Riemannian Metric Of Constant Curvature Research Articles

A random cover of a compact hyperbolic surface has relative spectral gap frac{3}{16}-varepsilon

Let X be a compact connected hyperbolic surface, that is, a closed connected orientable smooth surface with a Riemannian metric of constant curvature -1. For each nin {mathbf {N}}, let X_{n} be a random degree-n cover of X sampled uniformly from all degree-n Riemannian covering spaces of X. An eigenvalue of X or X_{n} is an eigenvalue of the associated Laplacian operator Delta _{X} or Delta _{X_{n}}. We say that an eigenvalue of X_{n} is new if it occurs with greater multiplicity than in X. We prove that for any varepsilon >0, with probability tending to 1 as nrightarrow infty , there are no new eigenvalues of X_{n} below frac{3}{16}-varepsilon . We conjecture that the same result holds with frac{3}{16} replaced by frac{1}{4}.

Area-minimizing properties of Pansu spheres in the sub-Riemannian 3-sphere

Abstract We consider the sub-Riemannian 3-sphere ( 𝕊 3 , g h ) {(\mathbb{S}^{3},g_{h})} obtained by restriction of the Riemannian metric of constant curvature 1 to the planar distribution orthogonal to the vertical Hopf vector field. It was shown in [A. Hurtado and C. Rosales, Area-stationary surfaces inside the sub-Riemannian three-sphere, Math. Ann. 340 2008, 3, 675–708] that ( 𝕊 3 , g h ) {(\mathbb{S}^{3},g_{h})} contains a family of spherical surfaces { 𝒮 λ } λ ⩾ 0 {\{\mathcal{S}_{\lambda}\}_{\lambda\geqslant 0}} with constant mean curvature λ. In this work, we first prove that the two closed half-spheres of 𝒮 0 {\mathcal{S}_{0}} with boundary C 0 = { 0 } × 𝕊 1 {C_{0}=\{0\}\times\mathbb{S}^{1}} minimize the sub-Riemannian area among compact C 1 {C^{1}} surfaces with the same boundary. We also see that the only C 2 {C^{2}} solutions to this Plateau problem are vertical translations of such half-spheres. Second, we establish that the closed 3-ball enclosed by a sphere 𝒮 λ {\mathcal{S}_{\lambda}} with λ > 0 {\lambda>0} uniquely solves the isoperimetric problem in ( 𝕊 3 , g h ) {(\mathbb{S}^{3},g_{h})} for C 1 {C^{1}} sets inside a vertical solid tube and containing a horizontal section of the tube. The proofs mainly rely on calibration arguments.

Isometry groups of infinite-genus hyperbolic surfaces

Given a 2-manifold, a fundamental question to ask is which groups can be realized as the isometry group of a Riemannian metric of constant curvature on the manifold. In this paper, we give a nearly complete classification of such groups for infinite-genus 2-manifolds with no planar ends. Surprisingly, we show there is an uncountable class of such 2-manifolds where every countable group can be realized as an isometry group (namely, those with self-similar end spaces). We apply this result to obtain obstructions to standard group theoretic properties for the groups of homeomorphisms, diffeomorphisms, and the mapping class groups of such 2-manifolds. For example, none of these groups satisfy the Tits Alternative; are coherent; are linear; are cyclically or linearly orderable; or are residually finite. As a second application, we give an algebraic rigidity result for mapping class groups.

Local isometric immersions of pseudo-spherical surfaces and kth order evolution equations

We consider the class of evolution equations of the form [Formula: see text], [Formula: see text], that describe pseudo-spherical surfaces. These were classified by Chern and Tenenblat in [Pseudospherical surfaces and evolution equations, Stud. Appl. Math 74 (1986) 55–83.]. This class of equations is characterized by the property that to each solution of such an equation, there corresponds a 2-dimensional Riemannian metric of constant curvature [Formula: see text]. Motivated by the special properties of the sine-Gordon equation, we investigate the following problem: given such a metric, is there a local isometric immersion in [Formula: see text] such that the coefficients of the second fundamental form of the immersed surface depend on a jet of finite order of [Formula: see text]? We extend our earlier results for second-order evolution equations [N. Kahouadji, N. Kamran and K. Tenenblat, Local isometric immersions of pseudo-spherical surfaces and evolution equations, Fields Inst. Commun. 75 (2015) 369–381; N. Kahouadji, N. Kamran and K. Tenenblat, Second-order equations and local isometric immersions of pseudo-spherical surfaces, Comm. Anal. Geom. 24(3) (2016) 605–643.] to [Formula: see text]th order equations by proving that there is only one type of equation that admit such an isometric immersion. More precisely, we prove under the condition of finite jet dependency that the coefficients of the second fundamental forms of the local isometric immersion determined by the solutions [Formula: see text] are universal, i.e. they are independent of [Formula: see text]. Moreover, we show that there exists a foliation of the domain of the parameters of the surface by straight lines with the property that the mean curvature of the surface is constant along the images of these straight lines under the isometric immersion.

Minimal rays on closed surfaces

Given an arbitrary Riemannian metric on a closed surface, we consider length-minimizing geodesics in the universal cover. Morse and Hedlund proved that such minimal geodesics lie in bounded distance of geodesics of a Riemannian metric of constant curvature. Knieper asked when two minimal geodesics in bounded distance of a single constant-curvature geodesic can intersect. In this paper we prove an asymptotic property of minimal rays, showing in particular that intersecting minimal geodesics as above can only occur as heteroclinic connections between pairs of homotopic closed minimal geodesics. A further application characterizes the boundary at infinity of the universal cover defined by Busemann functions. A third application shows that flat strips in the universal cover of a nonpositively curved surface are foliated by lifts of closed geodesics of a single homotopy class.

Spherical Quadrilaterals with Three Non-integer Angles

A spherical quadrilateral is a bordered surface homeomorphic to a closed disk, with four distinguished boundary points called corners, equipped with a Riemannian metric of constant curvature 1, except at the corners, and such that the boundary arcs between the corners are geodesic. We discuss the problem of classification of these quadrilaterals and perform the classification up to isometry in the case that one corner of a quadrilateral is integer (i.e., its angle is a multiple of �) while the angles at its other three corners are not multiples of �. The problem is equivalent to classification of Heun’s equations with real parameters and unitary monodromy, with the trivial monodromy at one of its four singular point. MSC 2010: 30C20,34M03.

A classification of locally homogeneous affine connections on compact surfaces

We classify the affine connections on compact orientable surfaces for which the pseudogroup of local isometries acts transitively. We prove that such a connection is either torsion-free and flat, the Levi–Civita connection of a Riemannian metric of constant curvature or the quotient of a translation-invariant connection in the plane. This refines previous results by Opozda.

Metrics with conic singularities and spherical polygons

A spherical n-gon is a bordered surface homeomorphic to a closed disk, with n distinguished boundary points called corners, equipped with a Riemannian metric of constant curvature 1, except at the corners, and such that the boundary arcs between the corners are geodesic. We discuss the problem of classification of these polygons and enumerate them in the case that two angles at the corners are not multiples of �. The problem is equivalent to classification of some second order linear differential equations with regular singularities, with real parameters and unitary monodromy. MSC 2010: 30C20, 34M03.

Explicit formula for the spectral counting function of the Laplace operator on a compact Riemannian surface of genus g > 1

Let the standard Riemannian metric of constant curvature K = −1 be given on a compact Riemannian surface of genus g > 1. Under this condition, for a class of strictly hyperbolic Fuchsian groups, we obtain an explicit expression for the spectral counting function of the Laplace operator in the form of a series over the zeros of the Selberg zeta function.

Gallot–Tanno theorem for pseudo-Riemannian metrics and a proof that decomposable cones over closed complete pseudo-Riemannian manifolds do not exist

We generalize for complete pseudo-Riemannian metrics a classical result of Gallot (1979) [3] and Tanno (1978) [13]: we show that if a closed complete manifold admits a nonconstant function λ satisfying ∇ k ∇ j ∇ i λ + 2 ∇ k λ ⋅ g i j + ∇ i λ ⋅ g j k + ∇ j λ ⋅ g i k = 0 , then the metric is the Riemannian metric of constant curvature +1. We use this result to give a simple proof of a recent result of Alekseevsky, Cortes, Galaev and Leistner (2009) [1]. Certain generalizations for higher Gallot equations are given.

On the role of external reference frames on visual judgements of parallelity

In a previous study we found large systematic errors (up to 40°) when subjects adjusted the orientation of a horizontal test bar until it appeared parallel to a horizontal reference bar, both bars rotating about their vertical axes. The deviations increased linearly with the separation angle but vanished when the orientation of the reference bar was either parallel or perpendicular to the median line. In order to test the assumption that external references caused these deviations to vanish, the same task was repeated in four different conditions: in the normal condition the horizontal aperture, formed by a cabin, and the facing wall of the room were frontoparallel to the subject; in the other conditions either the room, the cabin or both were oriented 30° to the right with respect to the subject. It was found that, depending on the subject, the occurrence of the vanishing deviations covaried with the orientation of the cabin or the room. Evidently, subjects are influenced by the external references provided by the walls of the room and the sides of the cabin. The results indicate that a description of visual space by a Riemannian metric of constant curvature is not valid in a visual environment containing external references.

A rigidity theorem for Haken manifolds

A compact, orientable 3-manifoldMis calledhyperbolicif intMadmits a complete hyperbolic structure (Riemannian metric of constant curvature − 1) of finite volume. Any hyperbolic 3-manifoldMis irreducible, and each component of∂Mis an incompressible torus. Letf:M→Nbe a proper, continuous map between hyperbolic 3-manifolds. By Mostow's Rigidity Theorem [8], iffis π1-isomorphic thenfis properly homotopic to a diffeomorphismg:M→Nsuch thatg| intM: intM→ intNis isometric. In particular, the topological type of intMdetermines uniquely the hyperbolic structure on intMup to isometry, so the volume vol (intM) of intMis well-defined. This Rigidity Theorem is generalized by Thurston[11, theorem 6·4] so that any proper, continuous mapf:M→Nbetween hyperbolic 3-manifolds with vol(intM) = deg(f) vol(intN) is properly homotopic to a deg(f)-fold coveringg:M→Nsuch thatg| intM: intM→ intNis locally isometric.

Finite group actions on 3-manifolds

If G is a finite group acting smoothly on a closed surface F, it is well known that G leaves invariant some Riemannian metric of constant curvature on F. Thus any action of G on the 2-sphere S 2 is conjugate in Diff(S 2) to an orthogonal action. If G acts on the torus SX• S 1, there is a G-invariant flat metric on S a • S 1, and if G acts on a surface F with negative Euler number, then F admits a G-invariant hyperbolic metric. Recently Thurston, [Th 1, Th 2, Th 3], has described the eight 3-dimensional geometries which provide geometric structures for closed 3-manifolds in the same way that the 2-sphere S 2, the Euclidean plane E 2 and the hyperbolic plane H 2 provide geometric structures for surfaces. See also the survey article by Scott [Sc4]. Thurston also conjectured that if M is a closed 3-manifold with a geometric structure modelled on one of these eight geometries, say X, then any smooth action of a finite group G on M should leave invariant some metric on M inducing the geometry X. We will say that G preserves the geometric structure on M in this case. It should be noted that the restriction to smooth actions of G on M is essential. For Bing [Bi] showed that there are involutions of S 3 whose fixed set is a wild 2-sphere. However, in dimension two, it was proved by Eilenberg [Ei] that any action of a finite group on a surface is conjugate to a smooth action. In this paper, our main result asserts that Thurston's conjecture holds for five of the eight geometries. The result is the following.

Foliations and laminations on hyperbolic surfaces

GEODESIC LAMINATIONS on surfaces have been introduced by Thurston in his study[l7] of 3-manifolds with a hyperbolic structure (i.e. a Riemannian metric of constant curvature -1). He has also noticed (unpublished) that, given a hyperbolic structure on a surface, there is a bijective correspondence between measured laminations and equivalence classes of measured foliations (a related result has been obtained by Aranson and Grines [ 11). But a general foliation with p-prong saddle singularities (p 3 3) on a surface is topologically conjugate to a measured foliation (in the sense of .[4, 161) if and only if its non-compact leaves are locally dense (see 94 below), and very few foliations (even of class Cr) satisfy this condition. The aim of the present paper is to show how Thurston’s construction can be generalized to non-measured foliations in order to yield a canonical representation of foliations with saddle singularities on a given hyperbolic surface. Let us first consider (non-singular) foliations 9 with no compact leaf on the 2-torus T2 = R2/Z2. As is well known since Denjoy[3], such a foliation, if C2, is topologically conjugate to an “irrational flow” (the projection onto T* of a foliation of R2 by lines of irrational slope). Denjoy also gave an example of a Co (even C’) foliation not conjugate to an irrational flow. Roughly speaking, his example can be obtained from an irrational flow by “opening up” a leaf f, that is replacing f by two leaves f’ and fwhose distance goes to 0 as-one goes out to infinity on either leaf, and pushing apart the other leaves to make room; the space between f’ and fis then filled in by new leaves, and the complement of these new leaves is a non-trivial compact minimal set. One way of distinguishing a Denjoy example from a C2 foliation is by considering the induced foliation 9 on the universal covering R2. Given a leaf f of @, the line defined by two points p and 4 of f has a limiting position rci> when p and 4 go to infinity in opposite directions on f. The line ru) has some irrational slope (Y, and conversely every line of slope (Y is obtained by “straightening” a leaf { If 9 is conjugate to an irrational flow, different leaves of .@ give rise to different lines. If 9 is a Denjoy example, corresponding lifts to R2 of the leaves f’ and f(and all “new” leaves in between) give rise to the same line. We call such a line “thick”. Projecting down to T2, we see that by straightening leaves we have attached to a foliation 9 an irrational flow r(F), which is a totally geodesic foliation for the canonical flat metric of T’; if 9 is not conjugate to an irrational flow, then y(9) has at least one thick leaf. Markley has shown[8] that foliations on T’ with no compact leaf can be classified up to isotopy by specifying an irrational flow on T* and a family (at most countably infinite) of thick lines; the foliation is obtained from the irrational flow by opening up these thick lines. A similar pattern works for a foliation 9 with saddle singularities on a hyperbolic

Regular Tessellations of Surfaces and (p, q, 2)-Triangle Groups

The purpose of this paper is a study of the existence and classification of certain regular tessellations of surfaces and their relationship with the Schwarz triangle groups of type (p, q, 2). These tessellations are the natural generalizations to arbitrary surfaces of the spherical tessellations corresponding to the five Platonic Solids. A regular tessellation of type {p, q}, a {p, q}-pattern, for short, on a surface M is a tessellation of M by p-sided faces such that the valence of each vertex is q. (A more precise definition appears in ? 1.) No global symmetry is assumed. A { p, q}-pattern on M is said to be geometric if M admits a Riemannian metric of constant curvature with respect to which the edges of the pattern are geodesic arcs of equal length and the interior angles at all vertices in all faces are (2 7/q). If the curvature happens to he zero then we further insist that the area of each face be 1. It is natural to call two {p, q}-patterns on M

Three dimensional manifolds, Kleinian groups and hyperbolic geometry

1. A conjectural picture of 3-manifolds. A major thrust of mathematics in the late 19th century, in which Poincare had a large role, was the uniformization theory for Riemann surfaces: that every conformai structure on a closed oriented surface is represented by a Riemannian metric of constant curvature. For the typical case of negative Euler characteristic (genus greater than 1) such a metric gives a hyperbolic structure: any small neighborhood in the surface is isometric to a neighborhood in the hyperbolic plane, and the surface itself is the quotient of the hyperbolic plane by a discrete group of motions. The exceptional cases, the sphere and the torus, have spherical and Euclidean structures. Three-manifolds are greatly more complicated than surfaces, and I think it is fair to say that until recently there was little reason to expect any analogous theory for manifolds of dimension 3 (or more)—except perhaps for the fact that so many 3-manifolds are beautiful. The situation has changed, so that I feel fairly confident in proposing the

Characterizations of the sphere by the curvature of the second fundamental form

On an ovaloid S S with Gaussian curvature K ( I) > 0 K({\text {I)}} > 0 in Euclidean three-space E 3 {E^3} the second fundamental form defines a nondegenerate Riemannian metric with curvature K ( II ) K({\text {II}}) . R. Schneider [7] proved that the spheres in Euclidean space E n + 1 {E^{n + 1}} are the only closed hypersurfaces on which the second fundamental form defines a nondegenerate Riemannian metric of constant curvature. For surfaces in E 3 {E^3} we give a common generalization of Schneider’s theorem and the classical theorem of Liebmann [6] (which states that any ovaloid in E 3 {E^3} with constant Gaussian curvature is a sphere).

Closed Convex Hypersurfaces with Second Fundamental Form of Constant Curvature

It is shown that the Euclidean spheres are the only closed hypersurfaces in Euclidean space on which the second fundamental form defines a (nondegenerate) Riemannian metric of constant curvature.

Closed convex hypersurfaces with second fundamental form of constant curvature

It is shown that the Euclidean spheres are the only closed hypersurfaces in Euclidean space on which the second fundamental form defines a (nondegenerate) Riemannian metric of constant curvature.

On real projective spaces as Finsler manifolds

Let M be a finite-dimensional C3 manifold supplied with a C2 Finsler metric ds = F(x, dx), which is not necessarily even in dx. Let p designate the induced oriented topological metric. For any p E M, the antipodal locus of p is the set A(p)= {qEMIp(p, q)_p(p, r) for all rEM}. For example, if M is a real projective space with the Riemannian metric of constant curvature 1, A (p) is a smooth hypersurface (in fact, a projective hyperplane) for every pEHM. One may ask, how close does this property come to characterizing real projective spaces among Finsler manifolds? We prove