# Constant angle surfaces in 4-dimensional Minkowski space

## TL;DR

This paper introduces the concept of constant angle surfaces in 4D Minkowski space, characterizes their geometric properties, and provides explicit constructions and classifications, extending Euclidean notions to Lorentzian geometry.

## Contribution

It defines a complex angle between spacelike planes in Minkowski space and studies the geometric and analytic properties of surfaces maintaining a constant such angle, including their curvatures and representations.

## Key findings

- Constant angle surfaces have zero Gauss and normal curvatures.
- Representation formulas are derived using PDE methods.
- Surfaces with non-zero constant angle are never complete.

## Abstract

We first define a complex angle between two oriented spacelike planes in 4-dimensional Minkowski space, and then study the constant angle surfaces in that space, i.e. the oriented spacelike surfaces whose tangent planes form a constant complex angle with respect to a fixed spacelike plane. This notion is the natural Lorentzian analogue of the notion of constant angle surfaces in 4-dimensional Euclidean space. We prove that these surfaces have vanishing Gauss and normal curvatures, obtain representation formulas for the constant angle surfaces with regular Gauss maps and construct constant angle surfaces using PDE's methods. We then describe their invariants of second order and show that a surface with regular Gauss map and constant angle $\psi\neq 0\ [\pi/2]$ is never complete. We finally study the special cases of surfaces with constant angle $\pi/2\ [\pi],$ with real or pure imaginary constant angle and describe the constant angle surfaces in hyperspheres and lightcones.