51. CMB 2007 (vol 50 pp. 24)
52. CMB 2006 (vol 49 pp. 321)
53. CMB 2006 (vol 49 pp. 226)
 Engman, Martin

The Spectrum and Isometric Embeddings of Surfaces of Revolution
A sharp upper bound on the first $S^{1}$ invariant eigenvalue of the Laplacian
for $S^1$ invariant metrics on $S^2$ is used to find obstructions to the existence
of $S^1$ equivariant isometric embeddings of such metrics in $(\R^3,\can)$. As a
corollary we prove: If the first four distinct eigenvalues have even multiplicities
then the metric cannot be equivariantly, isometrically embedded in $(\R^3,\can)$. This
leads to generalizations of some classical results in the theory of surfaces.
Categories:58J50, 58J53, 53C20, 35P15 

54. CMB 2006 (vol 49 pp. 134)
55. CMB 2006 (vol 49 pp. 152)
56. CMB 2006 (vol 49 pp. 36)
57. CMB 2005 (vol 48 pp. 561)
 Foth, Philip

A Note on Lagrangian Loci of Quotients
We study Hamiltonian actions of compact groups in the presence of
compatible involutions. We show that the Lagrangian fixed point set
on the symplectically reduced space is isomorphic to the disjoint
union of the involutively reduced spaces corresponding to
involutions on the group strongly inner to the given one.
Our techniques imply that the solution to the eigenvalues of a sum problem
for a given real form can be reduced to the quasisplit real form in the
same inner class. We also consider invariant quotients with respect to
the corresponding real form of the complexified group.
Keywords:Quotients, involutions, real forms, Lagrangian loci Category:53D20 

58. CMB 2005 (vol 48 pp. 112)
 Mo, Xiaohuan; Shen, Zhongmin

On Negatively Curved Finsler Manifolds of Scalar Curvature
In this paper, we prove a global rigidity theorem for negatively
curved Finsler metrics on a compact manifold of dimension $n \geq 3$.
We show that for such a Finsler manifold, if the flag curvature is a
scalar function on the tangent bundle, then the Finsler metric is of
Randers type. We also study the case when the Finsler metric is
locally projectively flat
Category:53C60 

59. CMB 2004 (vol 47 pp. 624)
 Zhang, Xi

A Compactness Theorem for YangMills Connections
In this paper, we consider YangMills connections
on a vector bundle $E$ over a compact Riemannian manifold $M$ of
dimension $m> 4$, and we show that any set of YangMills
connections with the uniformly bounded $L^{\frac{m}{2}}$norm of
curvature is compact in $C^{\infty}$ topology.
Keywords:YangMills connection, vector bundle, gauge transformation Categories:58E20, 53C21 

60. CMB 2004 (vol 47 pp. 492)
61. CMB 2004 (vol 47 pp. 354)
62. CMB 2004 (vol 47 pp. 314)
63. CMB 2003 (vol 46 pp. 617)
 Pak, Hong Kyung

On Harmonic Theory in Flows
Recently [8], a harmonic theory was developed for a compact
contact manifold from the viewpoint of the transversal geometry of
contact flow. A contact flow is a typical example of geodesible
flow. As a natural generalization of the contact flow, the present
paper develops a harmonic theory for various flows on compact
manifolds. We introduce the notions of $H$harmonic and
$H^*$harmonic spaces associated to a H\"ormander flow. We also
introduce the notions of basic harmonic spaces associated to a weak
basic flow. One of our main results is to show that in the special
case of isometric flow these harmonic spaces are isomorphic to the
cohomology spaces of certain complexes. Moreover, we find an
obstruction for a geodesible flow to be isometric.
Keywords:contact structure, geodesible flow, isometric flow, basic cohomology Categories:53C20, 57R30 

64. CMB 2003 (vol 46 pp. 277)
 Rochon, Frédéric

Rigidity of Hamiltonian Actions
This paper studies the following question: Given an
$\omega'$symplectic action of a Lie group on a manifold $M$ which
coincides, as a smooth action, with a Hamiltonian $\omega$action,
when is this action a Hamiltonian $\omega'$action? Using a result of
MorseBott theory presented in Section~2, we show in Section~3 of this
paper that such an action is in fact a Hamiltonian $\omega'$action,
provided that $M$ is compact and that the Lie group is compact and
connected. This result was first proved by LalondeMcDuffPolterovich
in 1999 as a consequence of a more general theory that made use of
hard geometric analysis. In this paper, we prove it using classical
methods only.
Categories:53D05, 37J25 

65. CMB 2003 (vol 46 pp. 130)
 Petersen, Peter; Wilhelm, Frederick

On Frankel's Theorem
In this paper we show that two minimal hypersurfaces in a manifold with
positive Ricci curvature must intersect. This is then generalized to show
that in manifolds with positive Ricci curvature in the integral sense two
minimal hypersurfaces must be close to each other. We also show
what happens if a manifold with nonnegative Ricci curvature admits
two nonintersecting minimal hypersurfaces.
Keywords:Frankel's Theorem Category:53C20 

66. CMB 2002 (vol 45 pp. 378)
 FernándezLópez, Manuel; GarcíaRío, Eduardo; Kupeli, Demir N.

The Local MÃ¶bius Equation and Decomposition Theorems in Riemannian Geometry
A partial differential equation, the local M\"obius equation, is
introduced in Riemannian geometry which completely characterizes the
local twisted product structure of a Riemannian manifold. Also the
characterizations of warped product and product structures of
Riemannian manifolds are made by the local M\"obius equation and an
additional partial differential equation.
Keywords:submersion, MÃ¶bius equation, twisted product, warped product, product Riemannian manifolds Categories:53C12, 58J99 

67. CMB 2002 (vol 45 pp. 161)
68. CMB 2002 (vol 45 pp. 232)
 Ji, Min; Shen, Zhongmin

On Strongly Convex Indicatrices in Minkowski Geometry
The geometry of indicatrices is the foundation of Minkowski geometry.
A strongly convex indicatrix in a vector space is a strongly convex
hypersurface. It admits a Riemannian metric and has a distinguished
invariant(Cartan) torsion. We prove the existence of nontrivial
strongly convex indicatrices with vanishing mean torsion and discuss
the relationship between the mean torsion and the Riemannian curvature
tensor for indicatrices of Randers type.
Categories:46B20, 53C21, 53A55, 52A20, 53B40, 53A35 

69. CMB 2001 (vol 44 pp. 408)
70. CMB 2001 (vol 44 pp. 376)
71. CMB 2001 (vol 44 pp. 129)
 CurrásBosch, Carlos

LinÃ©arisation symplectique en dimension 2
In this paper the germ of neighborhood of a compact leaf in a
Lagrangian foliation is symplectically classified when the compact
leaf is $\bT^2$, the affine structure induced by the Lagrangian
foliation on the leaf is complete, and the holonomy of $\bT^2$ in
the foliation linearizes. The germ of neighborhood is classified by a
function, depending on one transverse coordinate, this function is
related to the affine structure of the nearly compact leaves.
Keywords:symplectic manifold, Lagrangian foliation, affine connection Categories:53C12, 58F05 

72. CMB 2001 (vol 44 pp. 70)
73. CMB 2001 (vol 44 pp. 36)
 Kapovich, Michael; Millson, John J.

Quantization of Bending Deformations of Polygons In $\mathbb{E}^3$, Hypergeometric Integrals and the Gassner Representation
The Hamiltonian potentials of the bending deformations of $n$gons
in $\E^3$ studied in \cite{KM} and \cite{Kl} give rise to a Hamiltonian
action of the Malcev Lie algebra $\p_n$ of the pure braid group
$P_n$ on the moduli space $M_r$ of $n$gon linkages with the sidelengths
$r= (r_1,\dots, r_n)$ in $\E^3$. If $e\in M_r$ is a singular point we may
linearize the vector fields in $\p_n$ at $e$. This linearization yields a
flat connection $\nabla$ on the space $\C^n_*$ of $n$ distinct points on
$\C$. We show that the monodromy of $\nabla$ is the dual of a quotient
of a specialized reduced Gassner representation.
Categories:53D30, 53D50 

74. CMB 2000 (vol 43 pp. 440)
75. CMB 2000 (vol 43 pp. 427)
 Ivey, Thomas A.

Helices, Hasimoto Surfaces and BÃ¤cklund Transformations
Travelling wave solutions to the vortex filament flow generated by
elastica produce surfaces in $\R^3$ that carry mutually orthogonal
foliations by geodesics and by helices. These surfaces are classified
in the special cases where the helices are all congruent or are all
generated by a single screw motion. The first case yields a new
characterization for the B\"acklund transformation for constant
torsion curves in $\R^3$, previously derived from the wellknown
transformation for pseudospherical surfaces. A similar investigation
for surfaces in $H^3$ or $S^3$ leads to a new transformation for
constant torsion curves in those spaces that is also derived from
pseudospherical surfaces.
Keywords:surfaces, filament flow, BÃ¤cklund transformations Categories:53A05, 58F37, 52C42, 58A15 
