ON THE SECOND FUNDAMENTAL FORMS OF THE INTERSECTION OF SUBMANIFOLDS

Abstract

Let $G$ be a Lie group and $H$ its subgroup, and let $M^p$, $N^q$ be two submanifolds of dimensions $p$, $q$, respectively, in the Riemannian homogeneous space $G/H$. We study the relationships between the second fundamental forms of $M^p \cap gN^q$ and the second fundamental forms of $M^p$, $N^q$ for $g \in G$. We find that the second fundamental form of $M^p \cap gN^q$ can be expressed by the curvature functions of $M^p$, $N^q$ and the “angle” between $M^p$ and $N^q$. All results achieved are the generalizations of known results of the classical differential geometry in $\mathrm{R}^3$.