Constant Angle Surfaces in Product Spaces

TL;DR

This paper classifies surfaces in product spaces where the tangent plane makes a constant angle with a fixed factor, and also characterizes all totally geodesic surfaces within these spaces.

Contribution

It provides a complete classification of constant angle surfaces and totally geodesic surfaces in product spaces of two 2-dimensional space forms.

Findings

Classification of all constant angle surfaces in $M^2(c_1)\times M^2(c_2)$.

Complete description of totally geodesic surfaces in these product spaces.

Results applicable to non-flat space form products.

Abstract

We classify all the surfaces in $M^2(c_1)\times M^2(c_2)$ for which the tangent space $T_pM^2$ makes constant angles with $T_p(M^2(c_1)\times \{p_2\})$ (or equivalently with $T_p(\{p_1\}\times M^2(c_2))$ for every point $p=(p_1,p_2)$ of $M^2$. Here $M^2(c_1)$ and $M^2(c_2)$ are 2-dimensional space forms, not both flat. As a corollary we give a classification of all the totally geodesic surfaces in $M^2(c_1)\times M^2(c_2)$.