Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Minimal unit vector fields on oscillator groups

Alexander Yampolsky

TL;DR

The paper investigates minimal left-invariant unit vector fields on oscillator groups and their relation to harmonic maps into the unit tangent bundle with the Sasaki metric. By developing the extrinsic geometry of $V(M)$ inside $(T_1M,g_S)$ through the second fundamental form, Nomizu operator, and related curvature tools, it derives precise minimality conditions. A key finding is that when all structure constants are equal, every unit vector field defining a harmonic map into the Sasaki unit tangent bundle is minimal; more generally, the minimality of a left-invariant vector field on an oscillator group is characterized in terms of the structure constants $\lambda_i$ and the decomposition of the field relative to the Reeb direction. These results generalize and connect prior work on harmonic and minimal unit vector fields on Lie groups, providing explicit criteria for oscillator groups and clarifying the interplay between harmonicity and minimality in this setting.

Abstract

In this paper, we treat minimal left-invariant unit vector fields on oscillator group and their relations with the ones that define a harmonic map. Particularly, if all structure constants of the oscillator group are equal to each other, then all unit left invariant vector fields that define a harmonic map into the unit tangent bundle with Sasaki metric are minimal.

Minimal unit vector fields on oscillator groups

TL;DR

The paper investigates minimal left-invariant unit vector fields on oscillator groups and their relation to harmonic maps into the unit tangent bundle with the Sasaki metric. By developing the extrinsic geometry of inside through the second fundamental form, Nomizu operator, and related curvature tools, it derives precise minimality conditions. A key finding is that when all structure constants are equal, every unit vector field defining a harmonic map into the Sasaki unit tangent bundle is minimal; more generally, the minimality of a left-invariant vector field on an oscillator group is characterized in terms of the structure constants and the decomposition of the field relative to the Reeb direction. These results generalize and connect prior work on harmonic and minimal unit vector fields on Lie groups, providing explicit criteria for oscillator groups and clarifying the interplay between harmonicity and minimality in this setting.

Abstract

In this paper, we treat minimal left-invariant unit vector fields on oscillator group and their relations with the ones that define a harmonic map. Particularly, if all structure constants of the oscillator group are equal to each other, then all unit left invariant vector fields that define a harmonic map into the unit tangent bundle with Sasaki metric are minimal.
Paper Structure (5 sections, 10 theorems, 96 equations)

This paper contains 5 sections, 10 theorems, 96 equations.

Table of Contents

  1. Introduction and preliminaries
  2. Second fundamental form of $V (M)\subset (T_1M, g_S)$
  3. Minimal unit vector fields
  4. The Reeb vector field
  5. Minimal unit vector fields on oscillator group

Key Result

theorem thmcountertheorem

Ym-2022 The Reeb vector field on connected $(\alpha,\beta)$ trans-Sasakian manifold $M$ gives rise to totally geodesic submanifold $\xi(M)\subset (T_1M,g_S)$ only in the following cases

Theorems & Definitions (15)

  • theorem thmcountertheorem
  • corollary thmcountercorollary
  • theorem thmcountertheorem
  • lemma thmcounterlemma
  • proof
  • corollary thmcountercorollary
  • proof
  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma
  • ...and 5 more