Geometric classification of totally umbilical quasi-hemislant submanifolds

Abstract

UDC 514.76, 514.77

We study the geometric characteristics and classification of totally umbilical quasi-hemislant submanifolds in almost product Riemannian manifolds. We focus on three primary distributions: the slant distribution $D^\theta,$ the totally real distribution $D^\bot,$ and the invariant distribution $D,$ providing a detailed analysis of their integrability conditions and conditions of being totally geodesic. Through these investigations, we found that these submanifolds can be categorized as totally geodesic, CR submanifolds, or exhibits other interesting geometric properties, such as being extrinsic spheres or proper semislant submanifolds depending on specific geometric conditions. We establish necessary and sufficient conditions for each classification and describe possible structural configurations of these submanifolds. Our results contribute to a broader comprehension of quasi-hemislant submanifolds in differential geometry by providing new classifications and geometric perspective in almost product Riemannian structures.