A remark on the symmetries of the Riemann curvature tensor

A note on hyperbolic flows in sub-Riemannian Structures

The curvature and the reduced curvature are basic differential invariants of the pair (Hamiltonian system, Lagrange distribution) on the symplectic manifold. It is shown that the negativity of the reduced curvature implies the hyperbolicity of any compact invariant set of the Hamiltonian flow restricted to a prescribed energy level. We consider the Hamiltonian flows of the curve of least action of natural mechanical systems in sub-Riemannian structures with symmetries. We give sufficient conditions for the reduced flows (after reduction of the first integrals induced from the symmetries) to be…

Symmetries of Spaces with Lower Curvature Bounds

This article surveys results for Riemannian manifolds of positive and non-negative sectional curvature with symmetries.

Jacobi Equations and Comparison Theorems for Corank 1 sub-Riemannian Structures with Symmetries

The Jacobi curve of an extremal of optimal control problem is a curve in a Lagrangian Grassmannian defined up to a symplectic transformation and containing all information about the solutions of the Jacobi equations along this extremal. In our previous works we constructed the canonical bundle of moving frames and the complete system of symplectic invariants, called curvature maps, for parametrized curves in Lagrange Grassmannians satisfying very general assumptions. The structural equation for a canonical moving frame of the Jacobi curve of an extremal can be interpreted as the normal form fo…

Curvature-dimension inequalities and Ricci lower bounds for sub-Riemannian manifolds with transverse symmetries

Let $\M$ be a smooth connected manifold endowed with a smooth measure $μ$ and a smooth locally subelliptic diffusion operator $L$ satisfying $L1=0$, and which is symmetric with respect to $μ$. Associated with $L$ one has \textit{le carré du champ} $Γ$ and a canonical distance $d$, with respect to which we suppose that $(M,d)$ be complete. We assume that $\M$ is also equipped with another first-order differential bilinear form $Γ^Z$ and we assume that $Γ$ and $Γ^Z$ satisfy the Hypothesis below. With these forms we introduce in \eqref{cdi} below a generalization of the curvature-dimension inequa…

具有射影对称性的二维度量的Sophus Lie问题：局部分类的完成

Sophus Lie's problem on two-dimensional metrics with projective symmetries: completing the local classification

本文完成了Sophus Lie 1882年提出的二维伪黎曼度量的局部等距分类问题，补充了射影对称代数非正则情形的标准形，结合已有正则情形分类实现了完整求解。

Abstract Dynamical Systems: Remarks on Symmetries and Reduction

We review how an algebraic formulation for the dynamics of a physical system allows to describe a reduction procedure for both classical and quantum evolutions.

Remarks on the nonexistence of biharmonic maps

In this short note we study nonexistence result of biharmonic maps from a complete Riemannian manifold into a Riemannian manifold with nonpositive sectional curvature. Assume that $ϕ:(M,g)\to (N, h)$ is a biharmonic map, where $(M, g)$ is a complete Riemannian manifold and $(N,h)$ a Riemannian manifold with nonpositive sectional curvature, we will prove that $ϕ$ is a harmonic map if one of the following conditions holds: (i) $|dϕ|$ is bounded in $L^q(M)$ and $ \int_M|τ(ϕ)|^pdv_g<\infty, $ for some $1\leq q\leq\infty$, $1< p<\infty$; or (ii) $Vol(M)=\infty$ and $ \int_M|τ(ϕ)|^pdv_g<\infty, $ fo…

关于Kähler流形的Ricci对称性

On the Ricci symmetries of a Kähler manifold

本文研究Kähler流形的Ricci张量对称性，探讨Kähler-Einstein空间及其泛化，揭示holomorphic平面在Kähler几何中的作用，提出新的Einstein流形判据。

Ricci孤立子、几乎Theta-Yamabe孤立子及黎曼流形上秩一各向异性曲率向量场的有限阶张量对称性

Ricci Solitons, Almost Theta-Yamabe Solitons, and Finite-Order Tensor Symmetries of a Vector Field on Riemannian Manifolds with Rank-One Anisotropic Curvature

本文研究秩一各向异性曲率黎曼流形上，Ricci孤立子蕴含几乎θ-Yamabe孤立子，并证明向量场对称性条件等价于由算子D_X=X+6φ+2a控制的标量微分问题。

接触李系统在黎曼和洛伦兹空间上的应用：从缩放对称性到曲率依赖的降维

Contact Lie systems on Riemannian and Lorentzian spaces: from scaling symmetries to curvature-dependent reductions

本文提出了一种将缩放对称性应用于李-哈密顿系统的方法，用于构造接触李系统，并展示了其在不同空间中的应用及降维效果。

Some remarks on conformal symmetries and Bartnik's splitting conjecture

Inspired by the results in a recent paper by G. Galloway and C. Vega (see arXiv:1712.00785), we investigate a number of geometric consequences of the existence of a timelike conformal Killing vector field on a globally hyperbolic spacetime with compact Cauchy hypersurfaces, especially in connection with the so-called Bartnik's splitting conjecture. In particular we give a complementary result to the main theorem in Galloway and Vega's paper.