26. CMB 2011 (vol 56 pp. 184)
 Shen, Zhongmin

On Some NonRiemannian Quantities in Finsler Geometry
In this paper we study several nonRiemannian quantities in Finsler
geometry. These nonRiemannian quantities play an important role in
understanding the geometric properties of Finsler metrics. In
particular, we study a new nonRiemannian quantity defined by the
Scurvature. We show some relationships among the flag curvature,
the Scurvature, and the new nonRiemannian quantity.
Keywords:Finsler metric, Scurvature, nonRiemannian quantity Categories:53C60, 53B40 

27. CMB 2011 (vol 56 pp. 615)
 Sevim, Esra Sengelen; Shen, Zhongmin

Randers Metrics of Constant Scalar Curvature
Randers metrics are a special class of Finsler metrics. Every Randers
metric can be expressed in terms of a Riemannian metric and a vector
field via Zermelo navigation.
In this paper, we show that a Randers metric has constant scalar
curvature if the Riemannian metric has constant scalar curvature and
the vector field is homothetic.
Keywords:Randers metrics, scalar curvature, Scurvature Categories:53C60, 53B40 

28. CMB 2011 (vol 56 pp. 127)
 Li, Junfang

Evolution of Eigenvalues along Rescaled Ricci Flow
In this paper, we discuss monotonicity formulae of various entropy functionals under various
rescaled versions of Ricci flow. As an application, we prove that the lowest eigenvalue
of a family of geometric operators $4\Delta + kR$ is monotonic along the
normalized Ricci flow for all $k\ge 1$ provided the initial manifold has
nonpositive total scalar curvature.
Keywords:monotonicity formulas, Ricci flow Categories:58C40, 53C44 

29. CMB 2011 (vol 55 pp. 870)
 Wang, Hui; Deng, Shaoqiang

Left Invariant EinsteinRanders Metrics on Compact Lie Groups
In this paper we study left invariant EinsteinRanders metrics on compact Lie
groups. First, we give a method to construct left invariant nonRiemannian EinsteinRanders metrics
on a compact Lie group, using the Zermelo navigation data.
Then we prove that this gives a complete classification of left invariant EinsteinRanders metrics on compact simple
Lie groups with the underlying Riemannian metric naturally reductive.
Further, we completely determine the identity component of the group of
isometries for this type of metrics on simple groups. Finally, we study some
geometric properties of such metrics. In particular, we give the formulae of geodesics and flag curvature
of such metrics.
Keywords:EinsteinRanders metric, compact Lie groups, geodesic, flag curvature Categories:17B20, 22E46, 53C12 

30. CMB 2011 (vol 56 pp. 173)
 Sahin, Bayram

Semiinvariant Submersions from Almost Hermitian Manifolds
We introduce semiinvariant Riemannian submersions from almost
Hermitian manifolds onto Riemannian manifolds. We give examples,
investigate the geometry of foliations that arise from the
definition of a Riemannian submersion, and find necessary sufficient
conditions for total manifold to be a locally product Riemannian
manifold. We also find necessary and sufficient conditions for a
semiinvariant submersion to be totally geodesic. Moreover, we
obtain a classification for semiinvariant submersions with totally
umbilical fibers and show that such submersions put some
restrictions on total manifolds.
Keywords:Riemannian submersion, Hermitian manifold, antiinvariant Riemannian submersion, semiinvariant submersion Categories:53B20, 53C43 

31. CMB 2011 (vol 55 pp. 632)
 Pigola, S.; Rimoldi, M.

Characterizations of Model Manifolds by Means of Certain Differential Systems
We prove metric rigidity for complete manifolds supporting solutions of
certain second order differential systems, thus extending classical works on a
characterization of spaceforms. Along the way, we also discover
new characterizations of spaceforms. We next generalize results concerning metric
rigidity via equations involving vector fields.
Keywords:metric rigidity, model manifolds, Obata's type theorems Category:53C20 

32. CMB 2011 (vol 56 pp. 116)
 Krepski, Derek

Central Extensions of Loop Groups and Obstruction to PreQuantization
An explicit construction of a prequantum line bundle for the moduli
space of flat $G$bundles over a Riemann surface is given, where $G$
is any nonsimply connected compact simple Lie group. This work helps
to explain a curious coincidence previously observed between
Toledano Laredo's work classifying central extensions of loop groups
$LG$ and the author's previous work on the obstruction to
prequantization of the moduli space of flat $G$bundles.
Keywords:loop group, central extension, prequantization Categories:53D, 22E 

33. CMB 2011 (vol 55 pp. 723)
 Gigli, Nicola; Ohta, ShinIchi

First Variation Formula in Wasserstein Spaces over Compact Alexandrov Spaces
We extend results proved by the second author (Amer. J. Math., 2009)
for nonnegatively curved Alexandrov spaces
to general compact Alexandrov spaces $X$ with curvature bounded
below.
The gradient flow of a geodesically convex functional on the quadratic Wasserstein
space $(\mathcal P(X),W_2)$ satisfies the evolution variational inequality.
Moreover, the gradient flow enjoys uniqueness and contractivity.
These results are obtained by proving a first variation formula for
the Wasserstein distance.
Keywords:Alexandrov spaces, Wasserstein spaces, first variation formula, gradient flow Categories:53C23, 28A35, 49Q20, 58A35 

34. CMB 2011 (vol 55 pp. 663)
 Zhou, Chunqin

An Onofritype Inequality on the Sphere with Two Conical Singularities
In this paper, we give a new proof of the Onofritype inequality
\begin{equation*}
\int_S e^{2u} \,ds^2 \leq 4\pi(\beta+1) \exp \biggl\{
\frac{1}{4\pi(\beta+1)} \int_S \nabla u^2 \,ds^2 +
\frac{1}{2\pi(\beta+1)} \int_S u \,ds^2 \biggr\}
\end{equation*}
on the sphere $S$ with Gaussian curvature $1$ and with conical
singularities divisor $\mathcal A = \beta\cdot p_1 + \beta \cdot p_2$ for
$\beta\in (1,0)$; here $p_1$ and $p_2$ are antipodal.
Categories:53C21, 35J61, 53A30 

35. CMB 2011 (vol 56 pp. 44)
36. CMB 2011 (vol 55 pp. 611)
 Özgür, Cihan; Mihai, Adela

Chen Inequalities for Submanifolds of Real Space Forms with a SemiSymmetric NonMetric Connection
In this paper we prove Chen inequalities for submanifolds of real space
forms endowed with a semisymmetric nonmetric connection, i.e., relations
between the mean curvature associated with a semisymmetric nonmetric
connection, scalar and sectional curvatures, Ricci curvatures and the
sectional curvature of the ambient space. The equality cases are considered.
Keywords:real space form, semisymmetric nonmetric connection, Ricci curvature Categories:53C40, 53B05, 53B15 

37. CMB 2011 (vol 55 pp. 108)
38. CMB 2011 (vol 55 pp. 329)
 Kamiya, Shigeyasu; Parker, John R.; Thompson, James M.

NonDiscrete Complex Hyperbolic Triangle Groups of Type $(n,n, \infty;k)$
A complex hyperbolic triangle group is a group
generated by three involutions fixing complex lines in complex
hyperbolic space. Our purpose in this paper is to improve a previous result
and to discuss discreteness of complex hyperbolic
triangle groups of type $(n,n,\infty;k)$.
Keywords:complex hyperbolic triangle group Categories:51M10, 32M15, 53C55, 53C35 

39. CMB 2011 (vol 55 pp. 474)
 Chen, Bin; Zhao, Lili

A Note on Randers Metrics of Scalar Flag Curvature
Some families of Randers metrics of scalar flag curvature are
studied in this paper. Explicit examples that are neither locally
projectively flat nor of isotropic $S$curvature are given. Certain
Randers metrics with Einstein $\alpha$ are considered and proved to
be complex. Three dimensional Randers manifolds, with $\alpha$
having constant scalar curvature, are studied.
Keywords:Randers metrics, scalar flag curvature Categories:53B40, 53C60 

40. CMB 2011 (vol 55 pp. 138)
41. CMB 2011 (vol 55 pp. 114)
42. CMB 2011 (vol 54 pp. 716)
 Okassa, Eugène

Symplectic LieRinehartJacobi Algebras and Contact Manifolds
We give a characterization of contact manifolds in terms of symplectic
LieRinehartJacobi algebras. We also give a sufficient condition for a Jacobi
manifold to be a contact manifold.
Keywords:LieRinehart algebras, differential operators, Jacobi manifolds, symplectic manifolds, contact manifolds Categories:13N05, 53D05, 53D10 

43. CMB 2011 (vol 54 pp. 422)
44. CMB 2010 (vol 53 pp. 684)
 Proctor, Emily; Stanhope, Elizabeth

An Isospectral Deformation on an InfranilOrbifold
We construct a Laplace isospectral deformation of metrics on an
orbifold quotient of a nilmanifold. Each orbifold in the deformation
contains singular points with order two isotropy. Isospectrality is
obtained by modifying a generalization of Sunada's theorem due to
DeTurck and Gordon.
Keywords:spectral geometry, global Riemannian geometry, orbifold, nilmanifold Categories:58J53, 53C20 

45. 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 

46. CMB 2010 (vol 53 pp. 516)
47. CMB 2010 (vol 53 pp. 412)
48. 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 

49. 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 

50. 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 
