1. CMB Online first
 Nicola, Antonio De; Yudin, Ivan

Generalized Goldberg Formula
In this paper we prove a useful formula for the graded commutator
of the Hodge
codifferential with the left wedge multiplication by a fixed
$p$form acting on
the de Rham algebra of a Riemannian manifold. Our formula generalizes
a formula
stated by Samuel I. Goldberg for the case of 1forms. As first
examples of
application we obtain new identities on locally conformally KÃ¤hler
manifolds
and quasiSasakian manifolds. Moreover, we prove that under suitable
conditions
a certain subalgebra of differential forms in a compact manifold
is quasiisomorphic as a CDGA to the full de Rham algebra.
Keywords:graded commutator, Hodge codifferential, Hodge laplacian, de Rham cohomology, locally conformal Kaehler manifold, quasiSasakian manifold Categories:53C25, 53D35 

2. CMB 2015 (vol 59 pp. 50)
3. CMB 2015 (vol 58 pp. 692)
 Anona, F. M.; Randriambololondrantomalala, Princy; Ravelonirina, H. S. G.

Sur les algÃ¨bres de Lie associÃ©es Ã une connexion
Let $\Gamma$ be a connection on a smooth manifold
$M$, in this paper we give some properties of $\Gamma$ by studying
the corresponding Lie algebras. In particular, we compute the
first ChevalleyEilenberg cohomology space of the horizontal
vector fields Lie algebra on the tangent bundle of $M$, whose
the corresponding Lie derivative of $\Gamma$ is null, and of
the horizontal nullity curvature space.
Keywords:algÃ¨bre de Lie, connexion, cohomologie de ChevalleyEilenberg, champs dont la dÃ©rivÃ©e de Lie correspondante Ã une connexion est nulle, espace de nullitÃ© de la courbure Categories:17B66, 53B15 

4. CMB 2015 (vol 58 pp. 835)
5. CMB 2015 (vol 58 pp. 787)
 Kitabeppu, Yu; Lakzian, Sajjad

Nonbranching RCD$(0,N)$ Geodesic Spaces with Small Linear Diameter Growth have Finitely Generated Fundamental Groups
In this paper, we generalize the finite generation result of
Sormani
to nonbranching $RCD(0,N)$
geodesic spaces (and in particular, Alexandrov spaces) with full
support measures. This is a special case of the Milnor's Conjecture
for complete noncompact $RCD(0,N)$ spaces. One of the key tools
we use is the AbreschGromoll type excess estimates for nonsmooth
spaces obtained by GigliMosconi.
Keywords:Milnor conjecture, non negative Ricci curvature, curvature dimension condition, finitely generated, fundamental group, infinitesimally Hilbertian Categories:53C23, 30L99 

6. CMB 2015 (vol 58 pp. 530)
 Li, Benling; Shen, Zhongmin

Ricci Curvature Tensor and NonRiemannian Quantities
There are several notions of Ricci curvature tensor
in Finsler geometry and spray geometry. One of them is defined by the
Hessian of the wellknown Ricci curvature.
In this paper we will introduce a new notion of Ricci curvature
tensor and discuss its relationship with the Ricci curvature and some
nonRiemannian quantities. By this Ricci curvature tensor, we shall
have a better understanding on these nonRiemannian quantities.
Keywords:Finsler metrics, sprays, Ricci curvature, nonRiemanian quantity Categories:53B40, 53C60 

7. CMB 2015 (vol 58 pp. 575)
 MartinezTorres, David

The Diffeomorphism Type of Canonical Integrations Of Poisson Tensors on Surfaces
A surface $\Sigma$ endowed with a Poisson tensor
$\pi$ is known to admit
canonical integration, $\mathcal{G}(\pi)$,
which is a 4dimensional manifold with a (symplectic) Lie groupoid
structure.
In this short note we show that if $\pi$ is not an area
form on the 2sphere, then $\mathcal{G}(\pi)$ is diffeomorphic
to the cotangent bundle $T^*\Sigma$. This extends
results by the author and by Bonechi, Ciccoli, Staffolani, and Tarlini.
Keywords:Poisson tensor, Lie groupoid, cotangent bundle Categories:58H05, 55R10, 53D17 

8. CMB 2015 (vol 58 pp. 713)
 Brendle, Simon; Chodosh, Otis

On the Maximum Curvature of Closed Curves in Negatively Curved Manifolds
Motivated by Almgren's work on the isoperimetric inequality,
we prove a sharp inequality relating the length and maximum curvature
of a closed curve in a complete, simply connected manifold of
sectional curvature at most $1$. Moreover, if equality holds,
then the norm of the geodesic curvature is constant and the torsion
vanishes. The proof involves an application of the maximum principle
to a function defined on pairs of points.
Keywords:manifold, curvature Category:53C20 

9. CMB Online first
 Deng, Shaoqiang; Hu, Zhiguang; Li, Jifu

Cohomogeneity one Randers metrics
An action of a Lie group $G$ on a smooth manifold $M$ is called
cohomogeneity one if the orbit space $M/G$ is of dimension $1$.
A Finsler metric $F$ on $M$ is called invariant if $F$ is
invariant under the action of $G$. In this paper,
we study invariant
Randers metrics on cohomogeneity one manifolds. We first give a
sufficient and necessary condition for the existence of invariant
Randers metrics on cohomogeneity one manifolds. Then we obtain
some results on invariant Killing vector fields on the
cohomogeneity one manifolds and use that to deduce some
sufficient and necessary condition for a cohomogeneity one
Randers metric to be Einstein.
Keywords:cohomogeneity one actions, normal geodesics, invariant vector fields, Randers metrics Categories:53C30, 53C60 

10. CMB 2014 (vol 58 pp. 561)
 MartinezMaure, Yves

Plane Lorentzian and Fuchsian Hedgehogs
Parts of the BrunnMinkowski theory can be extended to hedgehogs, which are
envelopes of families of affine hyperplanes parametrized by their Gauss map.
F. Fillastre introduced Fuchsian convex bodies, which are the
closed convex sets of LorentzMinkowski space that are globally invariant
under the action of a Fuchsian group. In this paper, we undertake a study of
plane Lorentzian and Fuchsian hedgehogs. In particular, we prove the
Fuchsian analogues of classical geometrical inequalities (analogues which
are reversed as compared to classical ones).
Keywords:Fuchsian and Lorentzian hedgehogs, evolute, duality, convolution, reversed isoperimetric inequality, reversed Bonnesen inequality Categories:52A40, 52A55, 53A04, 53B30 

11. CMB 2014 (vol 58 pp. 158)
12. CMB 2014 (vol 57 pp. 765)
 da Silva, Rosângela Maria; Tenenblat, Keti

Helicoidal Minimal Surfaces in a Finsler Space of Randers Type
We consider the Finsler space $(\bar{M}^3, \bar{F})$ obtained by
perturbing the Euclidean metric of $\mathbb{R}^3$ by a rotation. It
is the open region of $\mathbb{R}^3$ bounded by a cylinder with a
Randers metric. Using the BusemannHausdorff volume form, we
obtain the differential equation that characterizes the helicoidal
minimal surfaces in $\bar{M}^3$. We prove that the helicoid is a
minimal surface in $\bar{M}^3$, only if the axis of the helicoid
is the axis of the cylinder. Moreover, we prove that, in the
Randers space $(\bar{M}^3, \bar{F})$, the only minimal
surfaces in the Bonnet family, with fixed axis $O\bar{x}^3$, are the catenoids
and the helicoids.
Keywords:minimal surfaces, helicoidal surfaces, Finsler space, Randers space Categories:53A10, 53B40 

13. CMB 2013 (vol 57 pp. 870)
 Parlier, Hugo

A Short Note on Short Pants
It is a theorem of Bers that any closed hyperbolic surface admits a pants decomposition consisting of curves of bounded length where the bound only depends on the topology of the surface. The question of the quantification of the optimal constants has been well studied and the best upper bounds to date are linear in genus, a theorem of Buser and SeppÃ¤lÃ¤. The goal of this note is to give a short proof of a linear upper bound which slightly improve the best known bound.
Keywords:hyperbolic surfaces, geodesics, pants decompositions Categories:30F10, 32G15, 53C22 

14. CMB 2013 (vol 57 pp. 821)
 Jeong, Imsoon; Kim, Seonhui; Suh, Young Jin

Real Hypersurfaces in Complex TwoPlane Grassmannians with Reeb Parallel Structure Jacobi Operator
In this paper we give a characterization of a real hypersurface of
Type~$(A)$ in complex twoplane Grassmannians ${ { {G_2({\mathbb
C}^{m+2})} } }$, which means a
tube over a totally geodesic $G_{2}(\mathbb C^{m+1})$ in
${G_2({\mathbb C}^{m+2})}$, by
the Reeb parallel structure Jacobi operator ${\nabla}_{\xi}R_{\xi}=0$.
Keywords:real hypersurfaces, complex twoplane Grassmannians, Hopf hypersurface, Reeb parallel, structure Jacobi operator Categories:53C40, 53C15 

15. CMB 2013 (vol 57 pp. 401)
 Perrone, Domenico

Curvature of $K$contact SemiRiemannian Manifolds
In this paper we characterize $K$contact semiRiemannian manifolds
and Sasakian semiRiemannian manifolds in terms of
curvature. Moreover, we show that any conformally flat $K$contact
semiRiemannian manifold is Sasakian and of constant sectional
curvature $\kappa=\varepsilon$, where $\varepsilon =\pm 1$ denotes
the causal character of the Reeb vector field. Finally, we give some
results about the curvature of a $K$contact Lorentzian manifold.
Keywords:contact semiRiemannian structures, $K$contact structures, conformally flat manifolds, Einstein LorentzianSasaki manifolds Categories:53C50, 53C25, 53B30 

16. CMB 2012 (vol 57 pp. 209)
17. CMB 2012 (vol 57 pp. 194)
18. CMB 2012 (vol 57 pp. 12)
19. CMB 2011 (vol 56 pp. 306)
20. 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 

21. 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 

22. 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 

23. 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 

24. 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 

25. 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 
