Prolongations of Geometric Overdetermined Systems

射影几何与PDE延拓

Projective Geometry and PDE Prolongation

本文研究射影几何与偏微分方程延拓，通过牵引子微积分恢复射影联络，并重点分析射影可度量化方程的延拓。

几何结构的延拓、不变量与基本恒等式

Prolongations, invariants, and fundamental identities of geometric structures

本文在幂零几何框架下统一处理几何结构等价问题，通过重构高阶几何结构与万有标架丛的阶延拓，将结构函数分解为κ、τ、σ并建立基本恒等式，进而利用广义Spencer上同调为常符号正规几何结构构造完全不变量系统，并刻画Cartan联络在阶延拓中的位置。

Geometrical structures on the prolongation of a pre-Lie algebroid on fibered manifolds and application to Partial Finsler geometry on foliated anchored bundle

A pre-Lie algebroid is an anchored bundle provided with an almost Lie bracket such that the anchor is compatible with the Lie bracket of vector fields. We firstly show how most geometrical structures intensively studied in the framework of Lie algebroid can easily be extended in the pre-Lie algebroid context. The principal purpose of this paper is to show that how all these results only depend of the foliated structure and do not depend of the pre-Lie algebroid structure that we can put on a foliated anchored bundle. As application, we obtain a Finsler connection on a foliated anchored bundle …

A note on jet and geometric approach to higher order connections

We compare two ways of interpreting higher order connections. The geometric approach lies in the decomposition of higher order tangent space into the horizontal and vertical structures while the jet--like approach considers a higher order connection as the section of a jet prolongation of a fibered manifold. Particularly, we use the Ehresmann prolongation of a general connection and study the result from the point of view of geometric theory. We pay attention to linear connections, too.

Prolongation of Tanaka structures: an alternative approach

The classical theory of prolongation of G-structures was generalized by N. Tanaka to a wide class of geometric structures (Tanaka structures), which are defined on a non-holonomic distribution. Examples of Tanaka structures include subriemannian, subconformal, CR-structures, structures associated to second order differential equations and structures defined by gradings of Lie algebras (in the setting of parabolic geometry). Tanaka's prolongation procedure associates to a Tanaka structure of finite order a manifold with an absolute parallelism. It is a very fruitful method for the description o…

Prolongations vs. Tulczyjew triples in Geometric Mechanics

In the scientific literature there are basically two schools of formulating Lagrangian (or Hamiltonian) mechanics in the (Lie) algebroid setting: in terms of prolongations and in terms of Tulczyjew triples. Despite the fact that in both approaches we describe the same phenomena, so far no comparison between prolongations and Tulczyjew triples was made. In this note we aim to fill this gap. More precisely, we will strip the prolongation approach to uncover the Tulczyjew triple reality hidden inside, thus proving that the latter approach is a more basic one.

Efficient construction of contact coordinates for partial prolongations

Let $\CV$ be a vector field distribution on manifold $M$. We give an efficient algorithm for the construction of local coordinates on $M$ such that $\CV$ may be locally expressed as some partial prolongation of the contact distribution $\Cal C^{(1)}_q$, on the first order jet bundle of maps from $\Bbb R$ to $\Bbb R^q$, $q\geq 1$. It is proven that if $\CV$ is locally equivalent to a partial prolongation of $\Cal C^{(1)}_q$ then the explicit construction of contact coordinates algorithmically depends upon the determination of certain first integrals in a sequence of geometrically defined and al…

Geometric Approaches for Generating Prolongations for Nonlinear Partial Differential Equations

The prolongation structure of a two-by-two problem is formulated very generally in terms of exterior differential forms on a standard representation of Pauli matrices. The differential system is general without making reference to any specific equation. An integrability condition is provided which gives by construction the equation to be investigated and whose components involve the structure constants of an SU(2) Lie algebra. Along side this, a related, different kind of prolongation, a type of Wahlquist-Estabrook prolongation, over a closed differential ideal is discussed and some applicatio…

广义Tanaka延拓与嵌入间形式等价的收敛性

Generalized Tanaka prolongation and convergence of formal equivalence between embeddings

本文证明了在半正法丛及特定几何条件下紧复流形嵌入间形式等价的收敛性，并提出了从属于向量分布的几何结构的广义Tanaka延拓过程作为关键工具。

Dynamical Equations and Lagrange--Ricci Flow Evolution on Prolongation Lie Algebroids

The approach to nonholonomic Ricci flows and geometric evolution of regular Lagrange systems [S. Vacaru: J. Math. Phys. \textbf{49} (2008) 043504 \& Rep. Math. Phys. \textbf{63} (2009) 95] is extended to include geometric mechanics and gravity models on Lie algebroids. We prove that such evolution scenarios of geometric mechanics and analogous gravity can be modelled as gradient flows characterized by generalized Perelman functionals if an equivalent geometrization of Lagrange mechanics [J. Kern, Arch. Math. (Basel) \textbf{25} (1974) 438] is considered. The R. Hamilton equations on Lie algebr…

半线性超定问题、散度公式与几何不等式

Semilinear overdetermined problems, a divergence formula, and geometric inequalities

本文研究满足Serrin型问题的黎曼流形边界面积界，推导相关散度公式，证明Minkowski型、面积-电荷不等式，推广Boucher-Gibbons-Horowitz经典结果。

A constructive generalised Goursat normal form

We provide necessary and sufficient conditions on the derived type of a vector field distribution $\Cal V$ in order that it be locally equivalent to a partial prolongation of the contact distribution $\Cal C^{(1)}_q$, on the first order jet bundle of maps from $\Bbb R$ to $\Bbb R^q$, $q\geq 1$. This result fully generalises the classical Goursat normal form. Our proof is constructive: it is proven that if $\Cal V$ is locally equivalent to a partial prolongation of $\Cal C^{(1)}_q$ then the explicit construction of contact coordinates algorithmically depends upon the integration of a sequence o…

一般几何结构上线性化边值问题的Hodge理论

Hodge Theory for Linearized Boundary-Value Problems on General Geometric Structures

本文发展了一个将线性化超定边值问题的可解性和唯一性条件系统转化为上同调术语的框架，通过引入椭圆预复形概念推广Hodge理论，并应用于外协变导数、Killing方程、Hessian方程和Riemann曲率方程等几何例子。

场论的雅可比方程与耗散的几何变分描述

Jacobi equation for field theories and a geometric variational description of dissipation

本文通过 $k$-余辛形式延拓拉格朗日量，几何化描述一阶场论的雅可比方程，并修正延拓拉格朗日量以建立耗散场论的变分表述。

外区域中齐次 $k$-Hessian 方程的一般单调公式及其应用

General monotone formula for homogeneous $k$-Hessian equation in the exterior domain and its applications

针对外区域齐次 $k$-Hessian 方程超定问题，论文结合积分恒等式与几何不等式提出新方法，并建立一般单调公式以证明球特征刻画及相关几何不等式。

相对代数胚与Cartan实现问题

Relative algebroids and Cartan realization problems

本文提出相对代数胚统一李代数胚与偏微分方程理论，通过延拓与形式可积性构造profinite李代数胚，以解决几何结构的存在与分类问题。

Discrete Nonholonomic Lagrangian Systems on Lie Groupoids

This paper studies the construction of geometric integrators for nonholonomic systems. We derive the nonholonomic discrete Euler-Lagrange equations in a setting which permits to deduce geometric integrators for continuous nonholonomic systems (reduced or not). The formalism is given in terms of Lie groupoids, specifying a discrete Lagrangian and a constraint submanifold on it. Additionally, it is necessary to fix a vector subbundle of the Lie algebroid associated to the Lie groupoid. We also discuss the existence of nonholonomic evolution operators in terms of the discrete nonholonomic Legendr…

Geometry of solutions to the c-projective metrizability equation

On an almost complex manifold, a quasi-Kähler metric, with canonical connection in the c-projective class of a given minimal complex connection, is equivalent to a non-degenerate solution of the c-projectively invariant metrizability equation. For this overdetermined equation, replacing this maximal rank condition on solutions with a nondegeneracy condition on the prolonged system yields a strictly wider class of solutions with non-vanishing (generalized) scalar curvature. We study the geometries induced by this class of solutions. For each solution, the strict point-wise signature partitions …

扭曲积流形中 Serrin 问题和 Alexandroff 定理的稳定性

Stability for Serrin's problem and Alexandroff's theorem in warped product manifolds

本文证明了几何偏微分方程多个结果的定量版本，包括空间形式中 Serrin 超定问题的双重稳定性定理及扭曲积空间中相关不等式与分类的稳定性定理。

Metrics in projective differential geometry: the geometry of solutions to the metrizability equation

Pseudo-Riemannian metrics with Levi-Civita connection in the projective class of a given torsion free affine connection can be obtained from (and are equivalent to) the maximal rank solutions of a certain overdetermined projectively invariant differential equation often called the metrizability equation. Dropping this rank assumption we study the solutions to this equation given less restrictive generic conditions on its prolonged system. In this setting we find that the solution stratifies the manifold according to the strict signature (pointwise) of the solution and does this in way that loc…

A new symmetry criterion based on the distance function and applications to PDE's

We prove that, if $Ω\subset \mathbb{R}^n$ is an open bounded starshaped domain of class $C^2$, the constancy over $\partial Ω$ of the function $$φ(y) = \int_0^{λ(y)} \prod_{j=1}^{n-1}[1-t κ_j(y)]\, dt$$ implies that $Ω$ is a ball. Here $k_j(y)$ and $λ(y)$ denote respectively the principal curvatures and the cut value of a boundary point $y \in \partial Ω$. We apply this geometric result to different symmetry questions for PDE's: an overdetermined system of Monge-Kantorovich type equations (which can be viewed as the limit as $p \to + \infty$ of Serrin's symmetry problem for the $p$-Laplacian),…

从二阶微分几何到随机几何力学

From Second-Order Differential Geometry to Stochastic Geometric Mechanics

本文借助Schwartz–Meyer二阶随机微分几何，重建随机情形下的辛、Lagrangian、Hamiltonian、HJB与Noether等结构，并以最优传输中的Schrödinger问题为贯穿示例。

$\mathbb{S}^2$上的极值领域：几何与分析方法

Extremal domains in $\mathbb{S}^2$: Geometric and Analytic methods

本文研究$\mathbb{S}^2$上支持过定问题的极值领域，通过几何与分析方法证明其对称性及特定非线性情况下的旋转对称性。