Maciej Dunajski, Michael Eastwood
Given a projective structure on a three-dimensional manifold, we find explicit obstructions to the local existence of a Levi-Civita connection in the projective class. These obstructions are given by projectively invariant tensors algebraically constructed from the projective Weyl curvature. We show, by examples, that their vanishing is necessary but not sufficient for local metrisability.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
On Kähler metrisability of two-dimensional complex projective structures
本文研究复曲面射影结构何时可由（伪）Kähler 度量的 Levi-Civita 联络诱导，通过延拓有限型一阶线性微分系统，建立了相容度量与 C^3 上秩至少为二的 Hermite 形式的一一对应。
A metric projective structure is a manifold equipped with the unparametrised geodesics of some pseudo-Riemannian metric. We make acomprehensive treatment of such structures in the case that there is a projective Weyl curvature nullity condition. The analysis is simplified by a fundamental and canonical 2-tensor invariant that we discover. It leads to a new canonical tractor connection for these geometries which is defined on a rank $(n+1)$-bundle. We show this connection is linked to the metrisability equations that govern the existence of metrics compatible with the structure. The fundamental…
Cone structures and path geometries with constant torsion
本文研究了具有非平凡最大秩挠率的三维路径几何，引入常挠率概念，证明其与齐次直纹曲面上的锥结构一一对应，并通过新可积系统的解进行描述。
We derive the equations of chains for path geometries on surfaces by solving the equivalence problem of a related structure: sub-Riemannian geometry of signature $(1,1)$ on a contact 3-manifold. This approach is significantly simpler than the standard method of solving the full equivalence problem for path geometry. We then use these equations to give a characterization of projective path geometries in terms of their chains (the chains projected to the surface coincide with the paths) and study the chains of four examples of homogeneous path geometries. In one of these examples (horocycles in …
Contact Path Geometries
接触路径几何是接触流形上的弯曲几何结构，包括在辛向量空间的投影化中的所有迷向直线的族上建模的光滑路径族。在局部上，这种结构等价于一个奇数二阶常微分方程组在一个极大不可积约束下的解族在自变量和相依变量空间中的图.接触路径几何的一个子类的特点是不变的接触扭转消失。对于这个子类，等价问题通过使用Tanaka-Morimoto-Čap-Schichl的方法构造规范化Cartan联络来解决。阐述了接触扭转的几何意义。如果二次接触扭转为零，则局部定义的接触路径空间允许分裂的四元数切触结构（类似于O. Biquard）。
Weyl structures for path geometries
本文提出了一种基于Weyl结构的路径几何研究方法，通过定义一类特殊连接和Schouten张量，为路径几何的tractor算术提供了基础，并探讨了其在不变算子构造中的应用。
Duality of quantum geometries
本文通过平行移动算子定义量子联络并引入度量以建立其对偶，证明了量子联络与其对偶在和乐性上的等价关系，并在酉条件下定义了类似信息几何的alpha联络族。
An almost complex structure J on a 4-manifold X may be described in terms of a rank 2 vector bundle E. A splitting of J consists of a pair of line bundles spanning E. A hypersurface M in X satisfying a nondegeneracy condition inherits a CR-structure from J and a path geometry from the splitting. Using the Cartan-Kähler theorem we show that locally every real analytic path geometry is induced by an embedding into C^2 equipped with the splitting generated by the real and imaginary part of the standard holomorphic volume form. As a corollary we obtain the well-known fact that every 3-dimensional …
Two Fefferman-type constructions involving almost Grassmann structures and path geometries
本研究提出两类费弗曼型构造，分别建立路径几何与近格拉斯曼结构、不同型近格拉斯曼结构间的对应，证明其正规性并给出等价刻画条件，丰富了Cartan几何领域的结构对应理论。
Recently, the behavior of different epidemic models and their relation both to different types of geometries and to some biological models has been revisited . Path equations representing the behavior of epidemic models and their corresponding deviation vectors are examined. A comparison between paths and their deviation vectors in Riemannian and Finslerian Geometries is presented.
Generalized Geometries Constructed from Differential Forms on Cotangent Bundle
该研究受Hu等人成果启发，采用新路径从非退化二维辛Monge-Ampère结构等构造新广义殆几何，证明其可生成二次曲面，并探讨了对应结构与方程的关联，拓展了广义几何的研究范围。
Brownian motion on manifolds with non-trivial diffusion coefficient can be constructed by stochastic development of Euclidean Brownian motions using the fiber bundle of linear frames. We provide a comprehensive study of paths for such processes that are most probable in the sense of Onsager-Machlup, however with path probability measured on the driving Euclidean processes. We obtain both a full characterization of the resulting family of most probable paths, reduced equation systems for the path dynamics where the effect of curvature is directly identifiable, and explicit equations in special …
We construct a family of canonical connections and surrounding basic theory for almost complex manifolds that are equipped with an affine connection. This framework provides a uniform approach to treating a range of geometries. In particular we are able to construct an invariant and efficient calculus for conformal almost Hermitian geometries, and also for almost complex structures that are equipped with a projective structure. In the latter case we find a projectively invariant tensor the vanishing of which is necessary and sufficient for the existence of an almost complex connection compatib…
Given a smooth family of unparameterized curves such that through every point in every direction there passes exactly one curve, does there exist a Lagrangian with extremals being precisely this family? It is known that in dimension 2 the answer is positive. In dimension 3, it follows from the work of Douglas that the answer is, in general, negative. We generalise this result to all higher dimensions and show that the answer is actually negative for almost every such a family of curves, also known as path structure or path geometry. On the other hand, we consider path geometries possessing inf…
On the Geometry of Chains
本文所研究的链推广了CR结构的Chern-Moser链。它们在具有抛物切触结构的流形中形成了一维子流形的一个特殊族。抛物型切触结构和链系统都可以等价地编码为Cartan几何（不同类型）。本文的目的是研究拉格朗日切触结构和部分可积的几乎CR结构的这两种Cartan几何之间的关系。我们开发了一个一般的方法，扩展Cartan几何推广的Cartan几何解释February man的建设与CR结构的共形结构。对于这两种结构，我们证明了当且仅当原来的抛物型切触结构是无挠的时，与链族相关联的Cartan几何可以用这种方式得到.特别是，该程序的工作原理完全（可积）CR结构的子类。这两个Cartan几何之间的紧密关系导致明确描述的Cartan曲率与家庭的链。一方面，这表明两个抛物切触结构的齐次模型产生了具有大自同构群的非平坦路径几何的例子。另一方面，我们表明，人们可以（几乎）重建基本的无挠的抛物切触结构从Cartan曲率相关的链。特别是，这导致了一个非常概念性的事实，即链保持接触的同构或反同构的抛物切触结构的证明。
The projective Finsler metrizability problem deals with the question whether a projective-equivalence class of sprays is the geodesic class of a (locally or globally defined) Finsler function. In this paper we use Hilbert-type forms to state a number of different ways of specifying necessary and sufficient conditions for this to be the case, and we show that they are equivalent. We also address several related issues of interest including path spaces, Jacobi fields, totally-geodesic submanifolds of a spray space, and the equivalence of path geometries and projective-equivalence classes of spra…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。