On focal stability in dimension two

MATHEMATICAL SCIENCES

On focal stability in dimension two

Mauricio M. Peixoto^I; Charles C. Pugh^II

^IInstituto de Matemática Pura e Aplicada (IMPA), Estrada Dona Castorina, 110, 22460-320 Rio de Janeiro, RJ, Brasil

^IIMathematics Department, University of Toronto, Toronto, Ontario, M5S 2E4, Canada

ABSTRACT

In Kupka et al. 2006 appears the Focal Stability Conjecture: the focal decomposition of the generic Riemann structure on a manifold M is stable under perturbations of the Riemann structure. In this paper, we prove the conjecture when M has dimension two, and there are no conjugate points.

Key words: geodesics, focal decomposition, focal stability, genericity, Teichmuller space.

RESUMO

Em Kupka et al. 2006, consideramos a Conjectura da Estabilidade Focal: a decomposição focal da estrutura Riemaniana genérica em uma variedade M é estável por perturbações dessa estrutura. No presente trabalho demonstramos essa conjectura quando M tem dimensão dois e não existem pontos conjugados.

Palavras-chave: geodésicas, decomposição focal, estabilidade focal, genetricidade, espaço de Teichmuller.

1 INTRODUCTION

Let M be a compact, smooth manifold of dimension m, and let = ^r be the space of C^r Riemann structures on M, equipped with the natural C^r topology, 2 < r < ¥. Fix p Î M. The k^thfocal component with respect to g Î at p is

where 1 < k < ¥, and ||, exp refer to the Riemann structure g.

The focal decomposition T_pM = _ks_k is said to be focally stable if a small perturbation of g has only a distant topological effect on s_k. Precisely, we require that given > 0 and given a compact set S Ì T_pM, there is a neighborhood of g in and there are balls B, B^'Ì T_pM such that for each g^'Î ,

(a) S Ì B Ç B^'.

(b) There is a homeomorphism h : B ® B^' that sends each s_k(g) Ç B onto s_k(g^') Ç B^'.

Thus, the focal decomposition s_k enjoys a kind of structural stability.

In Kupka et al. 2006 we investigated the concept of focal stability with an eye to proving the following

Focal Stability Conjecture: the generic Riemann structure is focally stable. (Since is an open subset of a complete metric space, genericity makes sense.) The main result of this paper concerns Riemann structures that have no conjugate points. It is most easily stated for the open set Ì of Riemann structures on TM whose Gauss curvature is everywhere negative. See Section 7 for a discussion of the more general case that g has no conjugate points.

THEOREM A. For a compact manifold of dimension two, the generic Riemann structure g Î is focally stable.

A different sort of result is also given. It concens surfaces of constant negative curvature. Fix a compact smooth surface of genus s > 2, such as the bitorus, and let denote the nonempty set of Riemann structures on M with curvature everywhere equal to -1. Since is a clsoed subset of , genericity in makes sense. Modulo isometric deformations is the Teichmuller space t_s.

THEOREM B.

(a) Fix g Î . For the generic p Î M, the focal decomposition of T_pM is stable with respect to perturbations of p in M.

(b) Fix p Î M. For the generic g Î , the focal decomposition of T_pM is stable with respect to perturbations of g within

.

2 MEDIATRICES

When the Riemann structure on M has non-positive curvature, there are no conjugate points and so exp_p : T_pM ® M is the universal covering space. Let be the lift of g to T_pM, and let be the corresponding metric on T_pM. The focal decomposition of T_pM can be described in terms of equidistance loci, called mediatrices by Bernhard and Veerman in Bernhard and Veerman 2006, as follows. A vector v₁Î s_k has k ''friends'' - vectors v₁, ... , v_k Î T_pM of equal length and equal exponential image. (A vector is always a friend of itself.) This implies that there are exactly k points in (p), one of which is the origin O_p of T_pM, and from which v₁ is equidistant with respect to the metric . See Figure 1.

3 A MULTITRANSVERSALITY RESULT

In Kupka et al. 2006, following Mather, we considered the multi-exponential map

where is the set of k-tuples of distinct nonzero vectors in T_pM, and exp, || refer to the Riemann structure g. The diagonal of (M ×)^k is

D = {(q, , ...,q, ): q Î M and Î }.

Theorem 6.1 of Kupka et al. 2006 states that if k > 3 then E^k is transverse to D. Here we need also the case k = 2. Although the proof becomes easier if we use a negative curvature hypothesis, we give the proof in general, since we hope to use the theorem as tool when M has conjugate points.

Theorem 3.1. E²: ×® (M ×)² is transverse to D.

PROOF. We give the proof in the case that M has dimension two, the main difference from the higher dimensional case being notational.

Lemma 6.3 of Kupka et al. 2006 states that, given L > 0, there is an open-dense set (p, L) Ì such that for g Î (p, L), there are at most a finite number of geodesic loops g at p having length < L, and that

(a) g is not a closed geodesic. (That is, the vectors tangent to g at its beginning and end are distinct.)

(b) g is ''single'' in the sense that it meets p only at its beginning and end, although other self-intersections are permitted.

(c) Under perturbation of g, g evolves continuously: it does not disappear or bifurcate.

Although some of the geodesic loops g may be self-conjugate in the sense that there is a transverse Jacobi field J along g that vanishes at both ends of g, a perturbation of g eliminates this feature. No such self-conjugacy can be created by a small perturbation of g, so we can restrict attention to Riemann structures in an open-dense subset ^*(p, L) Ì (p, L) that have no self-conjugate geodesic loops of length < L.

Let P = (v₁, v₂, g) Î × ^*(p, L) have E²-image Q = (q, , q, ) Î D. Let S be the sum S = Image(T_PE²) + T_QD. We must show

S = T_Q(M × )².

To do so we choose a basis of T_Q(M ×)² as follows.

The natural inclusions

induce isomorphisms

into the tangent space T_Q(M ×)².

We refer to the geodesics t ® exp(tv_j) as g_j, j = 1, 2 , and to their terminal tangent vectors as w₁ = (1), w₂ = (1). The time parameter t is always restricted to [0, 1]. Choose vectors u₁, u₂Î T_qM, normal to w₁, w₂. This gives bases {u₁, w₁}, {u₂, w₂} of T_qM, which the inclusions convert to a basis { e₁, f₁, h₁, e₂, f₂,h₂} of T_Q(M ×)²; namely

where h₁, h₂ are tangent to the appropriate factor in (M ×)².

CASE 1. The geodesics g₁, g₂ are unequal pointsets. We will show that E² is submersive at P, i.e., that

Image T_PE² = T_Q(M × )².

Because g₁, g₂ have the same length, neither contains the other, so there are ''free spots'' - points z₁Î g₁ \g₂ and z₂Î g₂ \g₁. (Note that even for the generic g, q may be conjugate to p along these geodesics.) See Figure 2.

Lemma 6.2 in Kupka et al. 2006 states that perturbation of g in the neighborhood of the free spots causes free and independent motion of the endpoints of g₁, g₂. Furthermore, perturbation of v₁ along itself makes ₁ = |v₁| vary linearly; and yoked to this variation of ₁, the endpoint q₁ = g₁(1) varies dependently along f₁. The corresponding facts hold for v₂, so we see that the image of T_PE² contains the vectors

e₁, f₁, f₁ + h₁, e₂, f₂, f₂ + h₂,

which is a basis for T_Q(M×)². This demonstrates that E² is submersive at P. Submersivity implies transversality.

CASE 2. g₁, g₂ are equal as point sets - they are merely the same geodesic loop g at p, traversed in opposite directions. See Figure 3. This implies that there are no free spots, so perturbation of the Riemann structure is futile.

Because g is a geodesic loop, but not a closed geodesic, the terminal vectors w₁, w₂ are linearly independent. Since they have equal length and are perpendicular to u₁, u₂, the coefficients b, d in the expression

satisfy

Because the loop g is not self-conjugate, variation of v₁ perpendicular to itself produces free and independent variation of the endpoint q₁ = g₁(1) perpendicular to w₁. The same is true for v₂. Thus, the image of T_PE² contains the vectors e₁, e₂. As in Case 1, variation of v₁, v₂ along themselves gives vectors f₁ + h₁ , f₂ +h₂ in the image of T_PE². Altogether, then, we have four vectors

e₁, f₁ + h₁, e₂, f₂ + h₂Î T_PE².

The curves

are contained in D, and hence T_QD contains their tangents at t = 1, namely, 4

The linear combination

d = (1) - b(1) = (1 - bd) f₁ + ae₂ - bce₁

of these vectors is tangent to D. By (1), this gives an explicit expression for f₁Î S = T_PE² +T_QD as

In the same way, f₂ belongs to S, and so do

Since S contains the whole basis {e₁, ... , h₂}, it equals T_Q(M ×)², which completes the proof in Case 2.

COROLLARY 3.2. For the generic g Î , and for all k > 1, the multiexponential : ® (M ×)^k is transverse to the diagonal D. When M has dimension two, the pre-image of D is empty for k > 4, is a discrete set of points for k = 3, and is a smooth 1-manifold for k = 2.

PROOF. The Abraham Transversality Theorem asserts that if a smooth map

F : X × ® Y É W

is transverse to W where is a Banach manifold and X, Y, W are finite dimensional, then for all a in a resdual subset of , the map

F(a, ) : X ® Y É W

is transverse to W. In our case, R is an open set of a Banach space, and we know that

E² : × V²® (M × )²É D

is transverse to D. Thus, for the generic g Î , is transverse to D.

When k = 1 , transversality is trivial because the diagonal coincides with M ×, while for k > 3, transversality is proved in Kupka et al. 2006, Theorem 6.1.

Now assume that M has dimension two. The codimension of D in (M×)^k is 3k -3, and the dimension of is 2k. Thus, if k > 4 then the codimension in the target exceeds the domain dimension, so transverse intersection implies empty intersection: () ÇD = Æ. Similarly, because transversality preserves codimension, the pre-image of D under is a discrete set of points in , and the pre-image of D under is a 1-manifold in .

4 FOCAL BRANCHES

Fix a g Î and let

n^k = ()^-1(D) = {(v, . . . , v_k) Î : (v₁, . . . , v_k) Î D}.

Clearly n^k is invariant under permutation of the factors T_pM in . Thus, if p_j : ® T_pM projects onto the j^th factor,

b^k = p_j (n^k)

is independent of j. Furthermore, b²É b³É b⁴É ... and,

PROPOSITION 4.1. When M has no conjugate points, the projection p_j: n^k® T_pM is a proper immersion onto a closed subset of T_pM.

PROOF. Properness means that the pre-image of a compact set is compact. Thus, from any given a sequence (v_1n, ... ,v_kn) in n^k such that for some fixed j, v_jn converges in T_pM as n ® ¥, we must extract a subsequence, convergent in n^k.

When k = 1 the assertion is trivial since the projection is the identity map. Thus we assume k > 2.

Convergence of v_jn, say to v_j Î T_pM, implies that |v_jn| ® |v_j|= . Since all the other v_in have the same length, there is a subsequence (unrelabeled) such that (v_1n, ... , v_kn) ® (v₁, ..., v_k). Each v_i has length . Fix i ¹ j. Then v_in ¹ v_jn. Since exp_p(v_in) = exp_p(v_jn), the facts that k > 2 and that exp is a local diffeomorphism from a neighborhood of the origin in T_pM to a neighborhood of p in M implies that ¹ 0. Also, since there are no conjugate points, exp is a local diffeomorphism at v_j, which implies that v_i ¹ v_j. Thus (v₁, ... , v_k) Î n^k, which completes the proof of properness.

A continuous proper map into a metric space necessarily has a closed range. Hence p_j(n^k) is closed in T_pM.

To check that p_j is an immersion, we must show that the projection v_j(t) of each nonsingular curve (v₁(t), ... , v_k(t)) in n^k is nonsingular in T_pM. Fix a t₀Î (a, b) where (a, b) is the curve's domain of definition. For at least one i, _i(t₀) ¹ 0. Thus, v_i(t) is nonsingular at t₀. Since there are no conjugate points, exp(v_i(t)) is also nonsingular at t₀. Since (v₁(t), ... , v_k(t)) Î n^k, exp(v_j(t)) = exp(v_i(t)) is also nonsingular at t₀. Therefore, v_j(t) is nonsingular at t₀.

5 PROOF OF THEOREM A

We assume that M is a compact surface of genus > 2, that p Î M is fixed, and we denote by the nonempty set of Riemann structures on TM having negative curvature. Clearly, is an open subset of and so it makes sense to speak of the generic g Î .

A Riemann structure with negative curvature has no conjugate points. Thus, according to Corollary 3.2 and Proposition 4.1, the focal decomposition of T_pM is quite simple for the generic g Î N. Namely:

(a) For all k > 4, s_k is empty.

(b) s₃ = b³ is a discrete subset of T_pM.

(c) s₂ = b² \b³ and b² consists of a closed set of immersed curves in T_pM.

Furthermore, in any fixed compact subset of T_pM, properties (a), (b), (c) remain valid for all small perturbations of g.

Consider a vector v₁Î b³. It has two friends v₂, v₃Î b³ with equal length and equal exponential image. Thus (v₁, v₂) Î n², and there are nonsingular curves v₁(t), v₂(t) with

Likewise there are nonsingular curves (t), v₃(t) with

We claim that

₁(0) and (0) are linearly independent. Suppose not. Nonsingularity implies that there is a c ¹ 0 such that

At time t = 0 the curve exp(v₃(ct)) has tangent

Similarly,

Thus, at t = 0

t(v₁(t), v₂(t), v₃(ct))

is tangent to the diagonal D Ì (M ×)³. The upshot is that the range of T_{(v1, v2, v3)} contains a nonzero vector tangent to the diagonal in (M×)³. This contradicts the fact that : ® (M×)³ is transverse to D, since D has codimension 6, which is the same as the dimension of .

Now we know that

₁(0) and (0) are linearly independent. This means that in addition to properties (a) - (c), above, we have

(d) Branches of s₂ meet transversally in pairs, they do so only at points of s₃, and every point of s₃ is such a crossing point,

where by a branch of s₂ we mean the projection of an arc in n². Since transversality is an open property, (d) also remains valid under perturbation of the Riemannn structure.

The remainder of the proof of focal stability follows the pattern of Theorem 5.1 in Kupka et al. 2006. Fix a compact set S Ì T_pM. Then choose a disc B in T_pM that contains S. We know that the focal decomposition amounts to a smooth one-dimensional graph, namely G = s₂Ès₃, which has transverse crossings of multiplicity two. We adjust B so that its boundary is transverse to G. Let g^' be a small perturbation of g and let G^' be the corresponding graph. Since all aspects of the graph depend continuously on the Riemann structure, and all are transverse, if g^' is sufficiently close to g, then there exists a homeomorphism from B to itself that sends GÇB to G^'ÇB.

6 PROOF OF THEOREM B

We assume that M is a compact surface of genus > 2 and we denote by Ì the nonempty set of Riemann structures whose curvature is identically equal to -1.

Fix g Î and p Î M. As described in section 2, we can lift g to a Riemann structure on T_pM and view the focal decomposition in terms of mediatrices for the corresponding metric . Since g has constant negative curvature, is isometric to the Poincaré metric r on the unit disc , and mediatrices are r-geodesics. As such, mediatrices are circular arcs meeting ¶ perpendicularly. Thus, any two mediatrices meet one another transversally, and they do so at most once.

Let P be the lattice of pre-images of p in T_pM, and denote the corresponding mediatrix set as

µ = {v Î T_pM : for some Î P \ {0} and |v| = r(v, )}.

Fix a compact set S Ì T_pM. At most finitely many µ-mediatrices meet S. Choose a constant R and let B denote the compact disc

T_pM(R) = {v Î T_pM : |v| = R}.

When R is large, B contains S in its interior and we can adjust R so that ¶B is transverse to the µ-mediatrices. Let t be the finite set of points in B at which the µ-mediatrices intersect one another. Thus

t = (s₃È s₄È· · · È ) Ç B

for some finite . If v Î s_k, there are k-1 µ-mediatrices that pass through it. They are pairwise transverse. A small change of p preserves all transversalities in the disc of radius R; such a perturbation of the base point can not increase the multiplicity of a vector in t, although it may lower it. (Here is where the argument uses the fact that the curvature is constant - mediatrices in the constant curvature case are always transverse to one another.) Thus, there is an open-dense set U Ì M such that if p Î U and p^' is sufficiently near p then all multiplicities of the µ^'-mediatrices in B^' = (R) are the same as those in B. (By µ^' we denote the mediatrices between the origin of and the other lifts of p^' in .) In other words, the graph of µ^'-mediatrices in B^' is homeomorphic to the graph of µ-mediatrices in B. Taking a sequence of compact sets S_n that exhausts T_pM leads to a sequence of such open dense sets U_n in M, and if p Î ÇU_n then the focal decomposition in T_pM is stable with respect to perturbation of the base point p. This completes the proof of the first assertion in Theorem B.

The second assertion in Theorem B is proved in the same way. Again, mediatrix transversality implies that perturbation of g can only decrease intersection multiplicity, it cannot increase it. Thus, there is an open dense set in Ì such that if g Î and g^'Î is near enough to g then the µ^'-mediatrix graph in B is homeomorphic to the µ-mediatrix graph in B. Again, choosing a sequence of compact sets S_n that exhausts T_pM leads to a sequence of open dense sets _n in , and if g Î Ç_n then the focal decomposition of T_pM is stable with respect to perturbation of g within .

REMARK. Theorem B does not assert the multiplicity of the focal decomposition of the generic g Î is at most three. We believe, however, that such an assertion is correct, and we can phrase our expectation as follows. If M has genus s > 2 then the Teichmuller space t_s of hyperbolic structures on M amounts to / where denotes isometric deformations. It is a space smoothly parameterized by 6(s - 1) real variables, and we expect that for a residual subset of these parameter values the corresponding hyperbolic structure has s_k = Æ for all k > 4. From this it would follow at once that the generic g Î is focally stable with respect to variation of g in , not just with respect variation of g within , as in Theorem B.

7 CONJUGATE POINTS

A Riemann structure with non-negative curvature has no conjugate points, but the set of such Riemann structures does not form an open subset of . For example, a flat Riemann structure on the torus has no conjugate points although it can be perturbed so that conjugate points appear. See Kupka et al. 2006, Proposition 4.8, where a bump is glued to a cylinder. Thus, the assertion of Theorem A^', below, should be viewed with caution, for a generic subset of need not be dense in .

THEOREM A^'. Focal stability (for a fixed p Î M) holds for the generic g Î .

PROOF. In the proof of Theorem A, we only used the assumption that exp_p is a local diffeomorphism, i.e., that there are no conjugate points, and the fact that the generic g Î stably possess the transversality properties (a)-(d).

The next result shows that Theorem A^' has wider scope than Theorem A.

PROPOSITION 7.1. The interior of

is strictly larger than

.

PROOF. Let M be the bitorus, or any surface with genus > 2. Equip it with a Riemann structure of constant curvature -1. Cut out a small disc in M and replace it with a small polar cap having unit positive curvature. A smoothing collar is used to attach the cap. This gives a new Riemann structure g on M, g Ï . Any g-geodesic spends relatively little time in the polar cap or collar. Most of the time the geodesic travels through the part of the surface with curvature -1. Thus, there are no conjugate points, so g belongs to , and the same holds for all nearby Riemann structures.

REMARK. The question remains as to whether the generic Riemann structure on a surface has the focal stability property - i.e., whether we can do without the no conjugate point assumption. Much of what we proved above does hold when there are conjugate points, and also some generic properties of conjugate points are known. For example, in Weinstein 1970, Alan Weinstein announces that in dimension two, the singularities of the generic exponential map are either folds or cusps. These are the Whitney singularities for maps of the plane to itself. If, in addition to this, we knew how the fold and cusp singularities relate to the foliation of T_pM by circles of constant radius, then we could probably resolve the Focal Stability Conjecture for surfaces. In higher dimensions the singularities of the generic exponential map are much more complicated than in dimension two, cf. Weinstein 1970, which leads us to think that the Focal Stability Conjecture will be quite hard to resolve in full generality.

ACKNOWLEDGMENTS

Both authors were partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq).

Manuscript received on August 3, 2006; accepted for publication on October 2, 2006; contributed by MAURICIO M. PEIXOTO* AND CHARLES C. PUGH*

* Member Academia Brasileira de Ciências

Correspondence to: Prof. Mauricio M. Peixoto

E-mail: [e-mail]

• KUPKA I, PEIXOTO MM AND PUGH C. 2006. Focal Stability of Riemann Metrics. J Reine Angew Math 593: 31-72.
• BERNHARD J AND VEERMAN JJP. 2006. Minimally Separating Sets, Mediatrices and Brillouin Spaces. Topol Appl 153: 1421-1433.
• WEINSTEIN A. 1970. The generic conjugate locus. Global Analysis, Proceedings of Symposia in Pure Mathematics. Am Math Soc 15: 299-301.

Publication Dates

• Publication in this collection
  23 Mar 2007
• Date of issue
  Mar 2007

History

• Accepted
  02 Oct 2006
• Received
  03 Aug 2006