Manifolds obtained by soldering together points, lines, etc

TL;DR

This paper explores a novel interpretation of soldering forms in differential geometry, proposing a new class of geometries with higher-dimensional basic objects, motivated by the mathematical needs of General Relativity and potential physics applications.

Contribution

It introduces an alternative interpretation of soldering forms leading to a new family of geometries with higher-dimensional basic objects, expanding the mathematical framework for physics.

Findings

Proposes a new interpretation of soldering forms in differential geometry.

Introduces a family of geometries with (p-1)-dimensional basic objects.

Discusses potential applications in physics and connections to General Relativity.

Abstract

This text is the extended version of a talk given at the conference Geometry, Topology, QFT and Cosmology hold from May 28 to May 30, 2008 at the Observatoire de Paris. We explore the notion of solder (or soldering form) in differential geometry and propose an alternative interpretation of it, motivated by the search of an accurate mathematical description of the General Relativity. This new interpretation leads naturally to imagine a rich family of new geometries which has not yet a satisfactory definition in general. We try however to communicate to the reader an intuition of such geometries through some examples and review quickly some possible applications in physics. The basic objects in this geometry are not points (i.e. 0-dimensional), but (p-1)-dimensional.