The intrinsic geometry of a convex surface in Galilean space

Abstract

This paper investigates the intrinsic geometry of a convex surface in the Galilean space. The Galilean space, as a special case of a pseudo-Euclidean space, possesses a degenerate metric. The angle between two directions is defined using a parabolic method, which is itself degenerate. The three-dimensional Galilean space, similar to the Euclidean space, is based on a three-dimensional affine space. While the fundamental geometric objects in these spaces coincide structurally, the geometric quantities associated with them differ significantly from those in Euclidean geometry. It becomes necessary to introduce and rigorously define various geometric characteristics of objects in Galilean space. Therefore, special attention in this work is given to the total angle around the vertex of a cone, the angle between curves on a convex surface, and the variation of curve turning on a convex surface. A geodesic on a convex surface is defined as a curve with bounded variation of turning. A triangle is defined as a curve homeomorphic to a circle, bounded by three geodesics. Using the concept of the total angle around the vertex of a cone, we define the intrinsic curvature of convex surfaces in Galilean space and obtain an analogue of the Gauss–Bonnet theorem for convex surfaces in Galilean geometry. The results obtained extend classical notions of intrinsic geometry under a degenerate metric.