1. CMB Online first
 Lang, Honglei; Sheng, Yunhe; Wade, Aissa

$\mathsf{VB}$Courant algebroids, $\mathsf{E}$Courant algebroids and generalized geometry
In this paper, we first discuss the relation between $\mathsf{VB}$Courant
algebroids and $\mathsf{E}$Courant algebroids and construct some examples
of $\mathsf{E}$Courant algebroids. Then we introduce the notion of
a generalized complex
structure on an $\mathsf{E}$Courant algebroid, unifying the usual
generalized complex structures on evendimensional manifolds
and
generalized contact structures on odddimensional manifolds.
Moreover, we study generalized complex structures on an omniLie
algebroid in detail. In particular, we show that generalized
complex structures on an omniLie algebra $\operatorname{gl}(V)\oplus V$
correspond
to complex Lie algebra structures on $V$.
Keywords:$\mathsf{VB}$Courant algebroid, $\mathsf{E}$Courant algebroid, omniLie algebroid, generalized complex structure, algebroidNijenhuis structure Categories:53D17, 18B40, 58H05 

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 60 pp. 122)
 Ghanei, Mohammad Reza; NasrIsfahani, Rasoul; Nemati, Mehdi

A Homological Property and Arens Regularity of Locally Compact Quantum Groups
We characterize two important notions of amenability and compactness
of
a locally compact quantum group ${\mathbb G}$ in terms of certain
homological
properties. For this, we show that ${\mathbb G}$ is character
amenable if and only if it is both amenable and coamenable.
We finally apply our results to
Arens regularity problems of the quantum group algebra
$L^1({\mathbb G})$; in particular, we improve an interesting result
by Hu, Neufang and Ruan.
Keywords:amenability, Arens regularity, coamenability, locally compact quantum group, homological property Categories:46L89, 43A07, 46H20, 46M10, 58B32 

4. CMB 2016 (vol 59 pp. 806)
 Izumiya, Shyuichi

Geometric Interpretation of Lagrangian Equivalence
As an application of the theory of
graphlike Legendrian unfoldings, relations of the hidden structures
of caustics and wave front propagations are revealed.
Keywords:wave front propagations, big wave fronts, graphlike Legendrian unfoldings, caustics Categories:58K05, 57R45, 58K60 

5. 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 

6. CMB 2016 (vol 59 pp. 760)
 Fichou, Goulwen; Quarez, Ronan; Shiota, Masahiro

Artin Approximation Compatible with a Change of Variables
We propose a version of the classical Artin
approximation
which allows to perturb the variables of the approximated solution. Namely, it is possible to approximate a formal solution of a
Nash equation by a Nash solution in a
compatible way with a given Nash change of variables.
This result is closely related to the socalled nested Artin
approximation and becomes false in the analytic setting. We provide
local and global versions of this approximation in real and complex
geometry together with an application to the RightLeft equivalence
of Nash maps.
Keywords:Artin approximation, global case, Nash functions Categories:14P20, 58A07 

7. 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 

8. CMB 2015 (vol 58 pp. 723)
 Castro, Alfonso; Fischer, Emily

Infinitely Many Rotationally Symmetric Solutions to a Class of Semilinear LaplaceBeltrami Equations on Spheres
We show that a class of semilinear LaplaceBeltrami equations
on the unit sphere
in $\mathbb{R}^n$ has infinitely many rotationally symmetric solutions.
The solutions to
these equations are the solutions to a two point boundary value
problem for a
singular ordinary differential equation. We prove the existence
of such solutions
using energy and phase plane analysis. We derive a
Pohozaevtype
identity
in
order to prove that the energy to an associated initial value
problem tends
to infinity as the energy at the singularity tends to infinity.
The nonlinearity is allowed to grow as fast as $s^{p1}s$ for
$s$ large
with $1 \lt p \lt (n+5)/(n3)$.
Keywords:LaplaceBeltrami operator, semilinear equation, rotational solution, superlinear nonlinearity, subsuper critical nonlinearity Categories:58J05, 35A24 

9. CMB 2015 (vol 58 pp. 846)
 Sundar, S.

A Computation with the ConnesThom Isomorphism
Let $A \in M_{n}(\mathbb{R})$ be an invertible matrix. Consider
the semidirect product $\mathbb{R}^{n} \rtimes \mathbb{Z}$ where
the action of $\mathbb{Z}$ on $\mathbb{R}^{n}$ is induced by
the left multiplication by $A$. Let $(\alpha,\tau)$ be a strongly
continuous action of $\mathbb{R}^{n} \rtimes \mathbb{Z}$ on a
$C^{*}$algebra $B$ where $\alpha$ is a strongly continuous action
of $\mathbb{R}^{n}$ and $\tau$ is an automorphism. The map $\tau$
induces a map $\widetilde{\tau}$ on $B \rtimes_{\alpha} \mathbb{R}^{n}$.
We show that, at the $K$theory level, $\tau$ commutes with the
ConnesThom map if $\det(A)\gt 0$ and anticommutes if $\det(A)\lt 0$.
As an application, we recompute the $K$groups of the CuntzLi
algebra associated to an integer dilation matrix.
Keywords:Ktheory, ConnesThom isomorphism, CuntzLi algebras Categories:46L80, 58B34 

10. 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 

11. CMB 2015 (vol 58 pp. 285)
 Karpukhin, Mikhail

Spectral Properties of a Family of Minimal Tori of Revolution in Fivedimensional Sphere
The normalized eigenvalues $\Lambda_i(M,g)$ of the LaplaceBeltrami
operator can be considered as functionals on the space of all
Riemannian metrics $g$ on a fixed surface $M$. In recent papers
several explicit examples of extremal metrics were provided.
These metrics are induced by minimal immersions of surfaces in
$\mathbb{S}^3$ or $\mathbb{S}^4$. In the present paper a family
of extremal metrics induced by minimal immersions in $\mathbb{S}^5$
is investigated.
Keywords:extremal metric, minimal surface Category:58J50 

12. CMB 2015 (vol 58 pp. 281)
 Kalus, Matthias

On the Relation of Real and Complex Lie Supergroups
A complex Lie supergroup can be described as a real Lie supergroup
with integrable almost complex structure. The necessary and
sufficient conditions on an almost complex structure on a real
Lie supergroup for defining a complex Lie supergroup are deduced.
The classification of real Lie supergroups with such almost
complex
structures yields a new approach to the known classification
of complex Lie supergroups by complex HarishChandra superpairs.
A universal complexification of a real Lie supergroup is
constructed.
Keywords:Lie supergroup, almost complex structure, HarishChandra pair, universal complexification Categories:32C11, 58A50 

13. CMB 2014 (vol 58 pp. 69)
 Fulp, Ronald Owen

Correction to "Infinite Dimensional DeWitt Supergroups and Their Bodies"
The Theorem below is a correction to Theorem
3.5 in the article
entitled " Infinite Dimensional DeWitt Supergroups and Their
Bodies" published
in Canad. Math. Bull. Vol. 57 (2) 2014 pp. 283288. Only part
(iii) of that Theorem
requires correction. The proof of Theorem 3.5 in the original
article failed to separate
the proof of (ii) from the proof of (iii). The proof of (ii)
is complete once it is established
that $ad_a$ is quasinilpotent for each $a$ since it immediately
follows that $K$
is quasinilpotent. The proof of (iii) is not complete
in the original article. The revision appears as the proof of
(iii) of the revised Theorem below.
Keywords:super groups, body of super groups, Banach Lie groups Categories:58B25, 17B65, 81R10, 57P99 

14. CMB 2013 (vol 57 pp. 357)
 Lauret, Emilio A.

Representation Equivalent Bieberbach Groups and Strongly Isospectral Flat Manifolds
Let $\Gamma_1$ and $\Gamma_2$ be Bieberbach groups contained in the
full isometry group $G$ of $\mathbb{R}^n$.
We prove that if the compact flat manifolds $\Gamma_1\backslash\mathbb{R}^n$ and
$\Gamma_2\backslash\mathbb{R}^n$ are strongly isospectral then the Bieberbach groups
$\Gamma_1$ and $\Gamma_2$ are representation equivalent, that is, the
right regular representations $L^2(\Gamma_1\backslash G)$ and
$L^2(\Gamma_2\backslash G)$ are unitarily equivalent.
Keywords:representation equivalent, strongly isospectrality, compact flat manifolds Categories:58J53, 22D10 

15. CMB 2012 (vol 56 pp. 814)
16. 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 

17. CMB 2011 (vol 56 pp. 366)
 Kyritsi, Sophia Th.; Papageorgiou, Nikolaos S.

Multiple Solutions for Nonlinear Periodic Problems
We consider a nonlinear periodic problem driven by a
nonlinear nonhomogeneous differential operator and a
CarathÃ©odory reaction term $f(t,x)$ that exhibits a
$(p1)$superlinear growth in $x \in \mathbb{R}$
near $\pm\infty$ and near zero.
A special case of the differential operator is the scalar
$p$Laplacian. Using a combination of variational methods based on
the critical point theory with Morse theory (critical groups), we
show that the problem has three nontrivial solutions, two of which
have constant sign (one positive, the other negative).
Keywords:$C$condition, mountain pass theorem, critical groups, strong deformation retract, contractible space, homotopy invariance Categories:34B15, 34B18, 34C25, 58E05 

18. CMB 2011 (vol 56 pp. 3)
 Aïssiou, Tayeb

Semiclassical Limits of Eigenfunctions on Flat $n$Dimensional Tori
We provide a proof of a conjecture by Jakobson, Nadirashvili, and
Toth stating
that on an $n$dimensional flat torus $\mathbb T^{n}$, and the Fourier transform
of squares of the eigenfunctions $\varphi_\lambda^2$ of the Laplacian have
uniform $l^n$ bounds that do not depend on the eigenvalue $\lambda$. The proof
is a generalization of an argument by Jakobson, et al. for the
lower dimensional cases. These results imply uniform bounds for semiclassical
limits on $\mathbb T^{n+2}$. We also prove a geometric lemma that bounds the number of
codimensionone simplices satisfying a certain restriction on an
$n$dimensional sphere $S^n(\lambda)$ of radius $\sqrt{\lambda}$, and we use it in
the proof.
Keywords:semiclassical limits, eigenfunctions of Laplacian on a torus, quantum limits Categories:58G25, 81Q50, 35P20, 42B05 

19. 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 

20. CMB 2011 (vol 54 pp. 396)
21. CMB 2011 (vol 54 pp. 693)
 Lusala, Tsasa; Śniatycki, Jędrzej

Stratified Subcartesian Spaces
We show that if the family $\mathcal{O}$ of orbits of all vector fields on
a subcartesian space $P$ is locally finite and each orbit in $\mathcal{O}$
is locally closed, then $\mathcal{O}$ defines a smooth Whitney A
stratification of $P$. We also show that the stratification by orbit type of
the space of orbits $M/G$ of a proper action of a Lie group $G$ on a smooth
manifold $M$ is given by orbits of the family of all vector fields on $M/G$.
Keywords:Subcartesian spaces, orbits of vector fields, stratifications, Whitney Conditions Categories:58A40, 57N80 

22. CMB 2011 (vol 54 pp. 249)
23. CMB 2010 (vol 53 pp. 674)
 Kristály, Alexandru; Papageorgiou, Nikolaos S.; Varga, Csaba

Multiple Solutions for a Class of Neumann Elliptic Problems on Compact Riemannian Manifolds with Boundary
We study a semilinear elliptic problem on a compact Riemannian
manifold with boundary, subject to an inhomogeneous Neumann
boundary condition. Under various hypotheses on the nonlinear
terms, depending on their behaviour in the origin and infinity, we
prove multiplicity of solutions by using variational arguments.
Keywords:Riemannian manifold with boundary, Neumann problem, sublinearity at infinity, multiple solutions Categories:58J05, 35P30 

24. 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 

25. CMB 2010 (vol 53 pp. 542)