微分几何

We give results on optimal constants of isoperimetric inequalities involving Steklov eigenvalues on surfaces with boundary. We both consider this question on Riemannian surfaces with a same given topology or more specifically belonging to…

We study three-dimensional path geometries with nontrivial torsion of maximal rank. We introduce the notion of constant torsion and show that such path geometries are in one-to-one correspondence with certain cone structures modeled on…

We show that given an element $X$ of the enhanced Teichm\"{u}ller space $\mathcal{T}^\pm(\mathbb{S}, \mathbb{M})$ and a type-preserving framed $\mathrm{PSL}_2(\mathbb{C})$-representation $\hat{\rho} = (\rho,\beta)$, there is a…

We introduce two pointwise subspace averages of sectional curvature on a d-dimensional plane Pi in T_p M: (i) the intrinsic mean Ricci (the average of sectional curvatures of 2-planes contained in Pi); and (ii) the normal (mixed) mean Ricci…

We present a thorough study of the differential geometry of weightings and develop the theory of weightings for vector bundles, Lie groupoids, and Lie algebroids. We begin by extending the work of Loizides and Meinrenken on weighted…

A Ricci soliton is a natural generalization of an Einstein metric. On a pseudo-Riemannian manifold (M, g), it is defined by : $LX g + \r{ho} = {\lambda} g, where X is a smooth vector field on M , LX denotes the Lie derivative in the…

In the comparison of nonholonomic mechanics and constrained variational mechanics, invariant affine subbundles arise in the determination of the initial conditions where the two methods yield the same trajectories. Motivated by this,…

Labourie raised the question of determining the possible asymptotics for the growth rate of compact $k$-surfaces, counted according to energy, in negatively curved $3$-manifolds, indicating the possibility of a theory of thermodynamical…

A four dimensional pseudo-Riemannian manifold of signature (2, 2) is called a Walker manifold if it admits a parallel degenerate plane field. In this paper, we study the curvature properties of such a class of four dimensional Walker…

We show that for a generic Riemannian metric on a compact manifold of dimension $n\ge 3$ all geodesic loops based at a fixed point have no self-intersections. We also show that for an open and dense subset of the space of Riemannian metrics…

We prove the continuity of bounded solutions to complex Monge-Amp\`{e}re equations on reduced, locally irreducible compact K\"{a}hler spaces. This in particular implies that any singular K\"{a}hler-Einstein potentials constructed in…

The hypersurface is one of the most important objects in a space. Many authors studied diffrent geometric aspects of hypersurfaces in a space. In this paper, we define three types of 2-ruled hypersurfaces in a Walker 4-manfold E 41 . We…

We produce finite energy BPS monopoles with arbitrary prescribed symmetry breaking from a new class of Nahm data.

The geometric description of incompressible hydrodynamics, as geodesic motion on the infinite-dimensional group of volume-preserving diffeomorphisms, enables notions of curvature in the study of fluids in order to study stability. Formulas…

The aim of this note is to prove that the index of the identity map on a quaternion space form of constant quaternion sectional curvature $c$ is zero, provided that $c<0$. As an immediate consequence, it is established that any compact…

Endo-Pajitnov manifolds are generalizations to higher dimensions of the Inoue surfaces $S^M$. We study the existence of complex submanifolds in Endo-Pajitnov manifolds. We identify a class of these manifolds that do contain compact complex…

In this article, we recapture the Smale conjecture on a Sasakian $3$-sphere via the Legendrian mean curvature flow. More precisely,~we deform the area-preserving contactomorphism (symplectomorphism) of Sasakian $3$-spheres to an isometry…

We prove the rigidity of rectifiable boundaries with constant distributional mean curvature in the Brendle class of warped product manifolds (which includes important models in General Relativity, like the deSitter--Schwarzschild and…

The interest of geometers in $f$-structures is motivated by the study of the dynamics of contact foliations, as well as their applications in physics. A weak $f$-structure on a smooth manifold, introduced by V. Rovenski and R. Wolak (2022),…

Linear connections satisfying the Einstein metricity condition are important in the study of generalized Riemannian manifolds $(M,G=g+F)$, where the symmetric part $g$ of $G$ is a non-degenerate $(0,2)$-tensor, and $F$ is the skew-symmetric…