Conelike radiant structures

Daniel J. F. Fox
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Abstract

Abstract Analogues of the classical affine-projective correspondence are developed in the context of statistical manifolds compatible with a radiant vector field. These utilize a formulation of Einstein equations for special statistical structures that generalizes the usual Einstein equations for pseudo-Riemannian metrics and is of independent interest. A conelike radiant structure is a not necessarily flat affine connection equipped with a family of surfaces that behave like the intersections of the planes through the origin with a convex cone in a real vector space. A radiant structure is a torsion-free affine connection and a vector field whose covariant derivative is the identity endomorphism. A radiant structure is conelike if for every point and every two-dimensional subspace containing the radiant vector field there is a totally geodesic surface passing through the point and tangent to the subspace. Such structures exist on the total space of any principal bundle with one-dimensional fiber and on any Lie group with a quadratic structure on its Lie algebra. The affine connection of a conelike radiant structure can be normalized in a canonical way to have antisymmetric Ricci tensor. Applied to a conelike radiant structure on the total space of a principal bundle with one-dimensional fiber this yields a generalization of the classical Thomas connection of a projective structure. The compatibility of radiant and conelike structures with metrics is investigated and yields a construction of connections for which the symmetrized Ricci curvature is a constant multiple of a compatible metric that generalizes well-known constructions of Riemannian and Lorentzian Einstein–Weyl structures over Kähler–Einstein manifolds having nonzero scalar curvature. A formulation of Einstein equations for special statistical manifolds is given that generalizes the Einstein–Weyl equations and encompasses these more general examples. There are constructed left-invariant conelike radiant structures on a Lie group endowed with a left-invariant nondegenerate bilinear form, and the case of three-dimensional unimodular Lie groups is described in detail.
锥形辐射结构
摘要在与辐射矢量场相容的统计流形的背景下,发展了经典仿射-射影对应的类似物。这些利用爱因斯坦方程的特殊统计结构的公式,推广了通常的伪黎曼度量的爱因斯坦方程,是独立的兴趣。锥形辐射结构不一定是一个平坦的仿射连接,它配备了一系列表面,这些表面的行为类似于在实向量空间中通过原点的平面与凸锥的相交。辐射结构是一个无扭仿射连接和一个协变导数为恒等自同态的向量场。如果对于包含辐射矢量场的每个点和每个二维子空间都有一个经过该点并与子空间相切的完全测地线面,则辐射结构是圆锥状的。这种结构存在于具有一维纤维的任何主束的总空间上,存在于其李代数上具有二次结构的任何李群上。锥形辐射结构的仿射连接可以正则化为具有反对称里奇张量。应用于具有一维纤维的主束总空间上的锥状辐射结构,得到了射影结构经典托马斯连接的推广。研究了辐射和锥状结构与度量的相容性,并得到了一种连接的构造,其中对称里奇曲率是相容度量的常数倍,该相容度量推广了具有非零标量曲率的Kähler-Einstein流形上著名的黎曼和洛伦兹爱因斯坦-魏尔结构的构造。本文给出了一个特殊统计流形的爱因斯坦方程的公式,它推广了爱因斯坦-魏尔方程,并包含了这些更一般的例子。在具有左不变非退化双线性形式的李群上构造了左不变锥状辐射结构,并详细描述了三维单模李群的情况。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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CiteScore
1.70
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0.00%
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