Invariants on projective space

Gover, A. Rod

doi:10.1090/s0894-0347-1994-1214703-8

Cited by 10 publications

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“…Thomas [109]. The invariant lift is a close relative of the notion of saturated cotractor used in [60]; for example in that language the application of Lemma 7.5 to the projective Weyl tensor as in Remark 7.6 is essentially [60, Equation 33]. In [62,Theorem 4.4] the result is proved essentially as stated here for…”

Section: Conelike Equiaffine Radiant Structure Invariant Under the Pr...mentioning

confidence: 99%

Conelike radiant structures

Fox¹

2021

Preprint

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The notion of conelike radiant structure formalizes the idea of 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 means a torsion-free affine connection and a radiant vector field, meaning its 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 geodesics 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 structure 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 statistical manifolds (and a conformal generalization of statistical structures) is given that generalizes the Einstein-Weyl equations and encompasses these more general examples.Finally, 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.

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Section: Conelike Equiaffine Radiant Structure Invariant Under the Pr...mentioning

confidence: 99%