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Surfaces with flat normal connection in 4-dimensional space forms

Naoya Ando, Ryusei Hatanaka

TL;DR

This work provides a comprehensive characterization of surfaces in 4D space forms with flat normal connection, unifying space-like and time-like cases across Riemannian, neutral, and Lorentzian signatures. By deriving the Gauss-Codazzi-Ricci equations for conformal immersions and analyzing the flat normal connection condition $R^{\perp}=0$, it establishes precise criteria linking the existence of parallel normal vector fields to a linearly dependent condition on the second fundamental form, with explicit distinctions depending on whether $K\neq L_0$ or $K\equiv L_0$. The authors give a generic characterization for the $K\equiv L_0$ case that permits no parallel normals, employing a network of auxiliary functions ($\lambda$, $\gamma$, $\theta$, $\psi$, $f_+$, $f_-$) and, in the flat ambient limit ($L_0=0$), an over-determined polynomial-type system that encodes the integrability conditions. The results extend the theory to neutral (paracomplex) and Lorentzian 4D space forms, yielding analogous criteria and constructions for space-like and time-like immersions, and thereby provide a complete geometric framework for flat normal connections in 4D space forms with varying signatures.

Abstract

Let $N$ be a Riemannian, Lorentzian or neutral $4$-dimensional space form with constant sectional curvature $L_0$. In this paper, noticing the linearly dependent condition, we obtain characterizations of space-like surfaces in $N$ with flat normal connection and parallel normal vector fields. In addition, we obtain a generic characterization of space-like surfaces in $N$ with flat normal connection and $K\equiv L_0$ which do not admit any parallel normal vector fields. For time-like surfaces in $N$ with flat normal connection, we obtain analogous results.

Surfaces with flat normal connection in 4-dimensional space forms

TL;DR

This work provides a comprehensive characterization of surfaces in 4D space forms with flat normal connection, unifying space-like and time-like cases across Riemannian, neutral, and Lorentzian signatures. By deriving the Gauss-Codazzi-Ricci equations for conformal immersions and analyzing the flat normal connection condition , it establishes precise criteria linking the existence of parallel normal vector fields to a linearly dependent condition on the second fundamental form, with explicit distinctions depending on whether or . The authors give a generic characterization for the case that permits no parallel normals, employing a network of auxiliary functions (, , , , , ) and, in the flat ambient limit (), an over-determined polynomial-type system that encodes the integrability conditions. The results extend the theory to neutral (paracomplex) and Lorentzian 4D space forms, yielding analogous criteria and constructions for space-like and time-like immersions, and thereby provide a complete geometric framework for flat normal connections in 4D space forms with varying signatures.

Abstract

Let be a Riemannian, Lorentzian or neutral -dimensional space form with constant sectional curvature . In this paper, noticing the linearly dependent condition, we obtain characterizations of space-like surfaces in with flat normal connection and parallel normal vector fields. In addition, we obtain a generic characterization of space-like surfaces in with flat normal connection and which do not admit any parallel normal vector fields. For time-like surfaces in with flat normal connection, we obtain analogous results.
Paper Structure (16 sections, 27 theorems, 175 equations)

This paper contains 16 sections, 27 theorems, 175 equations.

Table of Contents

  1. Introduction
  2. Surfaces with flat normal connection in Riemannian 4-dimensional space forms
  3. The Gauss-Codazzi-Ricci equations
  4. Surfaces with $R^{\perp} \equiv 0$ and parallel normal vector fields
  5. Surfaces with $R^{\perp} \equiv 0$ and $K\equiv L_0$ which do not admit any parallel normal vector fields
  6. Space-like surfaces with flat normal connection in neutral 4-dimensional space forms
  7. Time-like surfaces with flat normal connection in neutral 4-dimensional space forms
  8. The Gauss-Codazzi-Ricci equations
  9. Time-like surfaces with $R^{\perp} \equiv 0$ and parallel normal vector fields
  10. Time-like surfaces with $R^{\perp} \equiv 0$ and $K\equiv L_0$ which do not admit any parallel normal vector fields
  11. Space-like surfaces with flat normal connection in Lorentzian 4-dimensional space forms
  12. The Gauss-Codazzi-Ricci equations
  13. Space-like surfaces with $R^{\perp} \equiv 0$ and $K\equiv L_0$ which do not admit any parallel normal vector fields
  14. Time-like surfaces with flat normal connection in Lorentzian 4-dimensional space forms
  15. The Gauss-Codazzi-Ricci equations
  16. ...and 1 more sections

Key Result

Theorem 2.1

Suppose that $\nabla^{\perp}$ is flat. If $F$ has a parallel normal vector field, then it is given by $e^{-\lambda} (\cos \theta N_1 +\sin \theta N_2 )$.

Theorems & Definitions (34)

  • Theorem 2.1
  • Remark 2.1
  • Theorem 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Theorem 2.7
  • Theorem 2.8
  • Example 2.1
  • ...and 24 more