whose coordinates are nonnegative degenerate triples by , and their negatives by .

In Figure 1, the shaded cone is , and the interior of its convex hull is .

Say a point is realized in a model space if its coordinates are the sidelengths of a triangle. As usual, set if .

###### Lemma 2.1 (Realizability Lemma).

Points of have unique realizations, up to isometry of the model space, as follows:

• A point in is realized by a unique triangle in , provided the sum of its coordinates is . A point in is realized by a unique triangle in , provided the sum of its coordinates is .

• A point in is realized by unique triangles in and , provided the largest coordinate is . A point in is realized by unique triangles in and , provided the smallest coordinate is .

• A point in the complement of is realized by a unique triangle in . For , if the largest coordinate is , the point is realized by a unique triangle in . For , if the smallest coordinate is , the point is realized by a unique triangle in .

###### Proof.

Part 1 is standard, as is Part 2 for . Now consider a point not in , and denote its coordinates by .

To realize this point in , suppose and take a segment of length on the -axis. Since distance “circles” about a point are pairs of lines of slope through if the radius is , and hyperbolas asymptotic to these lines otherwise, it is easy to see that circles about the endpoints of intersect, either in two points or tangentially, subject only to the condition that if , namely, the point is not in . Thus our point may be realized in , uniquely up to an isometry of . On the other hand, if then , so by switching the sign of the metric, we have just shown there is a realization in .

For , is the simply connected cover of the quadric surface in Minkowski -space with signature . Suppose , and take a segment of length on the quadric’s equatorial circle of length in the -plane. A distance circle about an endpoint of is a hyperbola or pair of lines obtained by intersection with a -plane parallel to or coinciding with the tangent plane. Two circles about the endpoints of intersect, either in two points or tangentially, if the vertical line of intersection of their -planes cuts the quadric. This occurs subject only to the condition that if , namely, the point is not in . On the other hand, if then . Take a segment of length in the quadric, where is symmetric about the -plane. Circles of nonpositive radius about the endpoints of intersect if the horizontal line of intersection of their -planes cuts the quadric, and this occurs subject only to the condition that , namely, the point is not in .

Since , switching the sign of the metric completes the proof. ∎

Let us say the points of for which Lemma 2.1 gives model space realizations satisfy size bounds for (for , no size bounds apply). Such a point may be expressed as , where is a realizing triangle in a model space of curvature , the geodesic is a side parametrized by with , and we write . By the nonnormalized angle , we mean the inner product .

In our terminology, is the included, and and are the shoulder, nonnormalized angles for . This terminology is welldefined since the realizing model space and triangle are uniquely determined except for degenerate triples. The latter have only two realizations, which lie in geodesic segments in different model spaces but are isometric to each other.

An important ingredient of the Alexandrov theory is the Hinge Lemma for angles in , a monotonicity statement that follows directly from the law of cosines. Part 1 of the following lemma is its semi-Riemannian version. A new ingredient of our arguments is the use of nonnormalized shoulder angles, in which both the “angle” and one side vary simultaneously. Not only do we obtain a monotonicity statement that for is not directly apparent from the law of cosines (Part 2 of the following lemma), but we find that monotonicity persists even as the model space changes.

###### Lemma 2.2 (Hinge Lemma).

Suppose a point of satisfies size bounds for , and the third coordinate varies with the first two fixed. Denote the point by where lies in a possibly varying model space of curvature .

• The included nonnormalized angle is a decreasing function of .

• Each shoulder nonnormalized angle, or , is an increasing function of .

###### Proof.

Suppose . Then the model spaces are semi-Euclidean planes, and the sides of a triangle may be represented by vectors , and . Set and , so

 (2.1) c=a1+a2−2.

Since is an increasing function of its sidelength, Part in any fixed model space is immediate by taking and in (2.1) to be fixed. For Part in any fixed model space, it is only necessary to rewrite (2.1) as

 (2.2) c−a1+2=a2,

where and are fixed.

A change of model space occurs when the varying point in moves upward on a vertical line , and passes either into or out of by crossing (the same argument will hold for and ). See Figure 1.

Thus is the union of three closed segments, intersecting only at their two endpoints on . We have just seen that the included angle function is decreasing on each segment, since the realizing triangles are in the same model space (by choice at the endpoints and by necessity elsewhere). Since the values at the endpoints are the same from left or right, the included angle function is decreasing on all of . Similarly, each shoulder angle function is increasing.

Suppose . The vertices of a triangle in the quadric model space are also the vertices of a triangle in an ambient -plane, whose sides are the chords of the original sides. The length of the chord is an increasing function of the original sidelength. Thus to derive the lemma for from (2.1) and (2.2), we must verify the following: If a triangle in a quadric model space varies with fixed sidelengths adjacent to one vertex, and are the tangent vectors to the sides at that vertex, then is an increasing function of where the are the chordal vectors of the two sides. Indeed, all points of a distance circle of nonzero radius in the quadric model space lie at a fixed nonzero ambient distance from the tangent plane at the centerpoint. Thus is a linear combination of and a fixed normal vector to the tangent plane, where the coefficients depend only on the sidelength . The desired correlation follows.

By switching the sign of the metric, we obtain the claim for . ∎

###### Remark 2.3.

The Law of Cosines in a semi-Riemannian model space with is (2.1). If , the Law of Cosines for may be written in unified form as follows:

 (2.3) cos√KE(γpr)=cos√KE(γpq) cos√KE(γqr) −K∠pqrsin√KE(γpq)√KE(γpq)sin√KE(γqr)√KE(γqr).

Here we assume satisfies the size bounds for . Then each sidelength is if , and if . Part 1 of Lemma 2.2 can be derived from (2.3) as follows. Fix and , and observe that is decreasing in if , regardless of the sign of and even as passes through , and increasing in if . The size bounds imply that the factors become either for , or , depending on the signs of and , and hence are nonnegative.

Now we are ready to prove a semi-Riemannian version of Alexandrov’s Straightening Lemma, according to which a triangle inherits comparison properties from two smaller triangles that subdivide it. It turns out that the comparisons we need are on nonnormalized shoulder angles. Moreover, the original and “subdividing” triangles may lie in varying model spaces, so that geometrically we have come a long way from the original interpretation in terms of hinged rods.

Since geodesics are parametrized by , a point on a directed side of a triangle inherits an affine parameter .

###### Lemma 2.4 (Straightening Lemma for Shoulder Angles).

Suppose is a triangle satisfying size bounds for in a model space of curvature . Let be a point on side , and set . Let and be triangles in respective model spaces of curvature , where , , , , and . Assume if , and if . If

 (1−λ)∠p1m1q1+λ∠r2m2q2≥0,

then

 ∠˜q˜p˜m≥∠q1p1m1 and ∠˜q˜r˜m≥∠q2r2m2.

The same statement holds with all inequalities reversed.

###### Proof.

By the definition of nonnormalized angles, . Therefore, by hypothesis, either or . By Lemma 2.2.2, the inequality holds for either or , and hence for both. But then by Lemma 2.2.1, the claim follows. ∎

## 3. Modified distance functions on model spaces

In this section we give a unified proof that in the model spaces of curvature , the restrictions to geodesics of the modified distance functions defined by (1.6) satisfy the differential equation

 (3.1) (hK,q∘γ)′′+K⟨γ′,γ′⟩hK,q∘γ=⟨γ′,γ′⟩.

We begin by constructing the -affine functions on the model spaces. For intrinsic metric spaces the notion of a -affine function was considered in [AB1] and their structural implications were pursued in [AB2]. For semi-Riemannian manifolds the definition should be formulated to account for the causal character of geodesics, as follows.

###### Definition 3.1.

A -affine function on a semi-Riemannian manifold is a real-valued function such that for every geodesic the restriction satisfies

 (3.2) (f∘γ)′′+K⟨γ′,γ′⟩f=0.

We say is -concave if “” holds in (3.2), and -convex if “” holds.

(Elsewhere we have called the latter classes -concave/convex.)

As in the Riemannian case, the -dimensional model spaces of curvature carry an -dimensional vector space of -affine functions, namely, the space of restrictions of linear functionals in the ambient semi-Euclidean space of a quadric surface model.

Specifically, let be the semi-Euclidean space of index . For , set , with the induced semi-Riemannian metric, so that is an -dimensional space of constant curvature . (The -dimensional model spaces are the universal covers of such quadric surfaces.) For , let be the restriction to of the linear functional on dual to the element , namely, . Define on by (1.5).

###### Proposition 3.2.

For , the function on is -affine. For any that is joined to by a geodesic in ,

 ℓK,q(p)=1Kcos√KEq(p),

where the argument of cosine may be imaginary.

###### Proof.

We use the customary identification of elements of with tangent vectors to and . Then the gradient of the linear functional on is , viewed as a parallel vector field. For , projection is given by . In particular, . It is easily checked that .

The connection of is related to the connection of by projection, that is, for . Writing , for a geodesic of , then

 (ℓK,q∘γ)′′(t)=⟨∇vgradℓK,q,v⟩ =⟨πpDvπpq,v⟩ =⟨πpDv(q−K⟨q,p⟩p),v⟩ =⟨πp(0−K⟨q,v⟩p−K⟨q,p⟩∇vp),v⟩ (3.3) =−K⟨v,v⟩ℓK,q(γ(t)).

Thus is -affine.

Since is orthogonal to the tangent plane , the derivatives of at are all . Along a geodesic in that starts at , the initial conditions for are , , so the formula for is . ∎

For the case we consider the quadric surface model to be a hyperplane not through the origin, so that the affine functions on it are trivially the restrictions of linear functionals.

On a model space of curvature , the modified distance function defined by (1.6) may be written on its domain as

 (3.4) hK,q=−ℓK,q+1/K,

and satisfies the same differential equation along geodesics as except for an additional constant term, that is, satisfies (3.1). It is trivial to check that this equation holds when and .

## 4. Ricatti comparisons for modified shape operators

In a given semi-Riemannian manifold , set (as in (1.6)) for some fixed choice of and . Define the modified shape operator , on the region where is smooth, to be the self-adjoint operator associated with the Hessian of , namely,

The form of was chosen so that in a model space , is always a scalar multiple of the identity. Indeed, at any point in ,

 (4.2) S={I,if \,K=0,KℓK,q⋅I,if \,K≠0,

where the latter equality is by Proposition 3.2 and (3.3).

Below, our Riccati equation (4.3) along radial geodesics from differs from the standard one in [AH] and [Kr], being adjusted to facilitate the proof of Theorem 1.1. Thus it applies even if is null; it concerns an operator that is defined on the whole tangent space; when is nonnull, the restriction of to the normal space of does not agree with the second fundamental form of the equidistant hypersurface but rather with a rescaling of it; and we do not differentiate with respect to an affine parameter along , but rather use the integral curve parameter of .

The gradient vector field is tangent to the radial geodesics from . Note that is nonzero along null geodesics radiating from even though vanishes along such geodesics. Specifically, may be expressed in terms of on a normal coordinate neighborhood via (1.6). Here , where is the image under of the position vector field on (see [O’N, p. 128]). If , then , and an affine parameter on a radial geodesic from is given in terms of the integral curve parameter of by with at . If , then , so agrees with up to higher order terms, and the dominant term at in the integral curve expression is an exponential.

Let be the self-adjoint Ricci operator, . We are going to establish comparisons on modified shape operators, governed by comparisons on Ricci operators. Since we are interested in comparisons along two given geodesics, each radiating from a given basepoint, the effect of restricting to normal coordinate neighborhoods in the following proposition is merely to rule out conjugate points along both geodesics.

###### Proposition 4.1.

In a semi-Riemannian manifold , on a normal coordinate neighborhood of , the modified shape operator satisfies the first-order PDE

 (4.3) ∇GS+S2−(1−Kh)S+RG+Kdh⊗G=0.

Before verifying Proposition 4.1, we shift to the general setting of systems of ordinary differential equations in order to summarize all we need about Jacobi and Riccati equations.

###### Lemma 4.2.

For self-adjoint linear maps on a semi-Euclidean space, suppose satisfies

 (4.4) F′′(t)+R(t)F(t)=0

for , where ,  is invertible, and is invertible for all . For a given function with ,  , and on , define by

 (4.5) g(t)F′(t)=S(t)F(t) for t∈(0,b],

and

 (4.6) S(0)=I.

Then is self-adjoint, smooth on , and satisfies

 (4.7) gS′+S2−g′S+g2R=0.
###### Proof.

Self-adjointness of follows from (4.4) and self-adjointness of (see [AH, p. 839]). By (4.5) and (4.4), on we have

 S′F+g−1S2F =S′F+SF′=g′F′+gF′′ =g′F′−gRF=g′g−1SF−gRF.

Multiplying the first and last expressions by on the right yields (4.7).

On we have where , and where is invertible. Then (4.5) gives on . By (4.6), on , so is smooth there. ∎

Comparisons of solutions of (4.7) will be in terms of the notion of positive definite and positive semi-definite self-adjoint operators [AH, p. 838]. A linear operator on a semi-Euclidean space is positive definite if for every , positive semi-definite if . We then write if is positive definite, and similarly for . Note that the identity map is not positive definite if the index is positive; however, the eigenvalues of a positive definite operator are real. If and , then .

In [AH, p. 846-847], a comparison theorem for the shape operators of tubes in semi-Riemannian manifolds is stated without proof. For the proof of Theorem 1.1 we require a stronger version of the special case in which the central submanifolds are just points, so the shape operators of distance-spheres are compared; the strengthening comes from the extension to modified shape operators. Since it is a key result for us, we now show how this version can be derived from a modification of the comparison theorem proved in [AH, p. 838-841], together with a Taylor series argument to cover the behavior at the base-point singularity.

###### Theorem 4.3.

Let and () be as in Lemma 4.2, and assume . If for all , then on . If , then on .

###### Proof.

First we show that (4.7) and the initial data for imply

 (4.8) S′(0)=0

and

 (4.9) S′′(0)=13(g′′′(0)I−2R(0)).

To see this, differentiate (4.7), obtaining

 g′S′+gS′′+S′S+SS′−g′′S−g′S′+(g2R)′=0.

Applying the initial data for and gives (4.8). Now cancel the terms and differentiate again:

 g′S′′+gS′′′+2S′2+SS′′+S′′S−g′′′S−g′′S′+(g2R)′′=0.

Setting gives (4.9).

Now for , let , where is a positive definite self-adjoint operator, constant as a function of . The solutions of with and depend continuously on the parameter , approaching the solution of . In particular, is invertible for all if is sufficiently small. Define as in (4.4),  (4.5) with .

Since , setting and in (4.9) implies . Since , and by (4.8), then for all , where depends on .

But then for . Our argument for this follows [AH, p. 839], except for showing that the additional linear term in (4.7) is harmless. Namely, assume the statement is false. Then there exists for which , is not positive definite, and for . Hence there is a nonzero vector such that , and so . For , then by (4.7),

 g(t0)f′(t0) =⟨(g(t0)S′1(t0)−g(t0)S′δ(t0))x0,x0⟩ =⟨Sδ(t0)x0,Sδ(t0)x0⟩−⟨S1(t0)x0,S1(t0)x0⟩ +⟨g′(t0)(S1(t0)−Sδ(t0))x0,x0⟩+g(t0)2⟨(Rδ(t0)−R1(t0))x0,x0⟩ =g(t0)2⟨(Rδ(t0)−R1(t0))x0,x0⟩>0.

This contradicts , which is true because on and .

Since for all , and for all , we have . ∎

Returning to the geometric setting, let us verify Proposition 4.1.

Proof of Proposition 4.1. Let be the unit radial vector field tangent to nonnull geodesics from . By continuity, it suffices to verify (4.3) at every point that is joined to by a nonnull geodesic .

First we check that (4.3) holds when applied to . Note that the modified shape operator satisfies

 (4.10) SN=∇NG=(1−Kh)N.

Indeed, the form of along a unitspeed radial geodesic from the basepoint is the same in all manifolds, hence the same in as in a model space. But in a model space, (4.2) and (3.4) imply . Therefore

 (∇GS+ S2−(1−Kh)S+RG+Kdh⊗G)N =−K(Gh)N+(1−Kh)2N−(1−Kh)2N+0+K(Nh)G =−Kg(Nh)N+K(Nh)gN=0,

as required.

Now we verify that (4.3) holds on . If has dimension and index , consider an isometry . For a nonnull, unitspeed geodesic in radiating from , identify with by parallel translation to the base point composed with . Thus we identify linear operators on and , and likewise on and the corresponding -dimensional subspace of . If we restrict to , and set and , then (4.4) becomes the Jacobi equation for normal Jacobi fields, and the operator defined by (4.5) is , the Weingarten operator, for :

 Wv=∇vN,v∈V.

(See [AH], which uses the opposite sign convention for .) If instead we set as before but where , so that and for , then the operator defined by (4.5) and (4.6) is the restriction to of the modified shape operator, for . Indeed, (4.5) implies for , hence

 Sv=g∇vN=∇v(gN)=∇vG,

which agrees with the definition (4.1) of the modified shape operator. And the modified shape operator is the identity at by (4.10), since can be chosen to be any unit vector at . Then it is straightforward from (4.7) that the restriction to of the modified shape operator satisfies (4.3).

The proof of the rigidity statement proceeds just as in [AH, p. 840]. ∎

###### Remark 4.4.

To summarize, [AH, Theorem 3.2] applies to the Weingarten operator of the equidistant hypersurfaces from a hypersurface. In that case, both and are perturbed in order to obtain a strict inequality on operators; if instead we considered the modified Weingarten operator , so , we would perturb and . On the other hand, Theorem 4.3 above applies to , where is the Weingarten operator of the equidistant hypersurfaces from a point. Here we had and , and showed that merely perturbing implied a desired perturbation of and hence of for small . The theorem stated without proof in [AH, p.846-847] applies to the intermediate case of equidistant hypersurfaces from any submanifold . Except for changes in details, our proof above works for that case as well.

Now let us compare modified shape operators via Theorem 4.3. We say two geodesic segments and in semi-Riemannian manifolds and correspond if they are defined on the same affine parameter interval and satisfy .

###### Corollary 4.5.

For semi-Riemannian manifolds and of the same dimension and index, suppose and are corresponding nonnull geodesic segments radiating from the basepoints and and having no conjugate points. Identify linear operators on with those on by parallel translation to the basepoints, together with an isometry of and that identifies and . If at corresponding points of and , then the modified shape operators satisfy at corresponding points of and .

###### Proof.

The modified shape operators split into direct summands, corresponding to their action on the one-dimensional spaces tangent to the radial geodesics and on the orthogonal complements . The first summand is the same for both and . The second summand is as described in Lemma 4.2 with and . (Since our identification of and identifies and , we denote both of these by .) Furthermore, by (4.10), so by (1.6). Therefore the corollary follows from Theorem 4.3. ∎

###### Corollary 4.6.

Suppose is a semi-Riemannian manifold satisfying , and has the same dimension and index as and constant curvature . Then for any that is joined to by a geodesic that has no conjugate points and such that a corresponding geodesic segment in has no conjugate points, the modified shape operator satisfies

 (4.11) S(p)≤(1−KhK,q(p))⋅I.

The same statement holds with inequalities reversed.

###### Proof.

Let be the given geodesic from to , and be a corresponding geodesic from to . If is nonnull, then by Corollary 4.5, (4.2) and (3.4), we have

 S(p)≤˜S(˜p) =KℓK,˜q(˜p)⋅˜I =(1−K˜hK,˜q(˜p))⋅˜I,

where denotes the identity operator on , and are identified by parallel translation to followed by an isometry identifying