# Heterogeneous Riemannian Manifolds

Authors: James J. Hebda
Journal: International Journal of Mathematics and Mathematical Sciences
 (2010)

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## Abstract

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


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## Body

## 1. Introduction

We define a Riemannian metricg 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 manifoldM of dimension n≥2 is residual in the space of Riemannian metrics on M with the strong C∞ topology.A version of Theorem1.1 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.Proposition1.2 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 manifoldM 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 (M̅,g̅) be another complete, connected, simply connected, n-dimensional Riemannian manifold. Let p̅∈M̅, and let I:TpM→Tp̅M̅ be a linear isometry between the tangent spaces. For each geodesic γ:[0,1]→M satisfying γ(0)=p, there is a corresponding geodesic γ̅:[0,1]→M̅ satisfying γ̅(0)=p̅ characterized by the initial condition γ̅′(0)=I(γ'(0)). Given such a geodesic γ, Iγ=Pγ̅∘I∘Pγ-1 defines a linear isometry from Tγ(1)M onto Tγ̅(1)M̅ where Pγ and Pγ̅ denote parallel transport along γ and γ̅, respectively. Consider the hypothesis(*)Iγ(R(X,Y)Z)=R̅(Iγ(X),Iγ(Y))Iγ(Z)
for every such geodesic γ and for all vectors X,Y,Z∈Tγ(1)M, where R and R̅ 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→M̅ with Φ(p)=p̅ and dΦ=I at p (cf., [2, 5, 6]).In order to prove Proposition1.2, suppose that (M,g) is heterogeneous, and assume the hypothesis (*). We will first find an isometry Φ̅:M̅→M with Φ̅(p̅)=p and dΦ̅=I-1 at p̅. Thus we obtain the desired isometry Φ by setting Φ=Φ̅-1.To prove the existence ofΦ̅, let q̅ in M̅ be any nonconjugate cut point to p̅ of order two, and let γ̅1 and γ̅2 be the two minimizing geodesics joining p̅ to q̅, whose respective initial tangent vectors are X̅1=γ̅1′(0) and X̅2=γ̅2′(0). Thus expp̅(X̅1)=expp̅(X̅2)=q̅. Since q̅ is not conjugate to p̅ along γ̅1 or γ̅2, there are neighborhoods V1 of X̅1 and V2 of X̅2 in Tp̅M̅ for which expp̅ carries both V1 and V2 diffeomorphically onto the same neighborhood W of q̅. As noted in [6, page 561], for i∈{1,2}, the map f̅i=expp∘I-1∘(expp̅∣Vi)-1:W→M is an isometry (after possibly cutting down W) onto a neighborhood Ui of γi(1). It follows that γ1(1)=γ2(1), because otherwise f̅2∘f̅1-1: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 M̅ interchanged. Hence there exists an isometric immersion Φ̅:M̅→M with Φ̅(p̅)=p and dΦ̅=I-1. Actually Φ̅ 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 Proposition1.2.


## 3. Heterogeneity and Transversality

Let us say that twok-jets, Jpkg and Jqkh, 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 Jpk(f*(h))=Jpkg. 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.Letπ: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 πk:Xk→M denote the bundle of k-jets of Riemannian metrics on M. Following [8, 9], let 2πk:X2k→M2 denote the bundle of multijets of Riemannian metrics of order k and multiplicity 2. Thus(3.1)2Xk={(Jpkg,Jqkh)∈Xk×Xk:p≠q},2M={(p,q)∈M×M:p≠q},
and 2πk(Jpkg,Jqkh)=(πk(Jpkg),πk(Jqkh))=(p,q). Given a Riemannian metric g, the multijet extension(3.2)2Jkg:2M→X2r
is defined by the formula 2Jkg(p,q)=(Jpkg,Jqkg) for (p,q)∈M2.Since2Xk 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⊂2Xk consisting of pairs of equivalent k-jets. Obviously, g is heterogeneous of order k if and only if the image of its multijet extension 2Jkg(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.
Letn be the dimension of M. Then the set S is a union of finitely many submanifolds of codimension at least (n+12)(n+kk)-n(n+k+1k+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 Theorem1.1.Theorem 3.2.
The set of Riemannian metrics on a manifoldM 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.
SinceS 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 2Jkg is transverse to S forms a residual set in the space of Riemannian metrics with the strong C∞ topology. But because dim(M2)=2n and codim(S)>2n, it follows that 2Jkg 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 Theorem3.2. Theorem 1.1 is an immediate consequence.


## 4. The Structure ofS

Consider the collectionℳk(n) of k-jets of Riemannian metrics on Rn at 0 and the so-called jet group 𝒢k+1(n), consisting of the (k+1)-jets of diffeomorphisms f:Rn→Rn satisfying f(0)=0 ([12, page 128]). The jet group acts upon ℳk(n) on the left by the formula f·g=J0k((f-1)*g) for f∈𝒢k+1(n) and g∈ℳk(n). For any subgroup ℋ of 𝒢k+1(n), let ℳ[ℋ]k(n) denote the set of points on orbits of type 𝒢k+1(n)/ℋ, that is,(4.1)ℳ[ℋ]k(n)={g∈ℳk(n):Theisotropygroupofgisconjugatetoℋ}.
Finally, let ℳ̅[ℋ]k(n) denote the orbit space 𝒢k+1(n)∖ℳ[ℋ]k(n), and let Q denote the quotient projection Q:ℳ[ℋ]k(n)→ℳ̅[ℋ]k(n).Proposition 4.1.
ℳ[ℋ]k(n) and ℳ̅[ℋ]k(n) are smooth manifolds, and Q is a smooth fibration with fibers 𝒢k+1(n)/ℋ. Moreover, there are only finitely many distinct orbit types, and these finitely many submanifolds ℳ[ℋ]k(n) stratify ℳk(n).Proof.
If𝒢k+1(n) was compact, this proposition would follow from Theorem IV.3.3 in [10, page 182]. Although 𝒢k+1(n) is not compact, its action on ℳk(n) reduces to a compatible action of the orthogonal group O(n), which canonically includes as a subgroup of 𝒢k+1(n), on the subset 𝒩k(n) of ℳk(n). Here 𝒩k(n) denotes the set of k-jets of Riemannian metrics Rn at 0 for which the standard coordinates on Rn 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 𝒢k+1(n) orbit in ℳk(n) meets 𝒩k(n) in an O(n) orbit. Thus the inclusion of the orbit space O(n)∖𝒩k(n) into 𝒢k+1(n)∖ℳk(n) is a one-to-one correspondence. This also implies that every isotropy subgroup for the 𝒢k+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 𝒢k+1(n) action are compact. It also follows that every slice for the O(n) action on 𝒩nk is a slice for the 𝒢k+1(n) action on ℳk(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 𝒩k(n) also apply to the action of 𝒢k+1(n) on ℳk(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.Let𝒫k+1(M) denote the principal 𝒢k+1(n) bundle of the (k+1)th order frames on the n-dimensional manifold M [12, page 122]. Clearly, the bundle Xk of k-jets of Riemannian metrics on M is the associated bundle 𝒫k+1(M)×𝒢k+1(n)ℳk(n). Since the stratification by orbit types of ℳk(n) is invariant under 𝒢k+1(n), it induces a stratification of Xk where the typical stratum takes the form of the associated bundle(4.2)X[ℋ]k=𝒫k+1(M)×𝒢k+1(n)ℳ[ℋ]k(n).
Moreover, the quotient map Q induces a smooth submersion X[ℋ]k→ℳ̅[ℋ]k(n).BecauseS⊂Xk×Xk, we may set(4.3)S[ℋ]=S∩(X[ℋ]k×X[ℋ]k).
Since two equivalent jets automatically have the same orbit type, we have(4.4)S=⋃[ℋ]S[ℋ].
Clearly, S[ℋ] is just the inverse image of the diagonal in ℳ̅[ℋ]k(n)×ℳ̅[ℋ]k(n) under the product submersion X[ℋ]k×X[ℋ]k→ℳ̅[ℋ]k(n)×ℳ̅[ℋ]k(n) induced by Q. Therefore, S[ℋ] is a smooth submanifold of X[ℋ]k×X[ℋ]k whose codimension satisfies(4.5)codim(S[ℋ]⊂X[ℋ]k×X[ℋ]k)=dim(ℳ̅[ℋ]k(n)).
We may now compute the codimension of S[ℋ] in 2Xk:(4.6)codim(S[ℋ])=codim(S[ℋ]⊂Xk×Xk)=codim(S[ℋ]⊂X[ℋ]k×X[ℋ]k)+2codim(X[ℋ]k⊂Xk)=dim(ℳ̅[ℋ]k(n))+2codim(ℳ[ℋ]k(n)⊂ℳk(n))=dim(ℳ[ℋ]k(n))-dim(𝒢k+1(n)/ℋ)+2(dim(ℳk(n))-dim(ℳ[ℋ]k(n)))=dim(ℳk(n))-dim(𝒢k+1(n))+dim(ℋ)+(dim(ℳk(n))-dim(ℳ[ℋ]k(n)))≥dim(ℳk(n))-dim(𝒢k+1(n))=(n+12)(n+kk)-(n(n+k+1k+1)-n).
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, Theorem1.3 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 manifoldM is both open and closed in the space of Riemannian metrics on M with the strong C∞ topology.Proof.
Fix a Riemannian metricg0 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, let𝒰 be the set of all Riemannian metrics g such that
(5.1)12g0(X,X)<g(X,X)<2g0(X,X)
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
(5.2)12dist0(p,q)≤dist(p,q)≤2dist0(p,q)
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.



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