Research Article | Open Access
Heterogeneous Riemannian Manifolds
We solve Ambrose's Problem for a generic class of Riemannian metrics on a smooth manifold, namely, the class of heterogeneous metrics.
We define a Riemannian metric on a manifold to be heterogeneous if no two distinct points of 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 of dimension is residual in the space of Riemannian metrics on with the strong topology.
Ambrose  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 is a complete, connected, simply connected, heterogeneous Riemannian manifold of dimension , then, for every , is determined up to isometry by the behavior of curvature under parallel transport along geodesics emanating from .
Theorem 1.3. The set of complete metrics on a connected, simply connected smooth manifold of dimension which, for every point in , are determined up to isometry by the behavior of curvature under parallel transport along geodesics emanating from , is residual in the space of complete Riemannian metrics on with the strong 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, -dimensional Riemannian manifold to be determined up to isometry by the behavior of curvature under parallel transport along geodesics emanating from a point in . Let be another complete, connected, simply connected, -dimensional Riemannian manifold. Let , and let be a linear isometry between the tangent spaces. For each geodesic satisfying , there is a corresponding geodesic satisfying characterized by the initial condition . Given such a geodesic , defines a linear isometry from onto where and denote parallel transport along and , respectively. Consider the hypothesis
for every such geodesic and for all vectors , where and are the respective Riemann curvature tensors. Then is determined up to isometry by the behavior of curvature under parallel transport along geodesics emanating from if the hypothesis (*) implies that there exists an isometry with and at (cf., [2, 5, 6]).
To prove the existence of , let in be any nonconjugate cut point to of order two, and let and be the two minimizing geodesics joining to , whose respective initial tangent vectors are and . Thus . Since is not conjugate to along or , there are neighborhoods of and of in for which carries both and diffeomorphically onto the same neighborhood of . As noted in [6, page 561], for , the map is an isometry (after possibly cutting down ) onto a neighborhood of . It follows that , because otherwise would be an isometry between neighborhoods of two distinct points of , contradicting the assumed heterogeneity of . This verifies the hypothesis of Lemma in , with the roles of and interchanged. Hence there exists an isometric immersion with and . Actually is an isometry because () an isometric immersion between two complete Riemannian manifolds of the same dimension is a covering map by Theorem in [7, page 176], and () 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 -jets, and , of germs of Riemannian metrics and at points and in are equivalent if there is a germ of a diffeomorphism with such that . We then define a Riemannian metric on to be heterogeneous of order if the -jets of at any two distinct points of are not equivalent. Obviously, being heterogeneous of order for some is a stronger condition than being simply heterogeneous. Let us proceed to explain how to express heterogeneity of order in terms of transversality when is sufficiently large.
Let denote the the bundle of positive definite symmetric covariant 2-tensors over . Thus sections of are just Riemannian metrics on . Let denote the bundle of -jets of Riemannian metrics on . Following [8, 9], let denote the bundle of multijets of Riemannian metrics of order and multiplicity 2. Thus
and . Given a Riemannian metric , the multijet extension
is defined by the formula for .
Since is just the set of ordered pairs of -jets of Riemannian metrics over distinct points of , the equivalence relation on -jets of Riemannian metrics on defines a subset consisting of pairs of equivalent -jets. Obviously, is heterogeneous of order if and only if the image of its multijet extension misses the set . In the next section we investigate the structure of the set and prove the following proposition.
Proposition 3.1. Let be the dimension of . Then the set is a union of finitely many submanifolds of codimension at least . In particular in any of the three cases (i) and , (ii) and , or (iii) and .
As a corollary we obtain the following stronger version of Theorem 1.1.
Theorem 3.2. The set of Riemannian metrics on a manifold of dimension , which are heterogeneous of order , is residual in the space of Riemannian metrics on with the strong topology as long any one of the three cases listed in Proposition 3.1 for which holds.
Proof. Since is a union of submanifolds, application of the multijet transversality theorem (Corollary  or [9, page 739]), shows that the set of for which is transverse to forms a residual set in the space of Riemannian metrics with the strong topology. But because and , it follows that is transverse to if and only if its image misses , that is, if and only is heterogeneous of order .
This completes the proof of Theorem 3.2. Theorem 1.1 is an immediate consequence.
4. The Structure of
Consider the collection of -jets of Riemannian metrics on at and the so-called jet group , consisting of the -jets of diffeomorphisms satisfying ([12, page 128]). The jet group acts upon on the left by the formula for and . For any subgroup of , let denote the set of points on orbits of type , that is,
Finally, let denote the orbit space , and let denote the quotient projection .
Proposition 4.1. and are smooth manifolds, and is a smooth fibration with fibers . Moreover, there are only finitely many distinct orbit types, and these finitely many submanifolds stratify .
Proof. If was compact, this proposition would follow from Theorem in [10, page 182]. Although is not compact, its action on reduces to a compatible action of the orthogonal group , which canonically includes as a subgroup of , on the subset of . Here denotes the set of -jets of Riemannian metrics at for which the standard coordinates on form a normal coordinate system. These are the jets that satisfy the conditions () in  for . If one carries out the proof of Theorem in  for -jets, rather than for -jets, one concludes that each orbit in meets in an orbit. Thus the inclusion of the orbit space into is a one-to-one correspondence. This also implies that every isotropy subgroup for the action is conjugate to an isotropy subgroup of the 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 action are compact. It also follows that every slice for the action on is a slice for the action on which proves that slices exist for the latter action. Because of these observations, the conclusions of Theorem in  which apply to the action of on also apply to the action of on . 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 denote the principal bundle of the th order frames on the -dimensional manifold [12, page 122]. Clearly, the bundle of -jets of Riemannian metrics on is the associated bundle . Since the stratification by orbit types of is invariant under , it induces a stratification of where the typical stratum takes the form of the associated bundle
Moreover, the quotient map induces a smooth submersion .
Because , we may set
Since two equivalent jets automatically have the same orbit type, we have
Clearly, is just the inverse image of the diagonal in under the product submersion induced by . Therefore, is a smooth submanifold of whose codimension satisfies
We may now compute the codimension of in :
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 is compact. If is not compact, the space of complete metrics on is a proper subspace of the space of all Riemannian metrics on . 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 is both open and closed in the space of Riemannian metrics on with the strong topology.
Proof. Fix a Riemannian metric on . It suffices to show that there is a neighborhood of in the strong topology consisting either entirely of complete metrics if is complete or entirely of incomplete metrics if is incomplete.
To this end, let be the set of all Riemannian metrics such that
for all in . Obviously, is an open neighborhood of in the strong topology. Clearly, if , and if and are the associated distance functions corresponding to and , respectively, then we have for all and in . It follows that and have the same Cauchy sequences and the same convergent sequences with the same limits. Thus and are either both complete or both incomplete. This finishes the proof of Proposition 5.1.
- T. Sunada, “Riemannian coverings and isospectral manifolds,” Annals of Mathematics, vol. 121, no. 1, pp. 169–186, 1985.
- W. Ambrose, “Parallel translation of Riemannian curvature,” Annals of Mathematics, Second Series, vol. 64, pp. 337–363, 1956.
- J. Hebda, “Metric structure of cut loci in surfaces and Ambrose's problem,” Journal of Differential Geometry, vol. 40, no. 3, pp. 621–642, 1994.
- J. Itoh, “The length of a cut locus on a surface and Ambrose's problem,” Journal of Differential Geometry, vol. 43, no. 3, pp. 642–651, 1996.
- J. Hebda, “Conjugate and cut loci and the Cartan-Ambrose-Hicks theorem,” Indiana University Mathematics Journal, vol. 31, no. 1, pp. 17–26, 1982.
- J. Hebda, “Parallel translation of curvature along geodesics,” Transactions of the American Mathematical Society, vol. 299, no. 2, pp. 559–572, 1987.
- S. Kobayashi and K. Nomizu, Foundations of Differential Geometry. Vol I, John Wiley & Sons, New York, NY, USA, 1963.
- J. Mather, “Stability of mappings. V. Transversality,” Advances in Mathematics, vol. 4, pp. 301–336, 1970.
- C. Wall, “Geometric properties of generic differentiable manifolds,” in Geometry and Topology, vol. 597 of Lecture Notes in Mathematics, pp. 707–774, Springer, Berlin, Germany, 1977.
- G. Bredon, Introduction to Compact Transformation Groups, Academic Press, New York, NY, USA, 1972.
- D. B. A. Epstein, “Natural tensors on Riemannian manifolds,” Journal of Differential Geometry, vol. 10, no. 4, pp. 631–645, 1975.
- I. Kolář, P. W. Michor, and J. Slovák, Natural Operations in Differential Geometry, Springer, Berlin, Germany, 1993.
Copyright © 2010 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.