Heterogeneous Riemannian Manifolds

Abstract

We solve Ambrose's Problem for a generic class of Riemannian metrics on a smooth manifold, namely, the class of heterogeneous metrics.

1. Introduction

We define a Riemannian metric 𝑔 on a manifold 𝑀 to be heterogeneous if no two distinct points of 𝑀 have isometric neighborhoods. Intuitively, a heterogeneous metric is as far as possible from being homogeneous. Heterogeneity can be reformulated in terms of a multijet transversality condition so that by an application of the standard transversality theorems, the genericity of heterogeneous metrics is established.

Theorem 1.1. The set of heterogeneous metrics on a smooth manifold 𝑀 of dimension 𝑛≥2 is residual in the space of Riemannian metrics on 𝑀 with the strong 𝐶^∞ topology.

A version of Theorem 1.1 for compact manifolds was stated and proved by Sunada [1, Proposition 1].

Ambrose [2] asked whether or not a complete simply connected Riemannian manifold is determined up to isometry by the behavior of curvature under parallel transport along geodesics emanating from a point. For heterogeneous metrics the answer is always yes.

Proposition 1.2. If (𝑀,𝑔) is a complete, connected, simply connected, heterogeneous Riemannian manifold of dimension 𝑛≥2, then, for every 𝑝∈𝑀, (𝑀,𝑔) is determined up to isometry by the behavior of curvature under parallel transport along geodesics emanating from 𝑝.

Proposition 1.2 combines with Theorem 1.1 to produce a generic dense family of metrics on a manifold which answers Ambrose's Problem in the affirmative.

Theorem 1.3. The set of complete metrics on a connected, simply connected smooth manifold 𝑀 of dimension 𝑛≥2 which, for every point 𝑝 in 𝑀, are determined up to isometry by the behavior of curvature under parallel transport along geodesics emanating from 𝑝, is residual in the space of complete Riemannian metrics on 𝑀 with the strong 𝐶^∞ topology.

Although Ambrose's Problem has been completely settled in dimension 2 [3, 4], Theorem 1.3 does give a significant advance on the problem in higher dimensions, since earlier partial results in [5, 6] apply only to metrics with rather special properties.

2. Ambrose's Problem and Heterogeneity

Let us recall what it means for a complete, connected, simply connected, 𝑛-dimensional Riemannian manifold(𝑀,𝑔) to be determined up to isometry by the behavior of curvature under parallel transport along geodesics emanating from a point 𝑝 in 𝑀. Let (𝑀,𝑔) be another complete, connected, simply connected, 𝑛-dimensional Riemannian manifold. Let 𝑝∈𝑀, and let 𝐼∶𝑇_𝑝𝑀→𝑇_𝑝𝑀 be a linear isometry between the tangent spaces. For each geodesic 𝛾∶[0,1]→𝑀 satisfying 𝛾(0)=𝑝, there is a corresponding geodesic 𝛾∶[0,1]→𝑀 satisfying 𝛾(0)=𝑝 characterized by the initial condition 𝛾^′(0)=𝐼(𝛾^′(0)). Given such a geodesic 𝛾, 𝐼_𝛾=𝑃_𝛾∘𝐼∘𝑃_𝛾⁻¹ defines a linear isometry from 𝑇_𝛾(1)𝑀 onto 𝑇_𝛾(1)𝑀 where 𝑃_𝛾 and 𝑃_𝛾 denote parallel transport along 𝛾 and 𝛾, respectively. Consider the hypothesis

𝐼_𝛾(𝑅(𝑋,𝑌)𝑍)=𝑅(𝐼_𝛾(𝑋),𝐼_𝛾(𝑌))𝐼_𝛾(𝑍) (∗) for every such geodesic 𝛾 and for all vectors 𝑋,𝑌,𝑍∈𝑇_𝛾(1)𝑀, where 𝑅 and 𝑅 are the respective Riemann curvature tensors. Then (𝑀,𝑔) is determined up to isometry by the behavior of curvature under parallel transport along geodesics emanating from 𝑝 if the hypothesis (*) implies that there exists an isometry Φ∶𝑀→𝑀 with Φ(𝑝)=𝑝 and 𝑑Φ=𝐼 at 𝑝 (cf., [2, 5, 6]).

In order to prove Proposition 1.2, suppose that (𝑀,𝑔) is heterogeneous, and assume the hypothesis (*). We will first find an isometry Φ∶𝑀→𝑀 with Φ(𝑝)=𝑝 and 𝑑Φ=𝐼⁻¹ at 𝑝. Thus we obtain the desired isometry Φ by setting Φ=Φ⁻¹.

To prove the existence of Φ, let 𝑞 in 𝑀 be any nonconjugate cut point to 𝑝 of order two, and let 𝛾₁ and 𝛾₂ be the two minimizing geodesics joining 𝑝 to 𝑞, whose respective initial tangent vectors are 𝑋₁=𝛾^′₁(0) and 𝑋₂=𝛾^′₂(0). Thus exp_𝑝(𝑋₁)=exp_𝑝(𝑋₂)=𝑞. Since 𝑞 is not conjugate to 𝑝 along 𝛾₁ or 𝛾₂, there are neighborhoods 𝑉₁ of 𝑋₁ and 𝑉₂ of 𝑋₂ in 𝑇_𝑝𝑀 for which exp_𝑝 carries both 𝑉₁ and 𝑉₂ diffeomorphically onto the same neighborhood 𝑊 of 𝑞. As noted in [6, page 561], for 𝑖∈{1,2}, the map 𝑓_𝑖=exp_𝑝∘𝐼⁻¹∘(exp_𝑝∣𝑉_𝑖)⁻¹∶𝑊→𝑀 is an isometry (after possibly cutting down 𝑊) onto a neighborhood 𝑈_𝑖 of 𝛾_𝑖(1). It follows that 𝛾₁(1)=𝛾₂(1), because otherwise 𝑓₂∘𝑓₁⁻¹∶𝑈₁→𝑈₂ would be an isometry between neighborhoods of two distinct points of 𝑀, contradicting the assumed heterogeneity of 𝑀. This verifies the hypothesis of Lemma 2.1 in [6], with the roles of 𝑀 and 𝑀 interchanged. Hence there exists an isometric immersion Φ∶𝑀→𝑀 with Φ(𝑝)=𝑝 and 𝑑Φ=𝐼⁻¹. Actually Φ is an isometry because (1) an isometric immersion between two complete Riemannian manifolds of the same dimension is a covering map by Theorem IV.4.6 in [7, page 176], and (2) a smooth covering map between two simply connected manifolds is a diffeomorphism.

This completes the proof of Proposition 1.2.

3. Heterogeneity and Transversality

Let us say that two 𝑘-jets, 𝐽^𝑘_𝑝𝑔 and 𝐽^𝑘_𝑞ℎ, of germs of Riemannian metrics 𝑔 and ℎ at points 𝑝 and 𝑞 in 𝑀 are equivalent if there is a germ of a diffeomorphism 𝑓 with 𝑓(𝑝)=𝑞 such that 𝐽^𝑘_𝑝(𝑓^∗(ℎ))=𝐽^𝑘_𝑝𝑔. We then define a Riemannian metric 𝑔 on 𝑀 to be heterogeneous of order 𝑘 if the 𝑘-jets of 𝑔 at any two distinct points of 𝑀 are not equivalent. Obviously, being heterogeneous of order 𝑘 for some 𝑘≥2 is a stronger condition than being simply heterogeneous. Let us proceed to explain how to express heterogeneity of order 𝑘 in terms of transversality when 𝑘 is sufficiently large.

Let 𝜋∶𝑋→𝑀 denote the the bundle of positive definite symmetric covariant 2-tensors over 𝑀. Thus sections of 𝑋 are just Riemannian metrics on 𝑀. Let 𝜋^𝑘∶𝑋^𝑘→𝑀 denote the bundle of 𝑘-jets of Riemannian metrics on 𝑀. Following [8, 9], let ₂𝜋^𝑘∶₂𝑋^𝑘→₂𝑀 denote the bundle of multijets of Riemannian metrics of order 𝑘 and multiplicity 2. Thus

₂𝑋^𝑘={(𝐽^𝑘_𝑝𝑔,𝐽^𝑘_𝑞ℎ)∈𝑋^𝑘×𝑋^𝑘∶𝑝≠𝑞},₂𝑀={(𝑝,𝑞)∈𝑀×𝑀∶𝑝≠𝑞}, (3.1) and ₂𝜋^𝑘(𝐽^𝑘_𝑝𝑔,𝐽^𝑘_𝑞ℎ)=(𝜋^𝑘(𝐽^𝑘_𝑝𝑔),𝜋^𝑘(𝐽^𝑘_𝑞ℎ))=(𝑝,𝑞). Given a Riemannian metric 𝑔, the multijet extension

₂𝐽^𝑘𝑔∶₂𝑀→₂𝑋^𝑟 (3.2) is defined by the formula ₂𝐽^𝑘𝑔(𝑝,𝑞)=(𝐽^𝑘_𝑝𝑔,𝐽^𝑘_𝑞𝑔) for (𝑝,𝑞)∈₂𝑀.

Since ₂𝑋^𝑘 is just the set of ordered pairs of 𝑘-jets of Riemannian metrics over distinct points of 𝑀, the equivalence relation on 𝑘-jets of Riemannian metrics on 𝑀 defines a subset 𝑆⊂₂𝑋^𝑘 consisting of pairs of equivalent 𝑘-jets. Obviously, 𝑔 is heterogeneous of order 𝑘 if and only if the image of its multijet extension ₂𝐽^𝑘𝑔(₂𝑀) misses the set 𝑆. In the next section we investigate the structure of the set 𝑆 and prove the following proposition.

Proposition 3.1. Let 𝑛 be the dimension of 𝑀. Then the set 𝑆 is a union of finitely many submanifolds of codimension at least (₂^𝑛+1)(_𝑘^𝑛+𝑘)−𝑛(^𝑛+𝑘+1_𝑘+1)+𝑛. In particular codim(𝑆)>2𝑛 in any of the three cases (i)𝑛=2 and 𝑘≥4, (ii)𝑛=3 and 𝑘≥3, or (iii)𝑛≥4 and 𝑘≥2.

As a corollary we obtain the following stronger version of Theorem 1.1.

Theorem 3.2. The set of Riemannian metrics on a manifold 𝑀 of dimension 𝑛, which are heterogeneous of order 𝑘, is residual in the space of Riemannian metrics on 𝑀 with the strong 𝐶^∞ topology as long any one of the three cases listed in Proposition 3.1 for which codim(𝑆)>2𝑛 holds.

Proof. Since 𝑆 is a union of submanifolds, application of the multijet transversality theorem (Corollary 3.4 [8] or [9, page 739]), shows that the set of 𝑔 for which ₂𝐽^𝑘𝑔 is transverse to 𝑆 forms a residual set in the space of Riemannian metrics with the strong 𝐶^∞ topology. But because dim(₂𝑀)=2𝑛 and codim(𝑆)>2𝑛, it follows that ₂𝐽^𝑘𝑔 is transverse to 𝑆 if and only if its image misses 𝑆, that is, if and only 𝑔 is heterogeneous of order 𝑘.
This completes the proof of Theorem 3.2. Theorem 1.1 is an immediate consequence.

4. The Structure of 𝑆

Consider the collection ℳ^𝑘(𝑛) of 𝑘-jets of Riemannian metrics on 𝐑^𝑛 at 0 and the so-called jet group 𝒢^𝑘+1(𝑛), consisting of the (𝑘+1)-jets of diffeomorphisms 𝑓∶𝐑^𝑛→𝐑^𝑛 satisfying 𝑓(0)=0 ([12, page 128]). The jet group acts upon ℳ^𝑘(𝑛) on the left by the formula 𝑓⋅𝑔=𝐽^𝑘₀((𝑓⁻¹)^∗𝑔) for 𝑓∈𝒢^𝑘+1(𝑛) and 𝑔∈ℳ^𝑘(𝑛). For any subgroup ℋ of 𝒢^𝑘+1(𝑛), let ℳ^𝑘_[ℋ](𝑛) denote the set of points on orbits of type 𝒢^𝑘+1(𝑛)/ℋ, that is,

ℳ^𝑘_[ℋ](𝑛)={𝑔∈ℳ^𝑘(𝑛)∶Theisotropygroupof𝑔isconjugatetoℋ}. (4.1) Finally, let ℳ^𝑘_[ℋ](𝑛) denote the orbit space 𝒢^𝑘+1(𝑛)⧵ℳ^𝑘_[ℋ](𝑛), and let 𝑄 denote the quotient projection 𝑄∶ℳ^𝑘_[ℋ](𝑛)→ℳ^𝑘_[ℋ](𝑛).

Proposition 4.1. ℳ^𝑘_[ℋ](𝑛) and ℳ^𝑘_[ℋ](𝑛) are smooth manifolds, and 𝑄 is a smooth fibration with fibers 𝒢^𝑘+1(𝑛)/ℋ. Moreover, there are only finitely many distinct orbit types, and these finitely many submanifolds ℳ^𝑘_[ℋ](𝑛) stratify ℳ^𝑘(𝑛).

Proof. If 𝒢^𝑘+1(𝑛) was compact, this proposition would follow from Theorem IV.3.3 in [10, page 182]. Although 𝒢^𝑘+1(𝑛) is not compact, its action on ℳ^𝑘(𝑛) reduces to a compatible action of the orthogonal group 𝑂(𝑛), which canonically includes as a subgroup of 𝒢^𝑘+1(𝑛), on the subset 𝒩^𝑘(𝑛) of ℳ^𝑘(𝑛). Here 𝒩^𝑘(𝑛) denotes the set of 𝑘-jets of Riemannian metrics 𝐑^𝑛 at 0 for which the standard coordinates on 𝐑^𝑛 form a normal coordinate system. These are the jets that satisfy the conditions (2.5.1-3) in [11] for 1≤𝑟≤𝑘. If one carries out the proof of Theorem 2.6 in [11] for 𝑘-jets, rather than for ∞-jets, one concludes that each 𝒢^𝑘+1(𝑛) orbit in ℳ^𝑘(𝑛) meets 𝒩^𝑘(𝑛) in an 𝑂(𝑛) orbit. Thus the inclusion of the orbit space 𝑂(𝑛)⧵𝒩^𝑘(𝑛) into 𝒢^𝑘+1(𝑛)⧵ℳ^𝑘(𝑛) is a one-to-one correspondence. This also implies that every isotropy subgroup for the 𝒢^𝑘+1(𝑛) action is conjugate to an isotropy subgroup of the 𝑂(𝑛) action. Thus there is a one-to-one correspondence between the orbit types of the two actions. In addition, we see that the isotropy subgroups of the 𝒢^𝑘+1(𝑛) action are compact. It also follows that every slice for the 𝑂(𝑛) action on 𝒩^𝑘_𝑛 is a slice for the 𝒢^𝑘+1(𝑛) action on ℳ^𝑘(𝑛) which proves that slices exist for the latter action. Because of these observations, the conclusions of Theorem IV.3.3 in [10] which apply to the action of 𝑂(𝑛) on 𝒩^𝑘(𝑛) also apply to the action of 𝒢^𝑘+1(𝑛) on ℳ^𝑘(𝑛). That there are only finitely many of orbit types follows in the same way from the well-known finiteness of orbit types for orthogonal actions [10, page 112]. This completes the proof of Proposition 4.1.

Let 𝒫^𝑘+1(𝑀) denote the principal 𝒢^𝑘+1(𝑛) bundle of the (𝑘+1)th order frames on the 𝑛-dimensional manifold 𝑀 [12, page 122]. Clearly, the bundle 𝑋^𝑘 of 𝑘-jets of Riemannian metrics on 𝑀 is the associated bundle 𝒫^𝑘+1(𝑀)×_𝒢^𝑘+1_(𝑛)ℳ^𝑘(𝑛). Since the stratification by orbit types of ℳ^𝑘(𝑛) is invariant under 𝒢^𝑘+1(𝑛), it induces a stratification of 𝑋^𝑘 where the typical stratum takes the form of the associated bundle

𝑋^𝑘_[ℋ]=𝒫^𝑘+1(𝑀)×_𝒢^𝑘+1_(𝑛)ℳ^𝑘_[ℋ](𝑛). (4.2) Moreover, the quotient map 𝑄 induces a smooth submersion 𝑋^𝑘_[ℋ]→ℳ^𝑘_[ℋ](𝑛).

Because 𝑆⊂𝑋^𝑘×𝑋^𝑘, we may set

𝑆_[ℋ]=𝑆∩(𝑋^𝑘_[ℋ]×𝑋^𝑘_[ℋ]). (4.3) Since two equivalent jets automatically have the same orbit type, we have

𝑆=_[ℋ]𝑆_[ℋ]. (4.4) Clearly, 𝑆_[ℋ] is just the inverse image of the diagonal in ℳ^𝑘_[ℋ](𝑛)×ℳ^𝑘_[ℋ](𝑛) under the product submersion 𝑋^𝑘_[ℋ]×𝑋^𝑘_[ℋ]→ℳ^𝑘_[ℋ](𝑛)×ℳ^𝑘_[ℋ](𝑛) induced by 𝑄. Therefore, 𝑆_[ℋ] is a smooth submanifold of 𝑋^𝑘_[ℋ]×𝑋^𝑘_[ℋ] whose codimension satisfies

codim(𝑆_[ℋ]⊂𝑋^𝑘_[ℋ]×𝑋^𝑘_[ℋ])=dim(ℳ^𝑘_[ℋ](𝑛)). (4.5) We may now compute the codimension of 𝑆_[ℋ] in ₂𝑋^𝑘:

codim(𝑆_[ℋ])=codim(𝑆_[ℋ]⊂𝑋^𝑘×𝑋^𝑘)=codim(𝑆_[ℋ]⊂𝑋^𝑘_[ℋ]×𝑋^𝑘_[ℋ])+2codim(𝑋^𝑘_[ℋ]⊂𝑋^𝑘)=dim(ℳ^𝑘_[ℋ](𝑛))+2codim(ℳ^𝑘_[ℋ](𝑛)⊂ℳ^𝑘(𝑛))=dim(ℳ^𝑘_[ℋ](𝑛))−dim(𝒢^𝑘+1(𝑛)/ℋ)+2(dim(ℳ^𝑘(𝑛))−dim(ℳ^𝑘_[ℋ](𝑛)))=dim(ℳ^𝑘(𝑛))−dim(𝒢^𝑘+1(𝑛))+dim(ℋ)+(dim(ℳ^𝑘(𝑛))−dim(ℳ^𝑘_[ℋ](𝑛)))≥dim(ℳ^𝑘(𝑛))−dim(𝒢^𝑘+1=2𝑘(𝑛))𝑛+1𝑛+𝑘−(𝑛𝑛+𝑘+1𝑘+1−𝑛). (4.6) In view of Proposition 4.1, this completes the proof of first statement in Proposition 3.1. The second statement of Proposition 3.1 follows directly from the first.

5. The Space of Complete Metrics

Since every Riemannian metric on a compact manifold is complete, Theorem 1.3 is an immediate consequence of Theorem 1.1 and Proposition 1.2 when 𝑀 is compact. If 𝑀 is not compact, the space of complete metrics on 𝑀 is a proper subspace of the space of all Riemannian metrics on 𝑀. Thus to prove Theorem 1.3 in general, we need to show that the subspace of complete metrics inherits the property of being a Baire space from the space of all metrics, and that an open dense subset of the space of all metrics intersects the subspace of complete metrics in an open dense subset of the subspace. Both of these statements are immediate consequences of the next proposition.

Proposition 5.1. The set of all complete Riemannian metrics on a given smooth manifold 𝑀 is both open and closed in the space of Riemannian metrics on 𝑀 with the strong 𝐶^∞ topology.

Proof. Fix a Riemannian metric 𝑔₀ on 𝑀. It suffices to show that there is a neighborhood 𝒰 of 𝑔₀ in the strong 𝐶^∞ topology consisting either entirely of complete metrics if 𝑔₀ is complete or entirely of incomplete metrics if 𝑔₀ is incomplete.
To this end, let 𝒰 be the set of all Riemannian metrics 𝑔 such that
12𝑔₀(𝑋,𝑋)<𝑔(𝑋,𝑋)<2𝑔₀(𝑋,𝑋) (5.1) for all 𝑋≠0 in 𝑇𝑀. Obviously, 𝒰 is an open neighborhood of 𝑔₀ in the strong 𝐶^∞ topology. Clearly, if 𝑔∈𝒰, and if dist and dist₀ are the associated distance functions corresponding to 𝑔 and 𝑔₀, respectively, then we have 12dist₀(𝑝,𝑞)≤dist(𝑝,𝑞)≤2dist₀(𝑝,𝑞) (5.2) for all 𝑝 and 𝑞 in 𝑀. It follows that dist and dist₀ have the same Cauchy sequences and the same convergent sequences with the same limits. Thus 𝑔 and 𝑔₀ are either both complete or both incomplete. This finishes the proof of Proposition 5.1.