Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Some discussions on null helices in 3D semi-Riemannian manifold

Fatma Almaz

TL;DR

This work investigates null helices on a totally umbilical submanifold within a 3D semi-Riemannian manifold of index 2. By developing the null Frenet frame { $\zeta$, $N$, $W$ } and identifying curvature invariants $h$, $k_{1}$, $k_{2}$, it derives explicit Frenet relations and higher-derivative identities that characterize helices in this degenerate metric setting. A key result shows that a null helix with constant curvatures satisfies a specific higher-order relationship, and if such constant-curvature behavior is preserved in the ambient space, the submanifold must be totally geodesic. Overall, the paper provides a rigorous geometric framework for understanding lightlike helices in semi-Riemannian geometry and connects helix properties to the ambient submanifold's curvature structure.

Abstract

In this study, the geometric properties of null helices on a totally umbilical submanifold within a three-dimensional semi-Riemannian manifold are investigated. The pseudo-Riemannian metric structure of semi-Riemannian manifolds and the fact that the submanifold is totally umbilical complicate the differential geometry of null helices. The study uses the given null Frenet frame to reveal the local properties of null helices (the curvatures $% h,k_{1},k_{2}$ of the null curves). By considering the degenerate metric condition of null helices due to the null tangent vector and the structure of the totally umbilical submanifold, equations and invariants characterising null helices are obtained. This study aims to contribute to the theory of null curves on semi-Riemannian manifolds and the geometry of special submanifolds.

Some discussions on null helices in 3D semi-Riemannian manifold

TL;DR

This work investigates null helices on a totally umbilical submanifold within a 3D semi-Riemannian manifold of index 2. By developing the null Frenet frame { , , } and identifying curvature invariants , , , it derives explicit Frenet relations and higher-derivative identities that characterize helices in this degenerate metric setting. A key result shows that a null helix with constant curvatures satisfies a specific higher-order relationship, and if such constant-curvature behavior is preserved in the ambient space, the submanifold must be totally geodesic. Overall, the paper provides a rigorous geometric framework for understanding lightlike helices in semi-Riemannian geometry and connects helix properties to the ambient submanifold's curvature structure.

Abstract

In this study, the geometric properties of null helices on a totally umbilical submanifold within a three-dimensional semi-Riemannian manifold are investigated. The pseudo-Riemannian metric structure of semi-Riemannian manifolds and the fact that the submanifold is totally umbilical complicate the differential geometry of null helices. The study uses the given null Frenet frame to reveal the local properties of null helices (the curvatures of the null curves). By considering the degenerate metric condition of null helices due to the null tangent vector and the structure of the totally umbilical submanifold, equations and invariants characterising null helices are obtained. This study aims to contribute to the theory of null curves on semi-Riemannian manifolds and the geometry of special submanifolds.

Paper Structure

This paper contains 4 sections, 4 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The null helices in $3D$ semi-Riemannian manifold of index $2$
  4. Conclusion

Key Result

Proposition 1

Let $\gamma$ be a null curve of a semi-Riemannian manifold $(M,g)$ with index $q\geq 1$. Then, any screen vector bundle of $\gamma$ is semi-Riemannian with index $q-1,$4.

Theorems & Definitions (6)

  • Proposition 1
  • Theorem 1
  • Theorem 2
  • proof
  • Theorem 3
  • proof