Hineva Inequality on Some Submanifolds of Quaternionic Space forms

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

💡 Research Summary

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The paper “Hineva Inequality on Some Submanifolds of Quaternionic Space Forms” investigates curvature inequalities originally introduced by Hineva and extends them to a variety of submanifolds embedded in quaternionic space forms (QP^{m}(4c)). The authors begin by recalling the classical Nash immersion problem and Chen’s (\delta)-invariants, emphasizing the importance of relating intrinsic invariants (sectional, Ricci, scalar curvature) to extrinsic invariants (mean curvature vector (H) and the second fundamental form (\sigma)). They then review Hineva’s original lower and upper bounds for sectional and Ricci curvatures of submanifolds in a general Riemannian ambient space, and they reinterpret these bounds in the context of quaternionic Kaehler geometry, where three almost complex structures (I,J,K) satisfy the quaternionic relations.

In the preliminaries, the authors define quaternionic Kaehler manifolds, quaternionic space forms (constant quaternionic sectional curvature), and the associated curvature tensor (formula (2.1)). They introduce the decomposition of the ambient quaternionic structures into tangential components (P_i) and normal components (F_i), which later appear in the curvature estimates.

The core contributions are organized into three sections:

1. General Submanifolds – Theorem 3.2 provides a new lower bound for the Ricci curvature of an arbitrary (n)-dimensional submanifold (M\subset QP^{m}(4c)): \