Related papers: Canonical frames in contact 3-manifolds and applic…
We establish a gluing theorem for solutions of a Yamabe problem for manifolds with boundary studied by Escobar in the 90's. Given two scalar-flat Riemannian manifolds whose boundary has zero mean curvature and sharing a submanifold $K$, we…
The aim of the present paper is to establish a global theory of conformal changes in Finsler geometry. Under this change, we obtain the relationships between the most important geometric objects associated to $(M,L)$ and the corresponding…
We introduce symplectic Calabi-Yau caps to obtain new obstructions to exact fillings. In particular, it implies that any exact filling of the standard unit cotangent bundle of a hyperbolic surface has vanishing first Chern class and has the…
We study low-dimensional problems in topology and geometry via a study of contact and Cauchy-Riemann ($CR$) structures. A contact structure is called spherical if it admits a compatible spherical $CR$ structure. We will talk about spherical…
In this paper, we study the topological rigidity and its relationship with the positivity of scalar curvature. Precisely, we show that any complete contractible $3$-manifold with non-negative scalar curvature is homeomorphic to…
We provide an extension of Obata's theorem to Finsler geometry and establish some rigidity results based on a second order differential equation. Mainly, we prove that every complete connected Finsler manifold of positive constant flag…
This is a survey on known results and open problems about closed aspherical manifolds, i.e., connected closed manifolds whose universal coverings are contractible. Many examples come from certain kinds of non-positive curvature conditions.…
The question of whether a Sasakian metric can admit an additional compatible (K-)contact structure is addressed. In the complete case if the second structure is also assumed Sasakian, works of Tachibana-Yu and Tanno show that the manifold…
This is a paper based on a talk given at the conference on Conformal Geometry which held at Roscoff in France in the 2008 summer. We study some aspects of the equation arising from the problem of the existence on a given closed Riemannian…
In this article, we investigate the geometry of compact quasi-Einstein manifolds with boundary. We show that a $3$-dimensional simply connected compact quasi-Einstein manifold with boundary and constant scalar curvature is isometric, up to…
We study the topological structure of the symmetry group of the standard model, $G_{SM}=U(1)\times SU(2)\times SU(3)$. Locally, $G_{SM}\cong S^1\times (S^3)^2\times S^5$. For SU(3), which is an $S^3$ bundle over $S^5$ (and therefore a local…
Here is an English summary of the abstract: This research investigates a geometric dynamical mechanism within a specific class of domains that contain a fixed convex core. By using a radial structure that links the boundaries of the core…
In early study of Engel manifolds from R. Montgomery, the Cartan prolongation and the development map are central figures. However, the development map can be globally defined only if the characteristic foliation is "nice". In this paper,…
We compute a Bochner type formula for static three-manifolds and deduce some applications in the case of positive scalar curvature. We also explain in details the known general construction of the (Riemannian) Einstein (n+1)-manifold…
In this paper, we first provide an updated survey of the geometry of complex Cartan spaces. New characterizations for some particular classes of complex Cartan spaces are pointed out, e.g. Landsberg-Cartan, strongly Berwald-Cartan and…
Given a closed contact 3-manifold with a compatible Riemannian metric, we show that if the sectional curvature is 1/4-pinched, then the contact structure is universally tight. This result improves the Contact Sphere Theorem in [EKM12],…
Let M be a compact manifold with boundary. In this paper, we discuss some rigidity theorems of metrics in a same conformal class that fixes the boundary and satisfy certain integral conditions on the the scalar curvatures and the mean…
We present a geometric mechanism for the emergence of spherical $3$-manifolds from the superspace of Riemannian metrics associated with flat ${\rm{SU}}(2)$-bundles over closed orientable hyperbolic surfaces. Our main result shows that any…
We study complete $3$-manifolds with nonnegative scalar curvature under additional regularity assumptions. We prove that a contractible such manifold is diffeomorphic to $\mathbb{R}^3$, and that an open handlebody admitting such a metric…
We suggest a way to associate to a rational map of the Riemann sphere a three dimensional object called a hyperbolic orbifold 3-lamination. The relation of this object to the map is analogous to the relation of a hyperbolic 3-manifold to a…