On Focal stability in dimension two

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.


INTRODUCTION
Let M be a compact, smooth manifold of dimension m, and let R = R r be the space of C r Riemann structures on M, equipped with the natural C r topology, 2 ≤ r ≤ ∞.Fix p ∈ M. The k th focal component with respect to g ∈ R at p is where 1 ≤ k ≤ ∞, and | |, exp refer to the Riemann structure g.
The focal decomposition T p M = k σ k is said to be focally stable if a small perturbation of g has only a distant topological effect on σ k .Precisely, we require that given > 0 and given a compact set S ⊂ T p M, there is a neighborhood U of g in R and there are balls B, B ⊂ T p M such that for each g ∈ U, 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 R 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 N ⊂ R of Riemann structures on T M 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 ∈ N 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 H denote the nonempty set of Riemann structures on M with curvature everywhere equal to −1.Since H is a clsoed subset of R, genericity in H makes sense.Modulo isometric deformations H is the Teichmuller space τ s .

THEOREM B.
(a) Fix g ∈ H.For the generic p ∈ M, the focal decomposition of T p M is stable with respect to perturbations of p in M.
(b) Fix p ∈ M. For the generic g ∈ H, the focal decomposition of T p M is stable with respect to perturbations of g within H.

MEDIATRICES
When the Riemann structure on M has non-positive curvature, there are no conjugate points and so exp p : T p M → M is the universal covering space.Let ḡ be the lift of g to T p M, and let d be the corresponding metric on T p M. The focal decomposition of T p M can be described in terms of equidistance loci, called mediatrices by Bernhard and Veerman in Bernhard and Veerman 2006, as follows.A vector v 1 ∈ σ k has k " friends" -vectors v 1 , . . ., v k ∈ T p M of equal length and equal exponential image.(A vector is always a friend of itself.)This implies that there are exactly k points in exp −1 p ( p), one of which is the origin O p of T p M, and from which v 1 is equidistant with respect to the metric d.See Figure 1.

A MULTITRANSVERSALITY RESULT
In Kupka et al. 2006, following Mather, we considered the multi-exponential map where V k p is the set of k-tuples of distinct nonzero vectors in T p M, and exp, | | refer to the Riemann structure g.The diagonal of (M × R) k is = {(q, , . . ., q, ) : q ∈ M and ∈ R}.Theorem 6.1 of Kupka et al. 2006 states that if k ≥ 3 then E k is transverse to .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.
2 is transverse to .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 G( p, L) ⊂ R such that for g ∈ G( p, L), there are at most a fi nite number of geodesic loops γ at p having length ≤ L, and that (a) γ is not a closed geodesic.(That is, the vectors tangent to γ at its beginning and end are distinct.)(b) γ 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, γ evolves continuously: it does not disappear or bifurcate.
Although some of the geodesic loops γ may be self-conjugate in the sense that there is a transverse Jacobi fi eld J along γ that vanishes at both ends of γ , 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 G * ( p, L) ⊂ G( p, L) that have no self-conjugate geodesic loops of length ≤ L.
To do so we choose a basis of T Q (M × R) 2 as follows.
The natural inclusions We refer to the geodesics t → exp(tv j ) as γ j , j = 1, 2, and to their terminal tangent vectors as where h 1 , h 2 are tangent to the appropriate factor R in (M × R) 2 .CASE 1.The geodesics γ 1 , γ 2 are unequal pointsets.We will show that E 2 is submersive at P, i.e., that Because γ 1 , γ 2 have the same length, neither contains the other, so there are " free spots" -points z 1 ∈ γ 1 \γ 2 and z 2 ∈ γ 2 \ γ 1 .(Note that even for the generic g, q may be conjugate to p along these geodesics.)See Figure 2.
Fig. 2 -Varying g at a free spot z controls the endpoint q = γ (1) of the geodesic γ .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 γ 1 , γ 2 .Furthermore, perturbation of v 1 along itself makes 1 = |v 1 | vary linearly; and yoked to this variation of 1 , the endpoint q 1 = γ 1 (1) varies dependently along f 1 .The corresponding facts hold for v 2 , so we see that the image of T P E 2 contains the vectors which is a basis for T Q (M × R) 2 .This demonstrates that E 2 is submersive at P. Submersivity implies transversality.CASE 2. γ 1 , γ 2 are equal as point sets -they are merely the same geodesic loop γ 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 γ is a geodesic loop, but not a closed geodesic, the terminal vectors w 1 , w 2 are linearly independent.Since they have equal length and are perpendicular to u 1 , u 2 , the coeffi cients b, d in the expression Because the loop γ is not self-conjugate, variation of v 1 perpendicular to itself produces free and independent variation of the endpoint q 1 = γ 1 (1) perpendicular to w 1 .The same is true for v 2 .Thus, the image of T P E 2 contains the vectors e 1 , e 2 .As in Case 1, variation of v 1 , v 2 along themselves gives vectors f 1 + h 1 , f 2 + h 2 in the image of T P E 2 .Altogether, then, we have four vectors are contained in , and hence T Q contains their tangents at t = 1, namely, of these vectors is tangent to .By (1), this gives an explicit expression for In the same way, f 2 belongs to S, and so do Since S contains the whole basis {e 1 , . . ., h 2 }, it equals T Q (M × R) 2 , which completes the proof in Case 2.
COROLLARY 3.2.For the generic g ∈ R, and for all k ≥ 1, the multiexponential E k g : V k p → (M × R) k is transverse to the diagonal .When M has dimension two, the pre-image of 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 is transverse to W where A is a Banach manifold and X, Y, W are fi nite dimensional, then for all a in a resdual subset of A, the map is transverse to W .In our case, R is an open set of a Banach space, and we know that is transverse to .Thus, for the generic g ∈ R, E 2 g is transverse to .When k = 1, transversality is trivial because the diagonal coincides with M × R, 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 in (M × R) k is 3k − 3, and the dimension of V k p is 2k.Thus, if k ≥ 4 then the codimension in the target exceeds the domain dimension, so transverse intersection implies empty intersection: Similarly, because transversality preserves codimension, the pre-image of under E 3 g is a discrete set of points in V 3 p , and the pre-image of under E 2 g is a 1-manifold in V 2 p .

FOCAL BRANCHES
Fix a g ∈ R and let Clearly ν k is invariant under permutation of the factors T p M in V k p .Thus, if π j : V k p → T p M projects V k p onto the j th factor, is independent of j.Furthermore, β 2 ⊃ β 3 ⊃ β 4 ⊃ . . .and, PROPOSITION 4.1.When M has no conjugate points, the projection π j : ν k → T p M is a proper immersion onto a closed subset of T p M.
PROOF.Properness means that the pre-image of a compact set is compact.Thus, from any given a sequence (v 1n , . . ., v kn ) in ν k such that for some fi xed j, v jn converges in T p M as n → ∞, we must extract a subsequence, convergent in ν 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 ∈ T p M, 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 ) → (v 1 , . . ., v k ).Each v i has length .Fix i = j.Then v in = 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 p M to a neighborhood of p in M implies that = 0. Also, since there are no conjugate points, exp is a local diffeomorphism at v j , which implies that v i = v j .Thus (v 1 , . . ., v k ) ∈ ν k , which completes the proof of properness.
A continuous proper map into a metric space necessarily has a closed range.Hence π j (ν k ) is closed in T p M.
To check that π j is an immersion, we must show that the projection v j (t) of each nonsingular curve (v 1 (t), . . ., v k (t)) in ν k is nonsingular in T p M. Fix a t 0 ∈ (a, b) where (a, b) is the curve's domain of defi nition.For at least one i, vi (t 0 ) = 0. Thus, v i (t) is nonsingular at t 0 .Since there are no conjugate points, exp(v i (t)) is also nonsingular at t 0 .Since (v is also nonsingular at t 0 .Therefore, v j (t) is nonsingular at t 0 .

PROOF OF THEOREM A
We assume that M is a compact surface of genus ≥ 2, that p ∈ M is fi xed, and we denote by N the nonempty set of Riemann structures on T M having negative curvature.Clearly, N is an open subset of R and so it makes sense to speak of the generic g ∈ N .
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 p M is quite simple for the generic g ∈ N .Namely: (a) For all k ≥ 4, σ k is empty.
Furthermore, in any fi xed compact subset of T p M, properties (a), (b), (c) remain valid for all small perturbations of g.
Consider a vector v 1 ∈ β 3 .It has two friends v 2 , v 3 ∈ β 3 with equal length and equal exponential image.Thus (v 1 , v 2 ) ∈ ν 2 , and there are nonsingular curves v 1 (t), v 2 (t) with We claim that v1 (0) and v * 1 (0) are linearly independent.Suppose not.Nonsingularity implies that there is a c = 0 such that Similarly, Thus, at t = 0 is tangent to the diagonal ⊂ (M × R) 3 .The upshot is that the range of T (v 1 ,v 2 ,v 3 ) E 3 g contains a nonzero vector tangent to the diagonal in (M × R) 3 .This contradicts the fact that E 3 g : V 3 p → (M × R) 3 is transverse to , since has codimension 6, which is the same as the dimension of V 3 p .Now we know that v1 (0) and v * 1 (0) are linearly independent.This means that in addition to properties (a) -(c), above, we have (d) Branches of σ 2 meet transversally in pairs, they do so only at points of σ 3 , and every point of σ 3 is such a crossing point, where by a branch of σ 2 we mean the projection of an arc in ν 2 .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 ⊂ T p M. Then choose a disc B in T p M that contains S. We know that An Acad Bras Cienc (2007) 79 (1) the focal decomposition amounts to a smooth one-dimensional graph, namely = σ 2 ∪ σ 3 , which has transverse crossings of multiplicity two.We adjust B so that its boundary is transverse to .Let g be a small perturbation of g and let be the corresponding graph.Since all aspects of the graph depend continuously on the Riemann structure, and all are transverse, if g is suffi ciently close to g, then there exists a homeomorphism from B to itself that sends ∩ B to ∩ B.

PROOF OF THEOREM B
We assume that M is a compact surface of genus ≥ 2 and we denote by H ⊂ R the nonempty set of Riemann structures whose curvature is identically equal to −1.
Fix g ∈ H and p ∈ M. As described in section 2, we can lift g to a Riemann structure ḡ on T p M and view the focal decomposition in terms of mediatrices for the corresponding metric d.Since g has constant negative curvature, d is isometric to the Poincaré metric ρ on the unit disc D, and mediatrices are ρ-geodesics.As such, mediatrices are circular arcs meeting ∂D 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 p M, and denote the corresponding mediatrix set as µ = {v ∈ T p M : for some p ∈ P \ {0} and |v| = ρ(v, p)}.
Fix a compact set S ⊂ T p M. At most fi nitely many µ-mediatrices meet S. Choose a constant R and let B denote the compact disc 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 τ be the fi nite set of points in B at which the µ-mediatrices intersect one another.Thus for some fi nite .If v ∈ σ 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 τ , 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.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 U ⊂ H such that if g ∈ U and g ∈ H 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 p M leads to a sequence of open dense sets U n in H, and if g ∈ U n then the focal decomposition of T p M is stable with respect to perturbation of g within H. REMARK.Theorem B does not assert the multiplicity of the focal decomposition of the generic g ∈ H 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 τ s of hyperbolic structures on M amounts to H/D where D 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 σ k = ∅ for all k ≥ 4. From this it would follow at once that the generic g ∈ H is focally stable with respect to variation of g in R, not just with respect variation of g within H, as in Theorem B.

CONJUGATE POINTS
A Riemann structure with non-negative curvature has no conjugate points, but the set S of such Riemann structures does not form an open subset of R. 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 S need not be dense in S.
THEOREM A .Focal stability (for a fixed p ∈ M) holds for the generic g ∈ S.
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 ∈ R 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 S is strictly larger than N .
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 / ∈ N .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 S, 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 p M 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.
An Acad Bras Cienc (2007) 79 (1) (a) S ⊂ B ∩ B .(b) There is a homeomorphism h : B → B that sends each σ k (g) ∩ B onto σ k (g ) ∩ B . 2 MAURICIO M. PEIXOTO and CHARLES C. PUGH Thus, the focal decomposition σ k enjoys a kind of structural stability.

Fig. 1 -
Fig. 1 -Mediatrices µ corresponding to the focal decomposition.The d-distance from v 1 to O p , p2 , p3 is .The vectors v 1 , v 2 , v 3 have common exponential image q, while O p , p2 , p3 have exponential image p.
) Thus, there is an open-dense set U ⊂ M such that if p ∈ U and p is suffi ciently near p then all multiplicities of the µ -mediatrices in B = T p M(R) are the same as those in B. (By µ we denote the mediatrices between the origin of T p M and the other lifts of p in T p M.) 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 p M leads to a sequence of such open dense sets U n in M, and if p ∈ U n then the focal decomposition in T p M is stable with respect to perturbation of the base point p.This completes the proof of the fi rst assertion in Theorem B.