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On some optimal inequalities for bi-slant submanifolds in metallic Riemannian space forms

Harmandeep Kaur, Gauree Shanker

TL;DR

The work develops a comprehensive set of optimal curvature inequalities for bi-slant submanifolds of metallic Riemannian space forms, connecting intrinsic invariants ρ, ρ^⊥ and δ-invariants to extrinsic data such as the mean curvature and the shape operator. It introduces and leverages generalized Wintgen inequalities, δ-invariant bounds, Ricci–shape operator relations, and generalized normalized δ-Casorati curvature inequalities, with precise equality cases that characterize submanifold geometry. The results extend to special submanifold types (slant, semi-slant, hemi-slant, semi-invariant) within metallic product spaces, providing a unified framework for understanding the balance between intrinsic curvature and ambient geometry. These inequalities offer tools for identifying ideal immersions and understanding the geometric tension of submanifolds in metallic Riemannian spaces, with potential extensions to broader warped-product settings and soliton-related curvature studies.

Abstract

In this paper, we derive some important optimal relationships for bi-slant submanifolds in metallic Riemannian product space forms enriching the understanding of their geometric properties and deepening the connection between intrinsic and extrinsic curvature invariants. We establish generalized Wintgen inequality for bi-slant submanifolds in metallic Riemannian product space forms and discussed the equality case. Next we derive optimal inequalities involving $δ$-invariants, also known as Chen-invariants and discuss the conditions for Chen ideal submanifolds. Further, we derive optimal relationships involving Ricci curvature and shape operator invariants along with the discussion about the equality cases. In the last section, we establish optimal inequalities involving generalized normalized $δ$-Casorati curvatures for bi-slant submanifolds of metallic Riemannian product space form and discuss the conditions under which the equality holds. Furthermore, we examine how the main findings specialize to slant, semi-slant, hemi-slant, and semi-invariant submanifolds in metallic Riemannian product space forms, offering a better understanding of their geometric characteristics.

On some optimal inequalities for bi-slant submanifolds in metallic Riemannian space forms

TL;DR

The work develops a comprehensive set of optimal curvature inequalities for bi-slant submanifolds of metallic Riemannian space forms, connecting intrinsic invariants ρ, ρ^⊥ and δ-invariants to extrinsic data such as the mean curvature and the shape operator. It introduces and leverages generalized Wintgen inequalities, δ-invariant bounds, Ricci–shape operator relations, and generalized normalized δ-Casorati curvature inequalities, with precise equality cases that characterize submanifold geometry. The results extend to special submanifold types (slant, semi-slant, hemi-slant, semi-invariant) within metallic product spaces, providing a unified framework for understanding the balance between intrinsic curvature and ambient geometry. These inequalities offer tools for identifying ideal immersions and understanding the geometric tension of submanifolds in metallic Riemannian spaces, with potential extensions to broader warped-product settings and soliton-related curvature studies.

Abstract

In this paper, we derive some important optimal relationships for bi-slant submanifolds in metallic Riemannian product space forms enriching the understanding of their geometric properties and deepening the connection between intrinsic and extrinsic curvature invariants. We establish generalized Wintgen inequality for bi-slant submanifolds in metallic Riemannian product space forms and discussed the equality case. Next we derive optimal inequalities involving -invariants, also known as Chen-invariants and discuss the conditions for Chen ideal submanifolds. Further, we derive optimal relationships involving Ricci curvature and shape operator invariants along with the discussion about the equality cases. In the last section, we establish optimal inequalities involving generalized normalized -Casorati curvatures for bi-slant submanifolds of metallic Riemannian product space form and discuss the conditions under which the equality holds. Furthermore, we examine how the main findings specialize to slant, semi-slant, hemi-slant, and semi-invariant submanifolds in metallic Riemannian product space forms, offering a better understanding of their geometric characteristics.
Paper Structure (7 sections, 13 theorems, 74 equations, 6 tables)

This paper contains 7 sections, 13 theorems, 74 equations, 6 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main Results
  4. Generalized Wintgen inequality
  5. Optimal inequalities involving $\delta$-invariants
  6. Optimal inequalities involving Ricci curvature and shape operator
  7. Optimal inequalities involving generalized normalized $\delta$-Casorati curvatures

Key Result

Proposition 2.1

gw1 Every almost product structure $\mathcal{F}$ induces two metallic structures on a Riemannian manifold $M$ which are given as follows: Conversely, every metallic structure $\phi$ on $M$ induces two almost product structures on the manifold, given by where $\sigma_{p, q} = \dfrac{p + \sqrt{p^2 + 4q}}{2}$ represents the members of metallic means family. In particular, if the almost product stru

Theorems & Definitions (30)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.1
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • ...and 20 more