Premanifolds

TL;DR

This paper develops a differential geometry framework for premanifolds equipped with a Minkowski product, generalizing classical semi-Riemannian manifolds to include non-symmetric, non-bilinear products in space-time models.

Contribution

It introduces a method to analyze premanifolds with Minkowski products as hypersurfaces in generalized space-time models, extending differential geometric tools to this setting.

Findings

Defined fundamental forms and curvatures for premanifolds with Minkowski products.

Generalized semi-Riemannian manifolds include hyperbolic space, de Sitter sphere, and light cone.

Provided a framework for differential geometry in non-symmetric, non-bilinear product spaces.

Abstract

The tangent hyperplanes of the "manifolds" of this paper equipped a so-called Minkowski product. It is neither symmetric nor bilinear. We give a method to handing such an object as a locally hypersurface of a generalized space-time model and define the main tools of its differential geometry: its fundamental forms, its curvatures and so on. In the case, when the fixed space-time component of the embedding structure is a continuously differentiable semi-inner product space, we get a natural generalization of some important semi-Riemann manifolds as the hyperbolic space, the de Sitter sphere and the light cone of a Minkowski-Lorenz space, respectively.