A Simple Proof of a Rigidity Theorem for an Affine Kähler–Ricci Flat Graph

Xu, Runzhang; Zhu, Lingyun

doi:10.1007/s00025-015-0478-7

Cited by 3 publications

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“…In fact, this theorem along with many of the previously physically motivated result has been proved. It can be found in a recent work by Li and Xu [53,54,98] in which they generalized the classical Jörgens at al. theorem (Theor.…”

Section: Remark 14 Kählerian Characterization Of Non-relativistic Spmentioning

confidence: 87%

Affine Sphere Relativity

Minguzzi

2016

Commun. Math. Phys.

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We investigate spacetimes whose light cones could be anisotropic. We prove the equivalence of the structures: (a) Lorentz-Finsler manifold for which the mean Cartan torsion vanishes, (b) Lorentz-Finsler manifold for which the indicatrix (observer space) at each point is a convex hyperbolic affine sphere centered on the zero section, and (c) pair given by a spacetime volume and a sharp convex cone distribution. The equivalence suggests to describe (affine sphere) spacetimes with this structure, so that no algebraic-metrical concept enters the definition. As a result, this work shows how the metric features of spacetime emerge from elementary concepts such as measure and order. Non-relativistic spacetimes are obtained replacing proper spheres with improper spheres, so the distinction does not call for group theoretical elements. In physical terms, in affine sphere spacetimes the light cone distribution and the spacetime measure determine the motion of massive and massless particles (hence the dispersion relation). Furthermore, it is shown that, more generally, for Lorentz-Finsler theories non-differentiable at the cone, the lightlike geodesics and the transport of the particle momentum over them are well defined though the curve parametrization could be undefined. Causality theory is also well behaved. Several results for affine sphere spacetimes are presented. Some results in Finsler geometry, for instance in the characterization of Randers spaces, are also included.

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Section: Remark 14 Kählerian Characterization Of Non-relativistic Spmentioning

confidence: 87%