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Showing papers in "Geometriae Dedicata in 2010"


Journal ArticleDOI
TL;DR: In this article, a curvature-type tensor invariant called para contact (pc) curvature is defined on a paracontact manifold and it is shown that such a manifold is locally conformal to the hyperbolic Heisenberg group or to a hyperquadric of neutral signature iff the pc conformal curvature vanishes.
Abstract: A curvature-type tensor invariant called para contact (pc) conformal curvature is defined on a paracontact manifold. It is shown that a paracontact manifold is locally paracontact conformal to the hyperbolic Heisenberg group or to a hyperquadric of neutral signature iff the pc conformal curvature vanishes. In the three dimensional case the corresponding result is achieved through employing a certain symmetric (0,2) tensor. The well known result of Cartan–Chern–Moser giving necessary and sufficient condition a CR-structure to be CR equivalent to a hyperquadric in \({\mathbb{C}^{n+1}}\) is presented in-line with the paracontact case. An explicit formula for the regular part of a solution to the sub-ultrahyperbolic Yamabe equation on the hyperbolic Heisenberg group is shown.

66 citations


Journal ArticleDOI
Kensuke Onda1
TL;DR: Rahmani et al. as mentioned in this paper characterized the left-invariant Ricci soliton gcffff 1 as a Lorentz Ricci Soliton, which is a shrinking non-gradient RicciSoliton.
Abstract: The three-dimensional Heisenberg group H 3 has three left-invariant Lorentzian metrics g 1, g 2, and g 3 as in Rahmani (J. Geom. Phys. 9(3), 295–302 (1992)). They are not isometric to each other. In this paper, we characterize the left-invariant Lorentzian metric g 1 as a Lorentz Ricci Soliton. This Ricci Soliton g 1 is a shrinking non-gradient Ricci Soliton. We also prove that the group E(2) of rigid motions of Euclidean 2-space and the group E(1, 1) of rigid motions of Minkowski 2-space have Lorentz Ricci Solitons.

66 citations


Journal ArticleDOI
TL;DR: In this paper, the authors show that a left-invariant metric g on a nilpotent Lie group N is a soliton metric if and only if a matrix U and vector v associated the manifold (N, g) satisfy the matrix equation Uv = [1], where [1] is a vector with every entry a one.
Abstract: We show that a left-invariant metric g on a nilpotent Lie group N is a soliton metric if and only if a matrix U and vector v associated the manifold (N, g) satisfy the matrix equation Uv = [1], where [1] is a vector with every entry a one. We associate a generalized Cartan matrix to the matrix U and use the theory of Kac–Moody algebras to analyze the solution spaces for such linear systems. An application to the existence of soliton metrics on certain filiform Lie groups is given.

59 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that an isometric action of a Lie group on a Riemannian manifold admits a resolution preserving the transverse geometry if and only if the action is infinitesimally polar.
Abstract: We prove that an isometric action of a Lie group on a Riemannian manifold admits a resolution preserving the transverse geometry if and only if the action is infinitesimally polar. We provide applications concerning topological simplicity of several classes of isometric actions, including polar and variationally complete ones. All results are proven in the more general case of singular Riemannian foliations.

57 citations


Journal ArticleDOI
TL;DR: In this paper, the authors prove that the profinite topology on the fundamental group of π_1(M) is efficient with respect to the JSJ decomposition of M. They also prove that if M is a graph manifold then π(m) is conjugacy separable.
Abstract: Let M be a closed, orientable, irreducible, geometrizable 3-manifold. We prove that the profinite topology on the fundamental group of π_1(M) is efficient with respect to the JSJ decomposition of M. We go on to prove that π_1(M) is good, in the sense of Serre, if all the pieces of the JSJ decomposition are. We also prove that if M is a graph manifold then π_1(M) is conjugacy separable.

50 citations


Journal ArticleDOI
TL;DR: In this paper, an explicit diffeomorphism between the moduli space of a mirror projective polyhedron with fixed dihedral angles and the union of n copies of an ecimahedron was constructed.
Abstract: A projective mirror polyhedron is a projective polyhedron endowed with reflections across its faces. We construct an explicit diffeomorphism between the moduli space of a mirror projective polyhedron with fixed dihedral angles in \({(0,\frac{\pi}{2}]}\), and the union of n copies of \({\mathbb{R}^{d}}\), when the polyhedron has the combinatorics of an ecimahedron, an infinite class of combinatorial polyhedra we introduce here. Moreover, the integers n and d can be computed explicitly in terms of the combinatorics and the fixed dihedral angles.

40 citations


Journal ArticleDOI
TL;DR: In this paper, the local groups of germs associated with the higher dimensional R were calculated, and it was shown that the groups mV and nV cannot be isomorphic.
Abstract: We calculate the local groups of germs associated with the higher dimensional R. Thompson groups nV .F or ag ivenn ∈ N ∪ {ω}, these groups of germs are free abelian groups of rank r ,f orr ≤ n (there are some groups of germs associated with nV with rank precisely k for each index 1 ≤ k ≤ n). By Rubin's theorem, any conjectured isomorphism between higher dimensional R. Thompson groups induces an isomorphism between associ- ated groups of germs. Thus, if m �= n the groups mV and nV cannot be isomorphic. This answers a question of Brin.

39 citations


Journal ArticleDOI
TL;DR: In this article, intersection homology with general perversities was studied, and the authors obtained Poincare and Lefschetz duality results for intersection pairings that carry no restrictions on the input perversity.
Abstract: We study intersection homology with general perversities that assign integers to stratum components with none of the classical constraints of Goresky and MacPherson. We extend Goresky and MacPherson’s axiomatic treatment of Deligne sheaves, and use these to obtain Poincare and Lefschetz duality results for these general perversities. We also produce versions of both the sheaf-theoretic and the piecewise linear chain-theoretic intersection pairings that carry no restrictions on the input perversities.

37 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that if G is a discrete group that admits a metrically proper action on a finite-dimensional CAT(0) cube complex X, then G is weakly amenable.
Abstract: We prove that if G is a discrete group that admits a metrically proper action on a finite-dimensional CAT(0) cube complex X, then G is weakly amenable. We do this by constructing uniformly bounded Hilbert space representations πz for which the quantities zl(g) are matrix coefficients. Here l is a length function on G obtained from the combinatorial distance function on the complex X.

37 citations


Journal ArticleDOI
TL;DR: In this paper, the existence of higher open book structures defined by real map germs is studied and a general existence criterion for weighted-homogeneous maps with respect to weighted open books is proved.
Abstract: The paper focusses on the existence of higher open book structures defined by real map germs \({\psi : (\mathbb{R}^m ,0) \to (\mathbb{R}^p ,0)}\) such that Sing \({\psi \cap \psi^{-1}(0) \subset \{0\}}\). A general existence criterion is proved, with view to weighted-homogeneous maps.

36 citations


Journal ArticleDOI
TL;DR: In this article, the authors studied the asymptotic geometry of Teichmuller geodesic rays and showed that when the transverse measures to the vertical foliations of the quadratic differentials determining two different rays are topologically equivalent, but are not absolutely continuous with respect to each other, the rays diverge in the space in which they lie.
Abstract: We study the asymptotic geometry of Teichmuller geodesic rays. We show that, when the transverse measures to the vertical foliations of the quadratic differentials determining two different rays are topologically equivalent, but are not absolutely continuous with respect to each other, the rays diverge in Teichmuller space.

Journal ArticleDOI
TL;DR: In this paper, it was shown that the space of leaves of a singular Riemannian foliation is a Gromov-Hausdorff limit of a sequence of RiemANNian orbifolds.
Abstract: Let \({\mathcal{F}}\) be a singular Riemannian foliation on a compact Riemannian manifold M. By successive blow-ups along the strata of \({\mathcal{F}}\) we construct a regular Riemannian foliation \({\hat{\mathcal{F}}}\) on a compact Riemannian manifold \({\hat{M}}\) and a desingularization map \({\hat{\rho}:\hat{M}\rightarrow M}\) that projects leaves of \({\hat{\mathcal{F}}}\) into leaves of \({\mathcal{F}}\). This result generalizes a previous result due to Molino for the particular case of a singular Riemannian foliation whose leaves were the closure of leaves of a regular Riemannian foliation. We also prove that, if the leaves of \({\mathcal{F}}\) are compact, then, for each small \({\epsilon >0 }\), we can find \({\hat{M}}\) and \({\hat{\mathcal{F}}}\) so that the desingularization map induces an \({\epsilon}\)-isometry between \({M/\mathcal{F}}\) and \({\hat{M}/\hat{\mathcal{F}}}\). This implies in particular that the space of leaves \({M/\mathcal{F}}\) is a Gromov-Hausdorff limit of a sequence of Riemannian orbifolds \({\{(\hat{M}_{n}/\hat{\mathcal{F}}_{n})\}}\).

Journal ArticleDOI
TL;DR: In this article, the authors show that the bracket with a simple element has no cancellation and determines minimal intersection numbers, i.e., the signed sum over the intersection points of a and b of their loop product at the intersection point.
Abstract: This paper is a consequence of the close connection between combinatorial group theory and the topology of surfaces. In the eighties Goldman discovered a Lie algebra structure on the vector space generated by the free homotopy classes of oriented curves on an oriented surface. The Lie bracket [a, b] is defined as the signed sum over the intersection points of a and b of their loop product at the intersection points. If one of the classes has a simple representative we give a combinatorial group theory description of the terms of the Lie bracket and prove that this bracket has as many terms, counted with multiplicity, as the minimal number of intersection points of a and b. In other words the bracket with a simple element has no cancellation and determines minimal intersection numbers. We show that analogous results hold for the Lie bracket (also discovered by Goldman) of unoriented curves. We give three applications: a factorization of Thurston’s map defining the boundary of Teichmuller space, various decompositions of the underlying vector space of conjugacy classes into ad invariant subspaces and a connection between bijections of the set of conjugacy classes of curves on a surface preserving the Goldman bracket and the mapping class group.

Journal ArticleDOI
TL;DR: In this article, it was shown that any compact locally homogeneous Lorentz threefold M is isometric to a quotient of a LRC by a discrete subgroup Γ of G acting properly and freely on G/I.
Abstract: We classify three-dimensional Lorentz homogeneous spaces G/I having a compact manifold locally modeled on them. We prove a completeness result: any compact locally homogeneous Lorentz threefold M is isometric to a quotient of a Lorentz homogeneous space G/I by a discrete subgroup Γ of G acting properly and freely on G/I. Moreover, if I is noncompact, G/I is isometric to a Lie group L endowed with a left invariant Lorentz metric, where L is isomorphic to one of the following Lie groups: $${\bf R}^3, \widetilde{SL(2, {\bf R})}, He\,is \,{\rm or}\, SOL.$$ If L is not \({\widetilde{SL(2, {\bf R})}}\) , then M admits a finite cover which is a quotient of L by a lattice.

Journal ArticleDOI
TL;DR: In this article, it was shown that finite volume right-angled Coxeter polyhedra may exist in hyperbolic spaces only in dimension at most 12, which is a slight improvement on a theorem of Potyagailo and Vinberg stated in 2005.
Abstract: We show that (finite volume) right-angled Coxeter polyhedra may exist in hyperbolic spaces only in dimension at most 12 This is a slight improvement on a theorem of Potyagailo and Vinberg stated in Potyagailo and Vinberg (Comment Math Helv 80:1–12, 2005)

Journal ArticleDOI
TL;DR: In this article, it was shown that the dimension of the space of L 2 harmonic 1-forms on an n-dimensional complete non-compact oriented submanifold with finite total curvature is finite, where |H| and |A| are the mean curvatures and the squared length of the second fundamental form, respectively.
Abstract: Let Mn be an n-dimensional complete noncompact oriented submanifold with finite total curvature, i.e., \({\int_M(|A|^2-n|H|^2)^{\frac n2} < \infty}\), in an (n + p)-dimensional simply connected space form Nn+p(c) of constant curvature c, where |H| and |A|2 are the mean curvature and the squared length of the second fundamental form of M, respectively. We prove that if M satisfies one of the following: (i) n ≥ 3, c = 0 and \({\int_M|H|^n < \infty}\); (ii) n ≥ 5, c = −1 and \({|H| < 1-\frac{2}{\sqrt n}}\); (iii) n ≥ 3, c = 1 and |H| is bounded, then the dimension of the space of L2 harmonic 1-forms on M is finite. Moreover, in the case of (i) or (ii), M must have finitely many ends.

Journal ArticleDOI
TL;DR: In this paper, it was shown that every locally injective simplicial map between pants graphs is induced by a π 1-injective embedding between the corresponding surfaces.
Abstract: We prove that, except in some low-complexity cases, every locally injective simplicial map between pants graphs is induced by a π1-injective embedding between the corresponding surfaces.

Journal ArticleDOI
TL;DR: In this article, the waist of a knot in the 3-sphere S3 is defined as the set of all closed incompressible surfaces in S3−K−K, where K is the number of closed surfaces for which a disk can be compressed.
Abstract: Let K be a knot in the 3-sphere S3. We define the waist of K as $$waist (K) = \mathop{\rm max}\limits_{F\in\mathcal{F}} \mathop{\rm min}\limits_{D\in\mathcal{D}_{F}} |D \cap K|,$$ where \({\mathcal{F}}\) is the set of all closed incompressible surfaces in S3−K and \({\mathcal{D}_F}\) is the set of all compressing disks for F in S3. We define the trunk of K as $$trunk(K) = \mathop{\rm min}\limits_{h\in\mathcal{H}} \mathop{\rm max}\limits_{t\in\mathbb{R}} |h^{-1}(t) \cap K|,$$ where \({\mathcal{H}}\) is the set of all Morse function \({h : S^3 \to \mathbb{R}}\) with two critical points. We show that $$waist (K) \le \frac{trunk(K)}{3}$$ .

Journal ArticleDOI
TL;DR: In this paper, it was shown that the full braid group on n-strings of a surface S satisfies the R676 ∞ property in the cases where S is either the compact disk D, or the sphere S petertodd 2.
Abstract: We prove that the symplectic group $${Sp(2n,\mathbb{Z})}$$ and the mapping class group Mod S of a compact surface S satisfy the R ∞ property. We also show that B n (S), the full braid group on n-strings of a surface S, satisfies the R ∞ property in the cases where S is either the compact disk D, or the sphere S 2. This means that for any automorphism $${\phi}$$ of G, where G is one of the above groups, the number of twisted $${\phi}$$ -conjugacy classes is infinite.

Journal ArticleDOI
TL;DR: In this paper, it was shown that a Teichmuller quasi-geodesic in the thick part of the curve graph for a surface S is contained in a bounded neighborhood of a geodesic if and only if it induces a quasi-Geodesic.
Abstract: Let S be a surface S of genus g ≥ 0 with m ≥ 0 punctures and 3g − 3 + m ≥ 2. We show that a Teichmuller quasi-geodesic in the thick part of Teichmuller space for S is contained in a bounded neighborhood of a geodesic if and only if it induces a quasi-geodesic in the curve graph.

Journal ArticleDOI
TL;DR: In this paper, the authors investigated the numerical properties of relatively minimal isotrivial fibrations, where X is a smooth, projective surface and C is a curve, and showed that if g(C) ≥ 1 and X is neither ruled nor isomorphic to a quasi-bundle, then the inequality is sharp.
Abstract: In this paper we investigate the numerical properties of relatively minimal isotrivial fibrations $${\varphi : X \longrightarrow C}$$ , where X is a smooth, projective surface and C is a curve. In particular we prove that, if g(C) ≥ 1 and X is neither ruled nor isomorphic to a quasi-bundle, then $${K_X^2 \leq 8 \chi(\mathcal{O}_X)-2}$$ ; this inequality is sharp and if equality holds then X is a minimal surface of general type whose canonical model has precisely two ordinary double points as singularities. Under the further assumption that K X is ample, we obtain $${K_X^2 \leq 8\chi(\mathcal{O}_X)-5}$$ and the inequality is also sharp. This improves previous results of Serrano and Tan.

Journal ArticleDOI
TL;DR: In this paper, the authors show that the circumradius and asymptotic behavior of curves in cat(0) and cat(K) spaces (K > 0) are controlled by growth rates of total curvature.
Abstract: We show how circumradius and asymptotic behavior of curves in cat(0) and cat(K) spaces (K > 0) are controlled by growth rates of total curvature. We apply our results to pursuit and evasion games of capture type with simple pursuit motion, generalizing results that are known for convex Euclidean domains, and obtaining results that are new for convex Euclidean domains and hold on playing fields vastly more general than these.

Journal ArticleDOI
Atsumu Sasaki1
TL;DR: In this article, it was shown that a maximal compact subgroup of G/K acts on D in a strongly visible fashion in the sense of Kobayashi (Publ Res Inst Math Sci 41:497-549, 2005) if and only if G/k is of non-tube type.
Abstract: Let G/K be an irreducible Hermitian symmetric space of non-compact type, and \({G_{\mathbb{C}}/K_{\mathbb{C}}}\) its complexification by forgetting the original complex structure. Then, \({D :=G_{\mathbb{C}}/[K_{\mathbb{C}}, K_{\mathbb{C}}]}\) is a non-symmetric Stein manifold. We prove that a maximal compact subgroup of \({G_{\mathbb{C}}}\) acts on D in a strongly visible fashion in the sense of Kobayashi (Publ Res Inst Math Sci 41:497–549, 2005) if and only if G/K is of non-tube type. Our proof uses the theory of multiplicity-free representations and a construction of a slice and an anti-holomorphic involution on D.

Journal ArticleDOI
TL;DR: In this article, it was shown that generalized Mukai conjecture holds for Fano manifolds X of pseudoindex iX ≥ (dim X + 3)/3, and also gave different proofs of the conjecture for fourfolds and fivefolds.
Abstract: We prove that Generalized Mukai Conjecture holds for Fano manifolds X of pseudoindex iX ≥ (dim X + 3)/3. We also give different proofs of the conjecture for Fano fourfolds and fivefolds.

Journal ArticleDOI
TL;DR: In this article, the authors show that almost all quasi-polynomial flows on homogeneous spaces are not very well approximable in terms of their non-differential properties.
Abstract: Let Y 0 be a not very well approximable m × n matrix, and let $${\mathcal {M}}$$ be a connected analytic submanifold in the space of m × n matrices containing Y 0. Then almost all $${Y \in \mathcal {M}}$$ are not very well approximable. This and other similar statements are cast in terms of properties of certain orbits on homogeneous spaces and deduced from quantitative nondivergence estimates for‘quasi-polynomial’ flows on the space of lattices.

Journal ArticleDOI
TL;DR: The Ptolemy groupoid as mentioned in this paper is a combinatorial groupoid generated by elementary moves on marked trivalent fatgraphs with three types of relations, and is a mapping class group equivariant subgroupoid of the fundamental path groupoid.
Abstract: The Ptolemy groupoid is a combinatorial groupoid generated by elementary moves on marked trivalent fatgraphs with three types of relations. Through the fatgraph decomposition of Teichmuller space, the Ptolemy groupoid is a mapping class group equivariant subgroupoid of the fundamental path groupoid of Teichmuller space with a discrete set objects. In particular, it leads to an infinite, but combinatorially simple, presentation of the mapping class group of an orientable surface. In this note, we give a presentation of a full mapping class group equivariant subgroupoid of the Ptolemy groupoid of an orientable surface with one boundary component in terms of marked linear chord diagrams, with chord slides as generators and five types of relations. We also introduce a dual version of this presentation which has advantages for certain applications, one of which is given.

Journal ArticleDOI
TL;DR: In this article, it was shown that a manifold is insecure if there exists an insecure point pair, and totally insecure if all point pairs are insecure, and that a closed surface of genus greater than one with any riemannian metric is totally insecure.
Abstract: A pair of points in a riemannian manifold M is secure if the geodesics between the points can be blocked by a finite number of point obstacles; otherwise the pair of points is insecure. A manifold is secure if all pairs of points in M are secure. A manifold is insecure if there exists an insecure point pair, and totally insecure if all point pairs are insecure. Compact, flat manifolds are secure. A standing conjecture says that these are the only secure, compact riemannian manifolds. We prove this for surfaces of genus greater than zero. We also prove that a closed surface of genus greater than one with any riemannian metric and a closed surface of genus one with generic metric are totally insecure.

Journal ArticleDOI
TL;DR: In this article, the authors derived some basic properties of (p, n)-gonal surfaces considered as generalizations of hyperelliptic surfaces and also examined certain properties which do not generalize.
Abstract: A compact Riemann surface X is called a (p, n)-gonal surface if there exists a group of automorphisms C of X (called a (p, n)-gonal group) of prime order p such that the orbit space X/C has genus n. We derive some basic properties of (p, n)-gonal surfaces considered as generalizations of hyperelliptic surfaces and also examine certain properties which do not generalize. In particular, we find a condition which guarantees all (p, n)-gonal groups are conjugate in the full automorphism group of a (p, n)-gonal surface, and we find an upper bound for the size of the corresponding conjugacy class. Furthermore we give an upper bound for the number of conjugacy classes of (p, n)-gonal groups of a (p, n)-gonal surface in the general case. We finish by analyzing certain properties of quasiplatonic (p, n)-gonal surfaces. An open problem and two conjectures are formulated in the paper.

Journal ArticleDOI
TL;DR: In this paper, the authors gave various estimates of the minimal number of self-intersections of a nontrivial element of the kth term of the lower central series and derived series of the fundamental group of a surface.
Abstract: We give various estimates of the minimal number of self-intersections of a nontrivial element of the kth term of the lower central series and derived series of the fundamental group of a surface. As an application, we obtain a new topological proof of the fact that free groups and fundamental groups of closed surfaces are residually nilpotent. Along the way, we prove that a nontrivial element of the kth term of the lower central series of a nonabelian free group has to have word length at least k in a free generating set.

Journal ArticleDOI
TL;DR: In this article, the authors introduce the concepts of dual quermassintegral differences and width integral differences, and discuss the theory of dual Brunn-Minkowski type for them.
Abstract: In this paper, we introduce the concepts of dual quermassintegral differences and width-integral differences, and discuss the theory of dual Brunn–Minkowski type for them. One of the results implies that for two star bodies which are dilations of each other, the dual Brunn–Minkowski inequality still holds after two arbitrary star bodies included in them being excluded, respectively.