A topological metric in 2+1-dimensions

TL;DR

This paper introduces a topological metric in 2+1 dimensions derived from scalar fields, characterized by an integer degree, and formulated through harmonic maps between Riemannian manifolds.

Contribution

It presents a novel topological metric in 2+1 dimensions linked to scalar field configurations and harmonic map theory, expanding understanding of geometric structures in lower-dimensional gravity.

Findings

The metric tilts the scalar, not the light cone.

The metric is static and regular.

It is characterized by an integer degree .

Abstract

Real-valued triplet of scalar fields as source gives rise to a metric which tilts the scalar, not the light cone, in 2+1-dimensions. The topological metric is static, regular and it is characterized by an integer $\kappa =\pm 1,\pm 2,...$. The problem is formulated as a harmonic map of Riemannian manifolds in which the integer $\kappa $ equals to the degree of the map.