CR submanifolds of maximal CR dimension of a complex space form with recurrent shape operator

TL;DR

This paper investigates CR submanifolds with maximal CR dimension in complex space forms, demonstrating that if the shape operator of a specific vector field is recurrent, then the submanifold must be Euclidean space.

Contribution

It establishes a new geometric condition linking recurrent shape operators to the Euclidean nature of CR submanifolds in complex space forms.

Findings

Recurrent shape operator implies the submanifold is Euclidean.

The shape operator's recurrence condition is crucial for the classification.

The result extends understanding of CR submanifold geometry in complex space forms.

Abstract

Let M be a CR submanifold of maximal CR dimension of a complex space form M. The shape operator A of the distinguished vector field {\xi} is recurrent if there exists a 1-form v such that \nabla A = A \otimes v. We show that M is an Euclidean space under the condition that A is recurrent.