Surfaces with flat normal connection in 4-dimensional space forms

TL;DR

This paper characterizes space-like and time-like surfaces with flat normal connection in 4-dimensional space forms, including conditions for parallel normal vector fields and cases with constant Gaussian curvature.

Contribution

It provides new characterizations of surfaces with flat normal connection in various 4D space forms, including those without parallel normal vector fields.

Findings

Characterizations of space-like surfaces with flat normal connection and parallel normal vector fields.

Generic characterization of surfaces with flat normal connection and constant curvature without parallel fields.

Analogous results for time-like surfaces in the same setting.

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.