1. CMB Online first
2. CMB 2017 (vol 60 pp. 253)
 Chen, Bin; Zhao, Lili

On a Yamabe Type Problem in Finsler Geometry
In this paper, a new notion of scalar curvature for a Finsler
metric $F$ is introduced, and two conformal invariants $Y(M,F)$
and $C(M,F)$ are defined. We prove that there exists a Finsler
metric with constant scalar curvature in the conformal class
of $F$ if the Cartan torsion of $F$ is sufficiently small and
$Y(M,F)C(M,F)\lt Y(\mathbb{S}^n)$ where $Y(\mathbb{S}^n)$ is the
Yamabe constant of the standard sphere.
Keywords:Finsler metric, scalar curvature, Yamabe problem Categories:53C60, 58B20 

3. CMB 2016 (vol 59 pp. 813)
4. CMB 2016 (vol 60 pp. 77)
 Christ, Michael; Rieffel, Marc A.

Nilpotent Group C*algebras as Compact Quantum Metric Spaces
Let $\mathbb{L}$ be a length function on a group $G$, and let $M_\mathbb{L}$
denote the
operator of pointwise multiplication by $\mathbb{L}$ on $\lt(G)$.
Following Connes,
$M_\mathbb{L}$ can be used as a ``Dirac'' operator for the reduced
group C*algebra $C_r^*(G)$. It defines a
Lipschitz seminorm on $C_r^*(G)$, which defines a metric on the
state space of
$C_r^*(G)$. We show that
for any length function satisfying a strong form of polynomial
growth on a discrete group,
the topology from this metric
coincides with the
weak$*$ topology (a key property for the
definition of a ``compact quantum metric
space''). In particular, this holds for all wordlength functions
on finitely generated nilpotentbyfinite groups.
Keywords:group C*algebra, Dirac operator, quantum metric space, discrete nilpotent group, polynomial growth Categories:46L87, 20F65, 22D15, 53C23, 58B34 

5. CMB 2016 (vol 59 pp. 673)
 Bačák, Miroslav; Kovalev, Leonid V.

Lipschitz Retractions in Hadamard Spaces Via Gradient Flow Semigroups
Let $X(n),$ for $n\in\mathbb{N},$ be the set of all subsets of a metric
space $(X,d)$ of cardinality at most $n.$ The set $X(n)$ equipped
with the Hausdorff metric is called a finite subset space. In
this paper we are concerned with the existence of Lipschitz retractions
$r\colon X(n)\to X(n1)$ for $n\ge2.$ It is known that such retractions
do not exist if $X$ is the onedimensional sphere. On the other
hand L. Kovalev has recently established their existence in case $X$
is a Hilbert space and he also posed a question as to whether
or not such Lipschitz retractions exist for $X$ being a Hadamard
space. In the present paper we answer this question in the positive.
Keywords:finite subset space, gradient flow, Hadamard space, LieTrotterKato formula, Lipschitz retraction Categories:53C23, 47H20, 54E40, 58D07 

6. CMB 2016 (vol 59 pp. 721)
 Pérez, Juan de Dios; Lee, Hyunjin; Suh, Young Jin; Woo, Changhwa

Real Hypersurfaces in Complex Twoplane Grassmannians with Reeb Parallel Ricci Tensor in the GTW Connection
There are several kinds of classification problems for real hypersurfaces
in complex twoplane Grassmannians $G_2({\mathbb C}^{m+2})$.
Among them, Suh classified Hopf hypersurfaces $M$ in $G_2({\mathbb
C}^{m+2})$ with Reeb parallel Ricci tensor in LeviCivita connection.
In this paper, we introduce the notion of generalized TanakaWebster
(in shortly, GTW) Reeb parallel Ricci tensor for Hopf hypersurface
$M$ in $G_2({\mathbb C}^{m+2})$. Next, we give a complete classification
of Hopf hypersurfaces in $G_2({\mathbb C}^{m+2})$ with GTW Reeb
parallel Ricci tensor.
Keywords:Complex twoplane Grassmannian, real hypersurface, Hopf hypersurface, generalized TanakaWebster connection, parallelism, Reeb parallelism, Ricci tensor Categories:53C40, 53C15 

7. CMB 2016 (vol 59 pp. 575)
 Li, Jifu; Hu, Zhiguang; Deng, Shaoqiang

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 

8. CMB 2016 (vol 59 pp. 508)
 De Nicola, Antonio; 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 

9. CMB 2015 (vol 59 pp. 50)
10. CMB 2015 (vol 58 pp. 835)
11. 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 

12. 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 

13. 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 

14. CMB 2014 (vol 58 pp. 158)
15. 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 

16. 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 

17. 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 

18. CMB 2012 (vol 57 pp. 209)
19. CMB 2012 (vol 57 pp. 194)
20. CMB 2012 (vol 57 pp. 12)
21. CMB 2011 (vol 56 pp. 306)
22. 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 

23. 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 

24. 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 

25. 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 
