On totally real submanifolds

TL;DR

This paper proves several theorems classifying totally real submanifolds in K"ahler and complex space forms based on curvature, parallel structures, and geometric properties, revealing conditions for flatness, geodesicity, or product structures.

Contribution

It provides new classification results for totally real submanifolds with parallel structures and curvature conditions in complex manifolds.

Findings

Totally real semiparallel submanifolds are either flat or totally geodesic.

Minimal semiparallel submanifolds with constant second fundamental form are either totally geodesic or have positive scalar curvature.

Complete, compact totally real submanifolds with parallel mean curvature are either totally geodesic or products of circles.

Abstract

The following results are proved: Theorem 1. A totally real semiparallel submanifold of constant curvature with parallel f-structure in the normal bundle of a K\"ahler manifold N is flat or a totally geodesic submanifold of N. Theorem 2. A totally real minimal semiparallel submanifold M with parallel f-structure in the normal bundle and of constant length of the second fundamental form (or equivalently of constant scalar curvature) of a complex space form N is totally geodesic in N or of positive scalar curvature. Moreover, if the scalar curvature of M vanishes, then M is flat. Theorem 3. A complete, compact totally real submanifold with parallel mean curvature vector, parallel f-structure in the normal bundle and commutative second fundamental forms of a simply connected complete complex space form is totally geodesic or a pythagorean product of circles. Note that if M is a totally real submanifold of a K\"ahler manifold N and the dimension of N is twice the dimension of M, then the f-structure in the normal bundle vanishes.