Related papers: Bi-invariant metric on contact diffeomorphisms gro…

Let $G$ be a Lie group equipped with a left-invariant Riemannian metric. Let $K$ be a semisimple and normal subgroup of $G$ generating a left-invariant conformal foliation $\F$ of on $G$. We then show that the foliation $\F$ is Riemannian…

Let M be a compact spin manifold with a smooth action of the n-torus. Connes and Landi constructed theta-deformations M_{theta} of M, parameterized by n by n real skew-symmetric matrices theta. The M_{theta}'s together with the canonical…

This paper begins the study of relations between Riemannian geometry and global properties of contact structures on 3-manifolds. In particular we prove an analog of the sphere theorem from Riemannian geometry in the setting of contact…

A $n$-dimensional Lie algebra $g=(V,\mu)$ is called $2$-compatible if it is isomorphic to a quadratic deformation of a Lie algebra $g_0=(V,\mu_0)$. By quadratic deformation we means a formal deformation $\mu_t=\mu_0+t\varphi_1+t^2\varphi_2$…

We consider a $(2q+1)$-dimensional smooth manifold $M$ equipped with a $(q+1)$-dimensional, a priori non-integrable, distribution ${\cal D}$ and a $q$-vector field ${\bf T}=T_1\wedge\ldots\wedge T_q$, where $\{T_i\}$ are linearly…

The use of a diffeomorphism covariant star product enables us to construct diffeomorphism invariant gravities on noncommutative symplectic manifolds without twisting the symmetries. As an example, we construct noncommutative deformations of…

We prove that on closed Riemannian manifolds with infinite abelian, but not cyclic, fundamental group, any isometry that is homotopic to the identity possesses infinitely many invariant geodesics. We conjecture that the result remains true…

Let G be a Lie group equipped with a left-invariant semi-Riemannian metric. Let K be a semisimple subgroup of G generating a left-invariant conformal foliation F of codimension two on G. We then show that the foliation F is minimal. This…

We look at the functional Y(M) = int_M K(x) dV(x) for compact Riemannian 2d-manifolds M, where K(x) = (2d)! (d!)^-1 (4pi)^-d int_T prod_k=1^d K_t_2k,t_2k+1(x) dt involves products of d sectional curvatures K_ij(x) averaged over the space T…

On the symmetrized bidisc G2 with the Bergman metric, the holomorphic sectional curvature is negatively pinched and the holomorphic bisectional curvature is not. The consequences in invariant metrics are provided.

Let $N=(\Omega,\sigma)$ and $M=(\Omega^*,\rho)$ be doubly connected Riemann surfaces and assume that $\rho$ is a smooth metric with bounded Gauss curvature $\mathcal{K}$ and finite area. The paper establishes the existence of homeomorphisms…

We construct non-trivial continuous isospectral deformations of Riemannian metrics on the ball and on the sphere in $\R^n$ for every $n\geq 9$. The metrics on the sphere can be chosen arbitrarily close to the round metric; in particular,…

A consistent approach to the description of integral coordinate invariant functionals of the metric on manifolds ${\cal M}_{\alpha}$ with conical defects (or singularities) of the topology $C_{\alpha}\times\Sigma$ is developed. According to…

We compute the cylindrical contact homology of the links of the simple singularities. These manifolds are contactomorphic to $S^3/G$ for finite subgroups $G\subset SU(2)$. We perturb the degenerate contact form on $S^3/G$ with a Morse…

A dualistic structure on a smooth Riemaniann manifold $M$ is a triple $(M,g,\nabla)$ with $g$ a Riemaniann metric and $\nabla$ an affine connection, generally assumed to be torsionless. From $g$ and $\nabla$, the dual connection $\nabla^*$…

On the space of isometric embeddings $f_g$ of metrics $g$ on a manifold $M^n$ into the standard $(\mb{S}^{\tn=\tn(n)},\tg)$, we consider the total exterior scalar curvature $\Theta_{f_g}(M)$, and squared $L^2$ norm of the mean curvature…

Using the structural theorems developed in [Hua13], we study the deformation theory of coisotropic submanifolds in contact manifolds, under the assumption that the characteristic foliation is nonsingular. In the "middle" dimensions, we find…

We study left-invariant symmetric Killing 2-tensors on 2-step nilpotent Lie groups endowed with a left-invariant Riemannian metric, and construct genuine examples, which are not linear combinations of parallel tensors and symmetric products…

A Morse-Bott volume form on a manifold is a top-degree form which vanishes along a non-degenerate critical submanifold. We prove that two such forms are diffeomorphic (by a diffeomorphism fixed on the submanifold) provided that their…

A result from Gromov ensures the existence of a contact structure on any connected non-compact odd dimensional Lie group. But in general such structures are not invariant under left translations of the Lie group. The problem of finding…