An Approach for Hypersurface Family with Common Geodesic Curve in the 4D Galilean Space G4

TL;DR

This paper develops a method to construct hypersurface families in 4D Galilean space from a given geodesic curve, using Frenet frames and analyzing conditions for geodesicity.

Contribution

It introduces a new approach for generating hypersurfaces with a common geodesic in 4D Galilean space based on Frenet frames.

Findings

Derived conditions for a curve to be a geodesic in G4.

Constructed hypersurface families from isogeodesic curves.

Provided examples illustrating the method.

Abstract

In the present study, we derive the problem of constructing a hypersurface family from a given isogeodesic curve in the 4D Galilean space $\mathbf{G}_{4}.$ We obtain the hypersurface as a linear combination of the Frenet frame in $\mathbf{G}_{4}$ and examine the necessary and sufficient conditions for the curve as a geodesic curve$.$ Finally, some examples related to our method are given for the sake of clarity.