51. CJM 2007 (vol 59 pp. 845)
 Schaffhauser, Florent

Representations of the Fundamental Group of an $L$Punctured Sphere Generated by Products of Lagrangian Involutions
In this paper, we characterize unitary representations of $\pi:=\piS$ whose
generators $u_1, \dots, u_l$ (lying in conjugacy classes fixed initially)
can be decomposed as products of two Lagrangian involutions
$u_j=\s_j\s_{j+1}$ with $\s_{l+1}=\s_1$. Our main result is that such
representations are exactly the elements of the fixedpoint set of an
antisymplectic involution defined on the moduli space
$\Mod:=\Hom_{\mathcal C}(\pi,U(n))/U(n)$. Consequently, as this fixedpoint set is
nonempty, it is a Lagrangian submanifold of $\Mod$. To prove this, we use
the quasiHamiltonian description of the symplectic structure of $\Mod$ and
give conditions on an involution defined on a quasiHamiltonian $U$space
$(M, \w, \mu\from M \to U)$ for it to induce an antisymplectic involution on
the reduced space $M/\!/U := \mu^{1}(\{1\})/U$.
Keywords:momentum maps, moduli spaces, Lagrangian submanifolds, antisymplectic involutions, quasiHamiltonian Categories:53D20, 53D30 

52. CJM 2006 (vol 58 pp. 600)
 MartinezMaure, Yves

Geometric Study of Minkowski Differences of Plane Convex Bodies
In the Euclidean plane $\mathbb{R}^{2}$, we define the Minkowski difference
$\mathcal{K}\mathcal{L}$ of two arbitrary convex bodies $\mathcal{K}$,
$\mathcal{L}$ as a rectifiable closed curve $\mathcal{H}_{h}\subset \mathbb{R}
^{2}$ that is determined by the difference $h=h_{\mathcal{K}}h_{\mathcal{L}
} $ of their support functions. This curve $\mathcal{H}_{h}$ is
called the
hedgehog with support function $h$. More generally, the object of hedgehog
theory is to study the BrunnMinkowski theory in the vector space of
Minkowski differences of arbitrary convex bodies of Euclidean space $\mathbb{R}
^{n+1}$, defined as (possibly singular and selfintersecting) hypersurfaces
of $\mathbb{R}^{n+1}$. Hedgehog theory is useful for: (i)
studying convex bodies by splitting them into a sum in order to reveal their
structure; (ii) converting analytical problems into
geometrical ones by considering certain real functions as support
functions.
The purpose of this paper is to give a detailed study of plane
hedgehogs, which constitute the basis of the theory. In particular:
(i) we study their length measures and solve the extension of the
ChristoffelMinkowski problem to plane hedgehogs; (ii) we
characterize support functions of plane convex bodies among support
functions of plane hedgehogs and support functions of plane hedgehogs among
continuous functions; (iii) we study the mixed area of
hedgehogs in $\mathbb{R}^{2}$ and give an extension of the classical Minkowski
inequality (and thus of the isoperimetric inequality) to hedgehogs.
Categories:52A30, 52A10, 53A04, 52A38, 52A39, 52A40 

53. CJM 2006 (vol 58 pp. 282)
 Fels, M. E.; Renner, A. G.

Nonreductive Homogeneous PseudoRiemannian Manifolds of Dimension Four
A method, due to \'Elie Cartan, is used to give an algebraic
classification of the nonreductive homogeneous pseudoRiemannian
manifolds of dimension four. Only one case with Lorentz signature can
be Einstein without having constant curvature, and two cases with
$(2,2)$ signature are Einstein of which one is Ricciflat. If a
fourdimensional nonreductive homogeneous pseudoRiemannian manifold
is simply connected, then it is shown to be diffeomorphic to
$\reals^4$. All metrics for the simply connected nonreductive
Einstein spaces are given explicitly. There are no nonreductive
pseudoRiemannian homogeneous spaces of dimension two and none of
dimension three with connected isotropy subgroup.
Keywords:Homogeneous pseudoRiemannian, Einstein space Category:53C30 

54. CJM 2006 (vol 58 pp. 381)
 Jakobson, Dmitry; Nadirashvili, Nikolai; Polterovich, Iosif

Extremal Metric for the First Eigenvalue on a Klein Bottle
The first eigenvalue of the Laplacian on a surface can be viewed
as a functional on the space of Riemannian metrics of a given
area. Critical points of this functional are called extremal
metrics. The only known extremal metrics are a round sphere, a
standard projective plane, a Clifford torus and an equilateral
torus. We construct an extremal metric on a Klein bottle. It is a
metric of revolution, admitting a minimal isometric embedding into
a sphere ${\mathbb S}^4$ by the first eigenfunctions. Also, this
Klein bottle is a bipolar surface for Lawson's
$\tau_{3,1}$torus. We conjecture that an extremal metric for the
first eigenvalue on a Klein bottle is unique, and hence it
provides a sharp upper bound for $\lambda_1$ on a Klein bottle of
a given area. We present numerical evidence and prove the first
results towards this conjecture.
Keywords:Laplacian, eigenvalue, Klein bottle Categories:58J50, 53C42 

55. CJM 2006 (vol 58 pp. 362)
 Goldin, R. F.; Martin, S.

Cohomology Pairings on the Symplectic Reduction of Products
Let $M$ be the product of two compact Hamiltonian
$T$spaces $X$ and $Y$. We present a formula for evaluating
integrals on the symplectic reduction of $M$ by the diagonal $T$
action. At every regular value of the moment map for $X\times Y$, the
integral is the convolution of two distributions associated to the
symplectic reductions of $X$ by $T$ and of $Y$ by $T$. Several
examples illustrate the computational strength of this relationship.
We also prove a linear analogue which can be used to find cohomology
pairings on toric orbifolds.
Category:53D20 

56. CJM 2006 (vol 58 pp. 262)
57. 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 

58. 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 DarbouxGivental' 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 

59. 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 

60. 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 

61. CJM 2005 (vol 57 pp. 871)
 Zhang, Xi

Hermitian Yang_MillsHiggs 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 wellknown 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 YangMillsHiggs metric,, holomorphic vector bundle, KÃ¤hler manifolds Categories:58E15, 53C07 

62. CJM 2005 (vol 57 pp. 750)
 Sabourin, Hervé

Sur la structure transverse Ã une orbite nilpotente adjointe
We are interested in Poisson structures to
transverse nilpotent adjoint orbits in a complex semisimple 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 

63. 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 actionangle 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 

64. CJM 2004 (vol 56 pp. 1228)
65. 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
CartanHadamard 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, CartanHadamard manifold,, best approximation, minimal distance function, global tubular, neighborhood theorem, Schur complement, metric and spectral, geometric mean, Cayley transform Categories:15A48, 49R50, 15A18, 53C3 

66. 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 LaplaceBeltrami 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 

67. 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 

68. 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 TolmanWeitsman 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 HausmannKnutson are special cases of our result.
Categories:53D20, 53D30, 37J10, 37J15, 53D05 

69. 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 

70. 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 

71. 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 

72. 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
nontrivial 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 

73. CJM 2002 (vol 54 pp. 449)
 Akrout, H.

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 

74. CJM 2002 (vol 54 pp. 3)
 Alekseev, A.; KosmannSchwarzbach, Y.; Meinrenken, E.

QuasiPoisson Manifolds
A quasiPoisson 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 quasiPoisson manifolds to the previously
introduced quasiHamiltonian manifolds with groupvalued moment maps.
Category:53D 

75. 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 sidelengths 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 quasiPoisson $\SU(2)$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 gaugetheoretic 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 sidelengths in hyperbolic $3$space \cite{KMT}, and for
$M_r(\E^3)$, the moduli space of $n$gons with fixed sidelengths in
$\E^3$ \cite{KM1}.
Category:53D 
