Gauss-Bonnet theorems associated to deformed Schouten-Van Kampen connection in the affine group and the group of rigid motions of the Minkowski plane

TL;DR

This paper develops Gauss-Bonnet theorems for surfaces in affine and Minkowski groups using deformed Schouten-Van Kampen connections, linking curvature limits and geodesic curvature in sub-Riemannian geometry.

Contribution

It introduces deformed Schouten-Van Kampen connections and derives new Gauss-Bonnet theorems in affine and Minkowski group settings.

Findings

Computed sub-Riemannian limits of Gaussian curvature for surfaces.

Derived signed geodesic curvature for curves on surfaces.

Established Gauss-Bonnet theorems in the context of these groups.

Abstract

In this paper, we define deformed Schouten-Van Kampen connections which are metric connections and compute sub-Riemannian limits of Gaussian curvature for a Euclidean C2-smooth surface associated to deformed Schouten-Van Kampen connections with two kinds of distributions in the affine group and the group of rigid motions of the Minkowski plane away from characteristic points and signed geodesic curvature for Euclidean C2-smooth curves on surfaces. According to above results, we get Gauss-Bonnet theorems associated to two kinds of deformed Schouten-Van Kampen connections in the affine group and the group of rigid motions of the Minkowski plane.