The Clairaut's theorem on rotational surfaces in pseudo Euclidean 4-space with index 2

TL;DR

This paper extends Clairaut's theorem to rotational surfaces in semi Euclidean 4-space with index 2, providing new equations for geodesic curves on these surfaces.

Contribution

It generalizes Clairaut's theorem to pseudo Euclidean 4-space and characterizes geodesic equations on hyperbolic and elliptic rotational surfaces.

Findings

Clairaut's theorem is formulated for semi Euclidean 4-space surfaces.

Explicit equations for time-like geodesic curves are derived.

Results apply to hyperbolic and elliptic rotational surfaces.

Abstract

In this paper, Clairaut's theorem is expressed on the surfaces of rotation in semi Euclidean 4-space. Moreover, the general equations of time-like geodesic curves are characterized according to the results of Clairaut's theorem on the hyperbolic surfaces of rotation and the elliptic surface of rotation respectively.