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51. CJM 2005 (vol 57 pp. 1291)

Riveros, Carlos M. C.; Tenenblat, Keti
Dupin Hypersurfaces in $\mathbb R^5$
We study Dupin hypersurfaces in $\mathbb R^5$ parametrized by lines of curvature, with four distinct principal curvatures. We characterize locally a generic family of such hypersurfaces in terms of the principal curvatures and four vector valued functions of one variable. We show that these vector valued functions are invariant by inversions and homotheties.

Categories:53B25, 53C42, 35N10, 37K10

52. CJM 2005 (vol 57 pp. 1314)

Zhitomirskii, M.
Relative Darboux Theorem for Singular Manifolds and Local Contact Algebra
In 1999 V. Arnol'd introduced the local contact algebra: studying the problem of classification of singular curves in a contact space, he showed the existence of the ghost of the contact structure (invariants which are not related to the induced structure on the curve). Our main result implies that the only reason for existence of the local contact algebra and the ghost is the difference between the geometric and (defined in this paper) algebraic restriction of a $1$-form to a singular submanifold. We prove that a germ of any subset $N$ of a contact manifold is well defined, up to contactomorphisms, by the algebraic restriction to $N$ of the contact structure. This is a generalization of the Darboux-Givental' theorem for smooth submanifolds of a contact manifold. Studying the difference between the geometric and the algebraic restrictions gives a powerful tool for classification of stratified submanifolds of a contact manifold. This is illustrated by complete solution of three classification problems, including a simple explanation of V.~Arnold's results and further classification results for singular curves in a contact space. We also prove several results on the external geometry of a singular submanifold $N$ in terms of the algebraic restriction of the contact structure to $N$. In particular, the algebraic restriction is zero if and only if $N$ is contained in a smooth Legendrian submanifold of $M$.

Keywords:contact manifold, local contact algebra,, relative Darboux theorem, integral curves
Categories:53D10, 14B05, 58K50

53. CJM 2005 (vol 57 pp. 1012)

Karigiannis, Spiro
Deformations of $G_2$ and $\Spin(7)$ Structures
We consider some deformations of $G_2$-structures on $7$-manifolds. We discover a canonical way to deform a $G_2$-structure by a vector field in which the associated metric gets ``twisted'' in some way by the vector cross product. We present a system of partial differential equations for an unknown vector field $w$ whose solution would yield a manifold with holonomy $G_2$. Similarly we consider analogous constructions for $\Spin(7)$-structures on $8$-manifolds. Some of the results carry over directly, while others do not because of the increased complexity of the $\Spin(7)$ case.

Keywords:$G_2 \Spin(7)$, holonomy, metrics, cross product
Categories:53C26, 53C29

54. CJM 2005 (vol 57 pp. 708)

Finster, Felix; Kraus, Margarita
Curvature Estimates in Asymptotically Flat Lorentzian Manifolds
We consider an asymptotically flat Lorentzian manifold of dimension $(1,3)$. An inequality is derived which bounds the Riemannian curvature tensor in terms of the ADM energy in the general case with second fundamental form. The inequality quantifies in which sense the Lorentzian manifold becomes flat in the limit when the ADM energy tends to zero.

Categories:53C21, 53C27, 83C57

55. CJM 2005 (vol 57 pp. 871)

Zhang, Xi
Hermitian Yang-_Mills--Higgs Metrics on\\Complete Kähler Manifolds
In this paper, first, we will investigate the Dirichlet problem for one type of vortex equation, which generalizes the well-known Hermitian Einstein equation. Secondly, we will give existence results for solutions of these vortex equations over various complete noncompact K\"ahler manifolds.

Keywords:vortex equation, Hermitian Yang--Mills--Higgs metric,, holomorphic vector bundle, Kähler manifolds
Categories:58E15, 53C07

56. CJM 2005 (vol 57 pp. 750)

Sabourin, Hervé
Sur la structure transverse à une orbite nilpotente adjointe
We are interested in Poisson structures to transverse nilpotent adjoint orbits in a complex semi-simple Lie algebra, and we study their polynomial nature. Furthermore, in the case of $sl_n$, we construct some families of nilpotent orbits with quadratic transverse structures.

Keywords:nilpotent adjoint orbits, conormal orbits, Poisson transverse structure
Categories:22E, 53D

57. CJM 2005 (vol 57 pp. 114)

Flaschka, Hermann; Millson, John
Bending Flows for Sums of Rank One Matrices
We study certain symplectic quotients of $n$-fold products of complex projective $m$-space by the unitary group acting diagonally. After studying nonemptiness and smoothness of these quotients we construct the action-angle variables, defined on an open dense subset, of an integrable Hamiltonian system. The semiclassical quantization of this system reporduces formulas from the representation theory of the unitary group.

Category:53D20

58. CJM 2004 (vol 56 pp. 1228)

Ho, Nan-Kuo; Liu, Chiu-Chu Melissa
On the Connectedness of Moduli Spaces of Flat Connections over Compact Surfaces
We study the connectedness of the moduli space of gauge equivalence classes of flat $G$-connections on a compact orientable surface or a compact nonorientable surface for a class of compact connected Lie groups. This class includes all the compact, connected, simply connected Lie groups, and some non-semisimple classical groups.

Keywords:moduli space of flat $G$ connections
Category:53

59. CJM 2004 (vol 56 pp. 776)

Lim, Yongdo
Best Approximation in Riemannian Geodesic Submanifolds of Positive Definite Matrices
We explicitly describe the best approximation in geodesic submanifolds of positive definite matrices obtained from involutive congruence transformations on the Cartan-Hadamard manifold ${\mathrm{Sym}}(n,{\Bbb R})^{++}$ of positive definite matrices. An explicit calculation for the minimal distance function from the geodesic submanifold ${\mathrm{Sym}}(p,{\mathbb R})^{++}\times {\mathrm{Sym}}(q,{\mathbb R})^{++}$ block diagonally embedded in ${\mathrm{Sym}}(n,{\mathbb R})^{++}$ is given in terms of metric and spectral geometric means, Cayley transform, and Schur complements of positive definite matrices when $p\leq 2$ or $q\leq 2.$

Keywords:Matrix approximation, positive, definite matrix, geodesic submanifold, Cartan-Hadamard manifold,, best approximation, minimal distance function, global tubular, neighborhood theorem, Schur complement, metric and spectral, geometric mean, Cayley transform
Categories:15A48, 49R50, 15A18, 53C3

60. CJM 2004 (vol 56 pp. 566)

Ni, Yilong
Geodesics in a Manifold with Heisenberg Group as Boundary
The Heisenberg group is considered as the boundary of a manifold. A class of hypersurfaces in this manifold can be regarded as copies of the Heisenberg group. The properties of geodesics in the interior and on the hypersurfaces are worked out in detail. These properties are strongly related to those of the Heisenberg group.

Keywords:Heisenberg group, Hamiltonian mechanics, geodesic
Categories:53C22, 53C17

61. CJM 2004 (vol 56 pp. 590)

Ni, Yilong
The Heat Kernel and Green's Function on a Manifold with Heisenberg Group as Boundary
We study the Riemannian Laplace-Beltrami operator $L$ on a Riemannian manifold with Heisenberg group $H_1$ as boundary. We calculate the heat kernel and Green's function for $L$, and give global and small time estimates of the heat kernel. A class of hypersurfaces in this manifold can be regarded as approximations of $H_1$. We also restrict $L$ to each hypersurface and calculate the corresponding heat kernel and Green's function. We will see that the heat kernel and Green's function converge to the heat kernel and Green's function on the boundary.

Categories:35H20, 58J99, 53C17

62. CJM 2004 (vol 56 pp. 553)

Mohammadalikhani, Ramin
Cohomology Ring of Symplectic Quotients by Circle Actions
In this article we are concerned with how to compute the cohomology ring of a symplectic quotient by a circle action using the information we have about the cohomology of the original manifold and some data at the fixed point set of the action. Our method is based on the Tolman-Weitsman theorem which gives a characterization of the kernel of the Kirwan map. First we compute a generating set for the kernel of the Kirwan map for the case of product of compact connected manifolds such that the cohomology ring of each of them is generated by a degree two class. We assume the fixed point set is isolated; however the circle action only needs to be ``formally Hamiltonian''. By identifying the kernel, we obtain the cohomology ring of the symplectic quotient. Next we apply this result to some special cases and in particular to the case of products of two dimensional spheres. We show that the results of Kalkman and Hausmann-Knutson are special cases of our result.

Categories:53D20, 53D30, 37J10, 37J15, 53D05

63. CJM 2003 (vol 55 pp. 1000)

Graczyk, P.; Sawyer, P.
Some Convexity Results for the Cartan Decomposition
In this paper, we consider the set $\mathcal{S} = a(e^X K e^Y)$ where $a(g)$ is the abelian part in the Cartan decomposition of $g$. This is exactly the support of the measure intervening in the product formula for the spherical functions on symmetric spaces of noncompact type. We give a simple description of that support in the case of $\SL(3,\mathbf{F})$ where $\mathbf{F} = \mathbf{R}$, $\mathbf{C}$ or $\mathbf{H}$. In particular, we show that $\mathcal{S}$ is convex. We also give an application of our result to the description of singular values of a product of two arbitrary matrices with prescribed singular values.

Keywords:convexity theorems, Cartan decomposition, spherical functions, product formula, semisimple Lie groups, singular values
Categories:43A90, 53C35, 15A18

64. CJM 2003 (vol 55 pp. 1080)

Kellerhals, Ruth
Quaternions and Some Global Properties of Hyperbolic $5$-Manifolds
We provide an explicit thick and thin decomposition for oriented hyperbolic manifolds $M$ of dimension $5$. The result implies improved universal lower bounds for the volume $\rmvol_5(M)$ and, for $M$ compact, new estimates relating the injectivity radius and the diameter of $M$ with $\rmvol_5(M)$. The quantification of the thin part is based upon the identification of the isometry group of the universal space by the matrix group $\PS_\Delta {\rm L} (2,\mathbb{H})$ of quaternionic $2\times 2$-matrices with Dieudonn\'e determinant $\Delta$ equal to $1$ and isolation properties of $\PS_\Delta {\rm L} (2,\mathbb{H})$.

Categories:53C22, 53C25, 57N16, 57S30, 51N30, 20G20, 22E40

65. CJM 2003 (vol 55 pp. 266)

Kogan, Irina A.
Two Algorithms for a Moving Frame Construction
The method of moving frames, introduced by Elie Cartan, is a powerful tool for the solution of various equivalence problems. The practical implementation of Cartan's method, however, remains challenging, despite its later significant development and generalization. This paper presents two new variations on the Fels and Olver algorithm, which under some conditions on the group action, simplify a moving frame construction. In addition, the first algorithm leads to a better understanding of invariant differential forms on the jet bundles, while the second expresses the differential invariants for the entire group in terms of the differential invariants of its subgroup.

Categories:53A55, 58D19, 68U10

66. CJM 2003 (vol 55 pp. 112)

Shen, Zhongmin
Finsler Metrics with ${\bf K}=0$ and ${\bf S}=0$
In the paper, we study the shortest time problem on a Riemannian space with an external force. We show that such problem can be converted to a shortest path problem on a Randers space. By choosing an appropriate external force on the Euclidean space, we obtain a non-trivial Randers metric of zero flag curvature. We also show that any positively complete Randers metric with zero flag curvature must be locally Minkowskian.

Categories:53C60, 53B40

67. CJM 2002 (vol 54 pp. 449)

Akrout, H.
Théorème de Vorono\"\i\ dans les espaces symétriques
On d\'emontre un th\'eor\`eme de Vorono\"\i\ (caract\'erisation des maxima locaux de l'invariant d'Hermite) pour les familles de r\'eseaux param\'etr\'ees par les espaces sym\'etriques irr\'e\-ductibles non exceptionnels de type non compact. We prove a theorem of Vorono\"\i\ type (characterisation of local maxima of the Hermite invariant) for the lattices parametrized by irreducible nonexceptional symmetric spaces of noncompact type.

Keywords:réseaux, théorème de Vorono\"\i, espaces symétriques
Categories:11H06, 53C35

68. CJM 2002 (vol 54 pp. 3)

Alekseev, A.; Kosmann-Schwarzbach, Y.; Meinrenken, E.
Quasi-Poisson Manifolds
A quasi-Poisson manifold is a $G$-manifold equipped with an invariant bivector field whose Schouten bracket is the trivector field generated by the invariant element in $\wedge^3 \g$ associated to an invariant inner product. We introduce the concept of the fusion of such manifolds, and we relate the quasi-Poisson manifolds to the previously introduced quasi-Hamiltonian manifolds with group-valued moment maps.

Category:53D

69. CJM 2002 (vol 54 pp. 30)

Treloar, Thomas
The Symplectic Geometry of Polygons in the $3$-Sphere
We study the symplectic geometry of the moduli spaces $M_r=M_r(\s^3)$ of closed $n$-gons with fixed side-lengths in the $3$-sphere. We prove that these moduli spaces have symplectic structures obtained by reduction of the fusion product of $n$ conjugacy classes in $\SU(2)$ by the diagonal conjugation action of $\SU(2)$. Here the fusion product of $n$ conjugacy classes is a Hamiltonian quasi-Poisson $\SU(2)$-manifold in the sense of \cite{AKSM}. An integrable Hamiltonian system is constructed on $M_r$ in which the Hamiltonian flows are given by bending polygons along a maximal collection of nonintersecting diagonals. Finally, we show the symplectic structure on $M_r$ relates to the symplectic structure obtained from gauge-theoretic description of $M_r$. The results of this paper are analogues for the $3$-sphere of results obtained for $M_r(\h^3)$, the moduli space of $n$-gons with fixed side-lengths in hyperbolic $3$-space \cite{KMT}, and for $M_r(\E^3)$, the moduli space of $n$-gons with fixed side-lengths in $\E^3$ \cite{KM1}.

Category:53D

70. CJM 2001 (vol 53 pp. 780)

Nicolaescu, Liviu I.
Seiberg-Witten Invariants of Lens Spaces
We show that the Seiberg-Witten invariants of a lens space determine and are determined by its Casson-Walker invariant and its Reidemeister-Turaev torsion.

Keywords:lens spaces, Seifert manifolds, Seiberg-Witten invariants, Casson-Walker invariant, Reidemeister torsion, eta invariants, Dedekind-Rademacher sums
Categories:58D27, 57Q10, 57R15, 57R19, 53C20, 53C25

71. CJM 2000 (vol 52 pp. 757)

Hanani, Abdellah
Le problème de Neumann pour certaines équations du type de Monge-Ampère sur une variété riemannienne
Let $(M_n,g)$ be a strictly convex riemannian manifold with $C^{\infty}$ boundary. We prove the existence\break of classical solution for the nonlinear elliptic partial differential equation of Monge-Amp\`ere:\break $\det (-u\delta^i_j + \nabla^i_ju) = F(x,\nabla u;u)$ in $M$ with a Neumann condition on the boundary of the form $\frac{\partial u}{\partial \nu} = \varphi (x,u)$, where $F \in C^{\infty} (TM \times \bbR)$ is an everywhere strictly positive function satisfying some assumptions, $\nu$ stands for the unit normal vector field and $\varphi \in C^{\infty} (\partial M \times \bbR)$ is a non-decreasing function in $u$.

Keywords:connexion de Levi-Civita, équations de Monge-Ampère, problème de Neumann, estimées a priori, méthode de continuité
Categories:35J60, 53C55, 58G30

72. CJM 1999 (vol 51 pp. 1123)

Arnold, V. I.
First Steps of Local Contact Algebra
We consider germs of mappings of a line to contact space and classify the first simple singularities up to the action of contactomorphisms in the target space and diffeomorphisms of the line. Even in these first cases there arises a new interesting interaction of local commutative algebra with contact structure.

Keywords:contact manifolds, local contact algebra, Diracian, contactian
Categories:53D10, 14B05

73. CJM 1999 (vol 51 pp. 449)

Bahn, Hyoungsick; Ehrlich, Paul
A Brunn-Minkowski Type Theorem on the Minkowski Spacetime
In this article, we derive a Brunn-Minkowski type theorem for sets bearing some relation to the causal structure on the Minkowski spacetime $\mathbb{L}^{n+1}$. We also present an isoperimetric inequality in the Minkowski spacetime $\mathbb{L}^{n+1}$ as a consequence of this Brunn-Minkowski type theorem.

Keywords:Minkowski spacetime, Brunn-Minkowski inequality, isoperimetric inequality
Categories:53B30, 52A40, 52A38

74. CJM 1998 (vol 50 pp. 1298)

Milson, Robert
Imprimitively generated Lie-algebraic Hamiltonians and separation of variables
Turbiner's conjecture posits that a Lie-algebraic Hamiltonian operator whose domain is a subset of the Euclidean plane admits a separation of variables. A proof of this conjecture is given in those cases where the generating Lie-algebra acts imprimitively. The general form of the conjecture is false. A counter-example is given based on the trigonometric Olshanetsky-Perelomov potential corresponding to the $A_2$ root system.

Categories:35Q40, 53C30, 81R05

75. CJM 1997 (vol 49 pp. 1162)

Ku, Hsu-Tung; Ku, Mei-Chin; Zhang, Xin-Min
Isoperimetric inequalities on surfaces of constant curvature
In this paper we introduce the concepts of hyperbolic and elliptic areas and prove uncountably many new geometric isoperimetric inequalities on the surfaces of constant curvature.

Keywords:Gaussian curvature, Gauss-Bonnet theorem, polygon, pseudo-polygon, pseudo-perimeter, hyperbolic surface, Heron's formula, analytic and geometric isoperimetric inequalities
Categories:51M10, 51M25, 52A40, 53C20
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