Expand all Collapse all | Results 51 - 75 of 78 |
51. CJM 2005 (vol 57 pp. 1314)
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 |
52. CJM 2005 (vol 57 pp. 1012)
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 |
53. CJM 2005 (vol 57 pp. 708)
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 |
54. CJM 2005 (vol 57 pp. 871)
Hermitian Yang-_Mills--Higgs Metrics on\\Complete KÃ¤hler 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, KÃ¤hler manifolds Categories:58E15, 53C07 |
55. CJM 2005 (vol 57 pp. 750)
Sur la structure transverse Ã 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 |
56. CJM 2005 (vol 57 pp. 114)
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 |
57. CJM 2004 (vol 56 pp. 1228)
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 |
58. CJM 2004 (vol 56 pp. 776)
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 |
59. CJM 2004 (vol 56 pp. 566)
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 |
60. CJM 2004 (vol 56 pp. 590)
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 |
61. CJM 2004 (vol 56 pp. 553)
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 |
62. CJM 2003 (vol 55 pp. 1000)
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 |
63. CJM 2003 (vol 55 pp. 1080)
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 |
64. CJM 2003 (vol 55 pp. 266)
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 |
65. CJM 2003 (vol 55 pp. 112)
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 |
66. CJM 2002 (vol 54 pp. 449)
ThÃ©orÃ¨me de Vorono\"\i\ dans les espaces symÃ©triques 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:rÃ©seaux, thÃ©orÃ¨me de Vorono\"\i, espaces symÃ©triques Categories:11H06, 53C35 |
67. CJM 2002 (vol 54 pp. 3)
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 |
68. CJM 2002 (vol 54 pp. 30)
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 |
69. CJM 2001 (vol 53 pp. 780)
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 |
70. CJM 2000 (vol 52 pp. 757)
Le problÃ¨me de Neumann pour certaines Ã©quations du type de Monge-AmpÃ¨re sur une variÃ©tÃ© riemannienne |
Le problÃ¨me de Neumann pour certaines Ã©quations du type de Monge-AmpÃ¨re sur une variÃ©tÃ© 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, Ã©quations de Monge-AmpÃ¨re, problÃ¨me de Neumann, estimÃ©es a priori, mÃ©thode de continuitÃ© Categories:35J60, 53C55, 58G30 |
71. CJM 1999 (vol 51 pp. 1123)
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 |
72. CJM 1999 (vol 51 pp. 449)
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 |
73. CJM 1998 (vol 50 pp. 1298)
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 |
74. CJM 1997 (vol 49 pp. 1162)
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 |
75. CJM 1997 (vol 49 pp. 1323)
Stable parallelizability of partially oriented flag manifolds II In the first paper with the same title the authors
were able to determine all partially oriented flag
manifolds that are stably parallelizable or
parallelizable, apart from four infinite families
that were undecided. Here, using more delicate
techniques (mainly K-theory), we settle these
previously undecided families and show that none of
the manifolds in them is stably parallelizable,
apart from one 30-dimensional manifold which still
remains undecided.
Categories:57R25, 55N15, 53C30 |