51. CMB 2010 (vol 53 pp. 564)
 Watanabe, Yoshiyuki; Suh, Young Jin

On $6$Dimensional Nearly KÃ¤hler Manifolds
In this paper we give a sufficient condition for a complete, simply connected, and strict nearly KÃ¤hler manifold of dimension 6 to be a homogeneous nearly KÃ¤hler manifold. This result was announced in a previous paper by the first author.
Keywords:Nearly KÃ¤hler manifold, 6dimension, Homogeneous, The 1st Chern Class, Einstein manifolds Categories:53C40, 53C15 

52. CMB 2010 (vol 53 pp. 516)
53. CMB 2010 (vol 53 pp. 412)
54. CMB 2009 (vol 53 pp. 206)
 Atçeken, Mehmet

SemiSlant Submanifolds of an Almost Paracontact Metric Manifold
In this paper, we define and study the geometry of semislant submanifolds of an almost paracontact metric manifold. We give some characterizations for a submanifold to be semislant submanifold to be semislant product and obtain integrability conditions for the distributions involved in the definition of a semislant submanifold.
Keywords:paracontact metric manifold, slant distribution, semislant submanifold, semislant product Categories:53C15, 53C25, 53C40 

55. CMB 2009 (vol 40 pp. 257)
56. CMB 2009 (vol 40 pp. 108)
 Schaer, J.

Continuous Selfmaps of the Circle
Given a continuous map $\delta$ from the circle $S$ to itself we
want to find all selfmaps $\sigma\colon S\to S$ for which
$\delta\circ\sigma = \delta$. If the degree $r$ of $\delta$ is not
zero, the transformations $\sigma$ form a subgroup of the cyclic
group $C_r$. If $r=0$, all such invertible transformations form a
group isomorphic either to a cyclic group $C_n$ or to a dihedral
group $D_n$ depending on whether all such transformations are
orientation preserving or not. Applied to the tangent image of
planar closed curves, this generalizes a result of Bisztriczky and
Rival [1]. The proof rests on the theorem: {\it Let
$\Delta\colon\bbd R\to\bbd R$ be continuous, nowhere constant, and
$\lim_{x\to \infty}\Delta(x)=\infty$, $ \lim_{x\to+\infty}\Delta
(x)=+\infty$; then the only continuous map $\Sigma\colon\bbd R\to\bbd
R$ such that $\Delta\circ\Sigma=\Delta$ is the identity
$\Sigma=\id_{\bbd R}$.
Categories:53A04, 55M25, 55M35 

57. CMB 2009 (vol 40 pp. 204)
58. CMB 2009 (vol 52 pp. 18)
 Chinea, Domingo

Harmonicity of Holomorphic Maps Between Almost Hermitian Manifolds
In this paper we study holomorphic maps between almost Hermitian
manifolds. We obtain a new criterion for the harmonicity of such
holomorphic maps, and we deduce some applications to horizontally
conformal holomorphic submersions.
Keywords:almost Hermitian manifolds, harmonic maps, harmonic morphism Categories:53C15, 58E20 

59. CMB 2009 (vol 52 pp. 132)
 Shen, Zhongmin

On Projectively Flat $(\alpha,\beta)$metrics
The solutions to Hilbert's Fourth Problem in the regular case
are projectively flat Finsler metrics. In this paper,
we consider the socalled $(\alpha,\beta)$metrics defined by a
Riemannian metric $\alpha$ and a $1$form $\beta$, and find a
necessary and sufficient condition for such metrics to be projectively
flat in dimension $n \geq 3$.
Categories:53B40, 53C60 

60. CMB 2009 (vol 52 pp. 87)
 Lee, Junho

Holomorphic 2Forms and Vanishing Theorems for GromovWitten Invariants
On a compact K\"{a}hler manifold $X$ with a holomorphic 2form
$\a$, there is an almost complex structure associated with $\a$. We
show how this implies vanishing theorems for the GromovWitten
invariants of $X$. This extends the approach used by Parker and
the author for K\"{a}hler surfaces to higher dimensions.
Category:53D45 

61. CMB 2008 (vol 51 pp. 359)
62. CMB 2008 (vol 51 pp. 467)
 Wang, Yue

Coupled Vortex Equations on Complete KÃ¤hler Manifolds
In this paper, we first investigate the Dirichlet
problem for coupled vortex equations. Secondly, we give existence
results for solutions of the coupled vortex equations on a class
of complete noncompact K\"ahler manifolds which include
simplyconnected strictly negative curved manifolds, Hermitian
symmetric spaces of noncompact type and strictly pseudoconvex
domains equipped with the Bergmann metric.
Categories:58J05, 53C07 

63. CMB 2008 (vol 51 pp. 448)
64. CMB 2007 (vol 50 pp. 347)
65. 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 

66. 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 

67. 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 

68. CMB 2007 (vol 50 pp. 24)
69. 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 

70. CMB 2007 (vol 50 pp. 97)
71. CMB 2006 (vol 49 pp. 321)
72. 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 

73. CMB 2006 (vol 49 pp. 152)
74. CMB 2006 (vol 49 pp. 134)
75. CMB 2006 (vol 49 pp. 36)