Heterogeneous Riemannian Manifolds

Volume 2010, Article ID 187232,7pages doi:10.1155/2010/187232

Research Article

Heterogeneous Riemannian Manifolds

James J. Hebda

Department of Mathematics, Saint Louis University, St. Louis, MO 63103, USA

Correspondence should be addressed to James J. Hebda,[email protected]

Received 6 July 2009; Accepted 23 March 2010

Academic Editor: Wolfgang Kuehnel

Copyrightq2010 James J. Hebda. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 metricg on a manifoldMto be heterogeneous if no two distinct points ofMhave 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 onMwith the strongC∞topology.

A version ofTheorem 1.1for compact manifolds was stated and proved by Sunada1, Proposition 1.

Ambrose2asked 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. IfM, gis a complete, connected, simply connected, heterogeneous Riemannian manifold of dimension n ≥ 2, then, for everyp ∈ M,M, gis determined up to isometry by the behavior of curvature under parallel transport along geodesics emanating fromp.

Theorem 1.3. The set of complete metrics on a connected, simply connected smooth manifold M of dimensionn ≥ 2 which, for every pointp inM, 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 onMwith the strongC∞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 in5,6apply 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 manifoldM, gto 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 letI :TpM → TpMbe a linear isometry between the tangent spaces. For each

geodesicγ:0,1 → Msatisfyingγ0 p, there is a corresponding geodesicγ:0,1 → M

satisfying γ0 p characterized by the initial condition γ0 Iγ0. Given such a geodesic γ,Iγ Pγ ◦I ◦Pγ−1 defines a linear isometry from Tγ1M ontoTγ1M where Pγ

andPγdenote parallel transport alongγandγ, respectively. Consider the hypothesis

IγRX, YZ R

IγX, IγY

IγZ ∗

for every such geodesic γ and for all vectors X, Y, Z ∈ Tγ1M, 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 isometryΦ:M → MwithΦp panddΦ I

atpcf.,2,5,6.

In order to proveProposition 1.2, suppose thatM, gis heterogeneous, and assume the hypothesis∗. We will first find an isometryΦ:M → MwithΦp panddΦ I−1_at p. Thus we obtain the desired isometryΦby settingΦ Φ−1.

To prove the existence ofΦ, let qin Mbe any nonconjugate cut point topof order two, and letγ₁andγ₂be the two minimizing geodesics joiningptoq, whose respective initial tangent vectors areX1 γ10and X2 γ20. Thus exppX1 exppX2 q. Sinceqis

not conjugate topalongγ₁orγ₂, there are neighborhoodsV1ofX1andV2ofX2inTpMfor

which exp_pcarries bothV1 andV2 diffeomorphically onto the same neighborhoodW ofq.

As noted in6, page 561, fori ∈ {1,2}, the mapf_i exp_p◦I−1_◦exp

p |Vi−1 : W → M

is an isometryafter possibly cutting downWonto a neighborhoodUi ofγi1. It follows

thatγ11 γ21, because otherwisef₂◦f−₁1 : U1 → U2 would be an isometry between

neighborhoods of two distinct points ofM, contradicting the assumed heterogeneity ofM. This verifies the hypothesis of Lemma 2.1 in6, with the roles ofMandMinterchanged. Hence there exists an isometric immersionΦ:M → MwithΦp panddΦ 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 in7, page 176, and

3. Heterogeneity and Transversality

Let us say that twok-jets,Jk

pg andJqkh, of germs of Riemannian metricsg andhat points

p andqinM are equivalent if there is a germ of a diffeomorphism f with fp qsuch that Jk

pf∗h Jpkg. We then define a Riemannian metric g on M to be heterogeneous

of order k if thek-jets ofg at any two distinct points ofMare not equivalent. Obviously, being heterogeneous of orderk for some k ≥ 2 is a stronger condition than being simply heterogeneous. Let us proceed to explain how to express heterogeneity of orderkin terms of transversality whenkis sufficiently large.

Letπ : X → Mdenote the the bundle of positive definite symmetric covariant 2-tensors overM. Thus sections ofX are just Riemannian metrics onM. Letπk _{: X}k _{→ M}

denote the bundle ofk-jets of Riemannian metrics onM. Following8,9, let2πk : 2Xk → 2Mdenote the bundle of multijets of Riemannian metrics of orderkand multiplicity 2. Thus

2Xk

J_pkg, J_qkh∈Xk×Xk:p /q, 2M

p, q∈M×M:p /q , 3.1

and2πkJpkg, Jqkh πkJpkg, πkJqkh p, q. Given a Riemannian metricg, the multijet

extension

2Jkg: 2M−→ 2Xr 3.2

is defined by the formula2Jkgp, q Jpkg, Jqkgforp, q∈ 2M.

Since2Xkis just the set of ordered pairs ofk-jets of Riemannian metrics over distinct

points ofM, the equivalence relation onk-jets of Riemannian metrics onMdefines a subset

S ⊂ 2Xkconsisting of pairs of equivalentk-jets. Obviously,g is heterogeneous of orderkif

and only if the image of its multijet extension2Jkg2Mmisses the setS. In the next section

we investigate the structure of the setSand prove the following proposition.

Proposition 3.1. Letnbe the dimension ofM. Then the setSis a union of finitely many submanifolds of codimension at leastn1

2

_nk

k

−nnk1

k1

n. In particular codimS>2nin any of the three casesin2 andk≥4,iin3 andk≥3, oriiin≥4 andk≥2.

As a corollary we obtain the following stronger version ofTheorem 1.1.

Theorem 3.2. The set of Riemannian metrics on a manifold M of dimension n, which are heterogeneous of orderk, is residual in the space of Riemannian metrics onMwith the strongC∞ topology as long any one of the three cases listed inProposition 3.1for which codimS>2nholds.

Proof. SinceSis a union of submanifolds, application of the multijet transversality theorem

Corollary 3.4 8or9, page 739, shows that the set ofg for which2Jkg is transverse to Sforms a residual set in the space of Riemannian metrics with the strongC∞topology. But because dim2M 2nand codimS>2n, it follows that2Jkgis transverse toSif and only

if its image missesS, that is, if and onlygis heterogeneous of orderk.

4. The Structure of

S

Consider the collectionMk_{nofk-jets of Riemannian metrics onR}n_{at 0 and the so-called jet}

groupGk1_{n, consisting of thek1-jets of diffeomorphismsf :R}n _{→ R}n_satisfyingf0

012, page 128. The jet group acts uponMk_{non the left by the formulaf·gJ}k 0f−1

∗

g

forf∈ Gk1nandg∈ Mkn. For any subgroupHofGk1n, letMk_Hndenote the set of points on orbits of typeGk1_{n/H, that is,}

Mk Hn

g∈ Mkn: The isotropy group ofg is conjugate toH. 4.1

Finally, letMk_Hndenote the orbit spaceGk1n\ Mk_Hn, and letQdenote the quotient projectionQ:Mk

Hn → M k Hn.

Proposition 4.1. Mk

HnandM k

Hnare smooth manifolds, and Qis a smooth fibration with

fibersGk1_{n/H. Moreover, there are only finitely many distinct orbit types, and these finitely many}

submanifoldsMk_HnstratifyMkn.

Proof. If Gk1_{n was compact, this proposition would follow from Theorem IV.3.3 in 10,}

page 182. Although Gk1_{n is not compact, its action on M}k_{n reduces to a compatible}

action of the orthogonal groupOn, which canonically includes as a subgroup ofGk1_n,

on the subsetNk_nofMk_{n. HereN}k_{ndenotes the set ofk-jets of Riemannian metrics}

Rn_{at 0 for which the standard coordinates onR}n_{form a normal coordinate system. These are}

the jets that satisfy the conditions2.5.1-3in11for 1≤ r ≤ k. If one carries out the proof of Theorem 2.6 in11fork-jets, rather than for∞-jets, one concludes that eachGk1_norbit

inMk_nmeetsNk_{nin anOnorbit. Thus the inclusion of the orbit spaceOn\ N}k_n

intoGk1_{n\ M}k_{nis a one-to-one correspondence. This also implies that every isotropy}

subgroup for theGk1_{naction is conjugate to an isotropy subgroup of theOnaction. 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 Gk1_{naction are compact. It also follows that}

every slice for theOnaction onNk

nis a slice for theGk1naction onMknwhich proves

that slices exist for the latter action. Because of these observations, the conclusions of Theorem IV.3.3 in10which apply to the action ofOnonNk_{nalso apply to the action ofG}k1_n

onMk_{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 actions10, page 112. This completes the proof ofProposition 4.1.

LetPk1_{Mdenote the principalG}k1_{nbundle of thek1th order frames on the} n-dimensional manifoldM12, page 122. Clearly, the bundle Xk _{ofk-jets of Riemannian}

metrics onMis the associated bundlePk1M×Gk1_nMkn. Since the stratification by orbit

types ofMk_{nis invariant underG}k1_{n, it induces a stratification ofX}k_{where the typical}

stratum takes the form of the associated bundle

Xk_HPk1M×_Gk1_nMk_Hn. 4.2

BecauseS⊂Xk_×Xk_{, we may set}

S_HS∩Xk

H×XkH

. 4.3

Since two equivalent jets automatically have the same orbit type, we have

S

H

SH. _4.4

Clearly,SHis just the inverse image of the diagonal inMkHn×M k

Hnunder the product

submersionXk

H×XkH → M k

Hn× M k

Hninduced byQ. Therefore,SH is a smooth

submanifold ofXk

H×XkHwhose codimension satisfies

codimS_H⊂Xk_H×Xk_Hdim

Mk_Hn

. 4.5

We may now compute the codimension ofSHin2Xk:

codimSHcodim

SH⊂Xk×Xk

codimSH⊂Xk_H×Xk_H

2codimXk_H⊂Xk

dim

Mk_Hn

2 codimMk

Hn⊂ Mkn

dimMk_Hn−dimGk1n/H 2dimMkn−dimMk_Hn dimMkn−dimGk1ndimH

dimMkn−dimMk_Hn

≥dimMkn−dimGk1n

n1

2

nk

k − n

nk1

k1

−n

.

4.6

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 onMis a proper subspace of the space of all Riemannian metrics onM. Thus to proveTheorem 1.3in 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 manifoldMis both open and closed in the space of Riemannian metrics onMwith the strongC∞topology.

Proof. Fix a Riemannian metricg0onM. It suffices to show that there is a neighborhoodUof g0 in the strongC∞topology consisting either entirely of complete metrics ifg0is complete

or entirely of incomplete metrics ifg0is incomplete.

To this end, letUbe the set of all Riemannian metricsgsuch that 1

2g0X, X< gX, X<2g0X, X 5.1

for allX /0 inTM. Obviously,Uis an open neighborhood ofg0in the strongC∞topology.

Clearly, ifg ∈ U, and if dist and dist0are the associated distance functions corresponding to gandg0, respectively, then we have

1 2dist0

p, q≤distp, q≤2 dist0

p, q 5.2

for allpandqinM. It follows that dist and dist0have the same Cauchy sequences and the

same convergent sequences with the same limits. Thusg andg0are either both complete or

both incomplete. This finishes the proof ofProposition 5.1.

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