Hannover 2003 – scientific programme

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GR: Gravitation und Relativitätstheorie

GR 5: Grundlegende Probleme und alternative Ans
ätze

GR 5.3: Fachvortrag

Tuesday, March 25, 2003, 17:10–17:30, A310

Consequences of independent coordinatizations for points and hyperplanes — •Bertfried Fauser — Universität Konstant, Fachbereich Physik, Fach M678, 78457 Konstanz

Riemannian geometry is based on points as generic objects. Even if tangent and cotangent bundles are used, covectors are not employed as geometric objects, i.e. hyperplanes, but as linear forms. We have in the last years argued that a point-only view of geometry is not sufficiently handy for more advanced investigations, especially for projective relativity theory. Hehl and Obukhov currently made use of line coordinates (E,B) and (D,H) in R⁶ in their approach to axiomatic electrodynamics.

We will show that the coordinatization of point space and hyperplane space can be made independently. Due to such choices, certain symmetries will no longer be present. Using Poncelet’s r-linear polarity both coordinate systems can be connected but at the cost of incidence relations based on arbitrary not necessarily symmetric non-degenerate incidence matrices. We will exemplify this setting and draw some conclusions for differential geometry. Remarkably this process is related to Wick reordering in quantum field theory.

Part of this talk is based on unpublished results of Prof. H.-J. Stoß, Fachbereich Mathematik, Uni. Konstanz.