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Generalized Wintgen inequalities for submanifolds of conformally flat manifolds

Cihan Özgür, Adara M. Blaga

TL;DR

This work extends the classical Wintgen inequality to submanifolds embedded in conformally flat manifolds by establishing a generalized relation among intrinsic invariants $\rho$ and $\rho_N$ (or $\rho^\perp$), extrinsic mean curvature $\|H\|$, and ambient curvature data. The main result provides a sharp inequality $\rho + \rho_N \leq \|H\|^2 + \frac{2}{n(m-2)}\sum_{j=1}^n \widetilde{Ric}(e_j,e_j) - \frac{2\widetilde{\tau}}{(m-1)(m-2)}$, with explicit equality conditions on the second fundamental form; a parallel bound for $\rho + \rho^{\perp}$ is also obtained. In minimal submanifolds the inequality simplifies, and the paper demonstrates applications to quasi-constant curvature manifolds, yielding concrete bounds dependent on curvature parameters $p$ and $q$ and the tangential component of a distinguished vector field. The results specialize to warped-product and generalized Robertson–Walker spacetimes, offering practical geometric insights into how intrinsic and extrinsic curvatures constrain each other in these settings.

Abstract

We obtain generalized Wintgen inequalities for submanifolds in conformally flat manifolds. We give some applications for submanifolds in a Riemannian manifold of quasi-constant curvature. Equality cases are also considered.

Generalized Wintgen inequalities for submanifolds of conformally flat manifolds

TL;DR

This work extends the classical Wintgen inequality to submanifolds embedded in conformally flat manifolds by establishing a generalized relation among intrinsic invariants and (or ), extrinsic mean curvature , and ambient curvature data. The main result provides a sharp inequality , with explicit equality conditions on the second fundamental form; a parallel bound for is also obtained. In minimal submanifolds the inequality simplifies, and the paper demonstrates applications to quasi-constant curvature manifolds, yielding concrete bounds dependent on curvature parameters and and the tangential component of a distinguished vector field. The results specialize to warped-product and generalized Robertson–Walker spacetimes, offering practical geometric insights into how intrinsic and extrinsic curvatures constrain each other in these settings.

Abstract

We obtain generalized Wintgen inequalities for submanifolds in conformally flat manifolds. We give some applications for submanifolds in a Riemannian manifold of quasi-constant curvature. Equality cases are also considered.
Paper Structure (5 sections, 9 theorems, 44 equations)

This paper contains 5 sections, 9 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Generalized Wintgen Inequality
  4. Some Applications
  5. Conclusion

Key Result

Proposition 3.1

Let $M^{n},n\geq 3,$ be an $n$-dimensional submanifold of an $m$-dimensional, $m\geq 4$, conformally flat manifold $(\widetilde{M}^{m},\widetilde{g})$. Then The equality case holds identically if and only if, with respect to suitable orthonormal frames $\{e_{j}\}_{1\leq j\leq n}$ and $\{\upsilon _{r}\}_{1\leq r\leq m-n}$, the shape operators $\{A_{r}\}_{1\leq r\leq m-n}$ of $M^{n}$ in $(\widetild

Theorems & Definitions (15)

  • Conjecture 1
  • Proposition 3.1
  • proof
  • Corollary 3.2
  • Theorem 3.3
  • proof
  • Corollary 3.4
  • Corollary 4.1
  • proof
  • Corollary 4.2
  • ...and 5 more