Diffeomorphism invariant measure for finite dimensional geometries

TL;DR

This paper derives a unique, gauge-invariant measure for finite-dimensional geometries in D-dimensional space, ensuring independence from gauge fixing and parameterization, with implications for geometric analysis.

Contribution

It introduces a geometric invariant measure derived from the De Witt metric, invariant under gauge and parameter changes, and addresses the existence of geometries lacking orthogonal gauge fixing surfaces.

Findings

The measure is uniquely determined and gauge-invariant.

Existence of geometries without orthogonal gauge fixing surfaces.

The measure remains independent of the conformal factor parameter.

Abstract

We consider families of geometries of D--dimensional space, described by a finite number of parameters. Starting from the De Witt metric we extract a unique integration measure which turns out to be a geometric invariant, i.e. independent of the gauge fixed metric used for describing the geometries. The measure is also invariant in form under an arbitrary change of parameters describing the geometries. We prove the existence of geometries for which there are no related gauge fixing surfaces orthogonal to the gauge fibers. The additional functional integration on the conformal factor makes the measure independent of the free parameter intervening in the De Witt metric. The determinants appearing in the measure are mathematically well defined even though technically difficult to compute.