Two optimal inequalities for anti-holomorphic submanifolds and their applications

TL;DR

This paper establishes two new optimal inequalities involving the CR δ-invariant for anti-holomorphic submanifolds in complex space forms and provides classification results for cases of equality.

Contribution

It introduces two novel optimal inequalities for anti-holomorphic submanifolds involving the CR δ-invariant and classifies submanifolds satisfying equality cases.

Findings

Proved two new optimal inequalities involving CR δ-invariant.

Derived classification results for anti-holomorphic submanifolds satisfying equality.

Extended understanding of geometric properties of anti-holomorphic submanifolds.

Abstract

The CR $\delta$-invariant for CR-submanifolds was introduced in a recent article [B. Y. Chen, An optimal inequality for CR-warped products in complex space forms involving CR $\delta$-invariant, Internat. J. Math. 23} (2012), no. 3, 1250045 (17 pages)]. In this paper, we prove two new optimal inequalities for anti-holomorphic submanifolds in complex space forms involving the CR $\delta$-invariant. Moreover, we obtain some classification results for certain anti-holomorphic submanifolds in complex space forms which satisfy the equality case of either inequality.