Hineva Inequality on Some Submanifolds of Quaternionic Space forms

TL;DR

This work establishes sharp Hineva-type Ricci curvature inequalities for submanifolds of quaternionic space forms, including quaternionic projective spaces QP^m(4c). By exploiting the ambient curvature tensor in quaternionic geometry, the Gauss equation, and decompositions of the almost quaternionic structures, the authors derive both upper and lower bounds on Ric(X) in terms of the mean curvature H, the second fundamental form h, and projections P_i, with precise equality characterizations. The main contributions cover three submanifold classes: general submanifolds in QP^m(4c), quaternionic CR-submanifolds, and θ-slant submanifolds, providing necessary and sufficient conditions for equality through quasi-umbilical matrix patterns of h_{ij}^α and its eigenstructure. These results extend Chen-Hineva-type inequalities to quaternionic space forms, offering refined intrinsic-extrinsic curvature relations and potential rigidity insights for minimal or geodesic immersions in quaternionic geometry.

Abstract

In this article, we establish Hineva inequality for different types of submanifolds of Quaternionic Space forms

Theorems & Definitions (15)

• Theorem 1.1: A
• Theorem 1.2: B
• Theorem 1.3: C
• Theorem 1.4
• Theorem 1.5
• Theorem 2.1
• Theorem 3.1
• Theorem 3.2
• Corollary 3.1
• Theorem 4.1