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Hyperbolic generalized framed surfaces in hyperbolic 3-space

Donghe Pei, Masatomo Takahashi, Anjie Zhou

TL;DR

The paper develops hyperbolic generalized framed surfaces in $H^3$ by extending hyperbolic framed surfaces and one-parameter families of hyperbolic framed curves through a moving-frame formalism. It establishes necessary and sufficient conditions for a surface to be a hyperbolic generalized framed base surface, derives integrability criteria, analyzes singularities via invariants, and explores reductions to classical structures. It then connects these hyperbolic constructs to Euclidean generalized framed surfaces and lightcone framed surfaces through diffeomorphisms and projections, illustrating the unifying reach of the framework. As an application, the work treats horocyclic surfaces as hyperbolic generalized framed base surfaces, enriching the repertoire of explicit examples and clarifying the geometric interplay across hyperbolic, Euclidean, and Lorentzian settings.

Abstract

Generalizing both hyperbolic framed surfaces and one-parameter families of hyperbolic framed curves, we introduce the concept of hyperbolic generalized framed surfaces and establish their relations in hyperbolic 3-space. We provide the necessary and sufficient conditions for a smooth surface to be a hyperbolic generalized framed base surface, followed by an analysis of the singularities of hyperbolic generalized framed base surfaces. Additionally, relations between hyperbolic generalized framed surfaces, generalized framed surfaces and lightcone framed surfaces are explored. As an application of hyperbolic generalized framed surfaces, we investigate the properties of horocyclic surfaces.

Hyperbolic generalized framed surfaces in hyperbolic 3-space

TL;DR

The paper develops hyperbolic generalized framed surfaces in by extending hyperbolic framed surfaces and one-parameter families of hyperbolic framed curves through a moving-frame formalism. It establishes necessary and sufficient conditions for a surface to be a hyperbolic generalized framed base surface, derives integrability criteria, analyzes singularities via invariants, and explores reductions to classical structures. It then connects these hyperbolic constructs to Euclidean generalized framed surfaces and lightcone framed surfaces through diffeomorphisms and projections, illustrating the unifying reach of the framework. As an application, the work treats horocyclic surfaces as hyperbolic generalized framed base surfaces, enriching the repertoire of explicit examples and clarifying the geometric interplay across hyperbolic, Euclidean, and Lorentzian settings.

Abstract

Generalizing both hyperbolic framed surfaces and one-parameter families of hyperbolic framed curves, we introduce the concept of hyperbolic generalized framed surfaces and establish their relations in hyperbolic 3-space. We provide the necessary and sufficient conditions for a smooth surface to be a hyperbolic generalized framed base surface, followed by an analysis of the singularities of hyperbolic generalized framed base surfaces. Additionally, relations between hyperbolic generalized framed surfaces, generalized framed surfaces and lightcone framed surfaces are explored. As an application of hyperbolic generalized framed surfaces, we investigate the properties of horocyclic surfaces.
Paper Structure (17 sections, 20 theorems, 104 equations, 2 figures)

This paper contains 17 sections, 20 theorems, 104 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Preliminaries
  4. Hyperbolic framed surfaces
  5. One-parameter families of hyperbolic framed curves
  6. Generalized framed surfaces in Euclidean 3-space
  7. Lightcone framed surfaces in Lorentz-Minkowski 3-space
  8. Hyperbolic generalized framed surfaces
  9. Relations between hyperbolic generalized framed surfaces, hyperbolic framed surfaces and one-parameter families of hyperbolic framed curves.
  10. Reflection and rotation of frame as well as a parameter change in $U$.
  11. Relations between hyperbolic generalized framed base surfaces, hyperbolic framed base surfaces, one-parameter families of hyperbolic framed base curves and smooth surfaces.
  12. Singularities of hyperbolic generalized framed base surfaces
  13. Examples
  14. Relations between hyperbolic generalized framed surfaces in $H^3$, generalized framed surfaces in $\mathbb{R}^3$ and lightcone framed surfaces in $\mathbb{R}_1^3$
  15. Relations between hyperbolic generalized framed surfaces and generalized framed surfaces
  16. ...and 2 more sections

Key Result

Theorem 3.3

Let $a_i, b_i,c_i, e_i$, $f_i, g_i: U\rightarrow \mathbb{R}$, $i=1,2$ be smooth functions with $a_1b_2-a_2b_1=0$ and the integrability conditions $(eq3)$ and $(eq4)$. Then there exists a hyperbolic generalized framed surface $(\hbox{\boldmath $x$},\hbox{\boldmath $\nu$}_1,\hbox{\boldmath $\nu$}_2):

Figures (2)

  • Figure 1: A curve $\hbox{\boldmath $\gamma$}$ and a hyperbolic ruled surface $\hbox{\boldmath $x$}$ projected to Poincaré 3-disc. The two black points are cross cap singularities of $\hbox{\boldmath $x$}$.
  • Figure 2: A curve $\hbox{\boldmath $\gamma$}$ and a hyperbolic ruled surface $\hbox{\boldmath $x$}$ projected to Poincaré 3-disc. The two black points are swallowtail singularities of $\hbox{\boldmath $x$}$. The other points on $\hbox{\boldmath $\gamma$}$ are cuspidal edge singularities of $\hbox{\boldmath $x$}$.

Theorems & Definitions (46)

  • Definition 2.1: ZP
  • Definition 2.2
  • Definition 2.3: TY2024
  • Definition 2.4: LPT2025
  • Definition 3.1
  • Definition 3.2
  • Theorem 3.3: Existence theorem for hyperbolic generalized framed surfaces
  • proof
  • Theorem 3.4: Uniqueness theorem for hyperbolic generalized framed surfaces
  • proof
  • ...and 36 more