Constant Angle Surfaces in Product Spaces

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 $M^2(c_1)\times M^2(c_2)$.