51. CMB 2007 (vol 50 pp. 347)
52. CMB 2007 (vol 50 pp. 474)
 Zhou, Jiazu

On Willmore's Inequality for Submanifolds
Let $M$ be an $m$ dimensional submanifold in the Euclidean space
${\mathbf R}^n$ and $H$ be the mean curvature of $M$. We obtain
some low geometric estimates of the total square mean curvature
$\int_M H^2 d\sigma$. The low bounds are geometric invariants
involving the volume of $M$, the total scalar curvature of $M$,
the Euler characteristic and the circumscribed ball of $M$.
Keywords:submanifold, mean curvature, kinematic formul, scalar curvature Categories:52A22, 53C65, 51C16 

53. CMB 2007 (vol 50 pp. 365)
 Godinho, Leonor

Equivariant Cohomology of $S^{1}$Actions on $4$Manifolds
Let $M$ be a symplectic $4$dimensional manifold equipped with a
Hamiltonian circle action with isolated fixed points. We describe a
method for computing its integral equivariant cohomology in terms of
fixed point data. We give some examples of these computations.
Categories:53D20, 55N91, 57S15 

54. CMB 2007 (vol 50 pp. 321)
 Blair, David E.

On Lagrangian Catenoids
Recently I. Castro and F. Urbano introduced the
Lagrangian catenoid.
Topologically, it is $\mathbb R\times S^{n1}$ and its induced metric is
conformally flat,
but not cylindrical. Their result is that if a Lagrangian minimal
submanifold in
${\mathbb C}^n$ is foliated by round $(n1)$spheres, it is congruent to
a Lagrangian
catenoid. Here we study the question of conformally flat, minimal, Lagrangian
submanifolds in
${\mathbb C}^n$. The general problem is formidable, but we first show that such a
submanifold resembles a Lagrangian catenoid in that its Schouten tensor has an
eigenvalue of multiplicity one. Then, restricting to the case of at most two
eigenvalues, we show that the submanifold is either flat and totally
geodesic or is
homothetic to (a piece of) the Lagrangian catenoid.
Categories:53C42, 53D12 

55. CMB 2007 (vol 50 pp. 97)
56. CMB 2007 (vol 50 pp. 113)
 Li, ZhenYang; Zhang, Xi

Hermitian Harmonic Maps into Convex Balls
In this paper, we consider Hermitian harmonic maps from
Hermitian manifolds into convex balls. We prove that there exist
no nontrivial Hermitian harmonic maps from closed Hermitian
manifolds into convex balls, and we use the heat flow method to
solve the Dirichlet problem for Hermitian harmonic maps when the
domain is a compact Hermitian manifold with nonempty boundary.
Keywords:Hermitian harmonic map, Hermitian manifold, convex ball Categories:58E15, 53C07 

57. CMB 2007 (vol 50 pp. 24)
58. CMB 2006 (vol 49 pp. 321)
59. 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 

60. CMB 2006 (vol 49 pp. 134)
61. CMB 2006 (vol 49 pp. 152)
62. CMB 2006 (vol 49 pp. 36)
63. 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 

64. 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 

65. CMB 2004 (vol 47 pp. 492)
66. 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 

67. CMB 2004 (vol 47 pp. 354)
68. CMB 2004 (vol 47 pp. 314)
69. 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 

70. 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 

71. 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 

72. 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 

73. CMB 2002 (vol 45 pp. 161)
74. 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 

75. CMB 2001 (vol 44 pp. 408)