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Topological and geometric rigidity of nonnegatively curved submanifolds

Theodoros Vlachos

TL;DR

This work derives a sharp sectional-curvature pinching bound K_{ ext{min}} ≥ b(n,H,c) (with b(n,H,c)= rac{1}{2}igl(c+ rac{n}{4}H^2igr) for n≥5 and rac{1}{3}(c+H^2) for n=4) for compact submanifolds in space forms, tying intrinsic curvature to the mean curvature length H. For n≥5, Lawson–Simons type vanishing theorems yield that M^n is topologically a sphere, with diffeomorphism results in some dimensions; for n=4, the bound forces a sharp dichotomy: either M^4 ≅ S^4 or, when equality occurs, M^4 is isometric to CP_r^2 and immersed via an umbilical inclusion, with the immersion structure explicitly described. The 4D case is analyzed using isotropic curvature and Bochner techniques, leveraging the decomposition of 2-forms and harmonic forms to classify equality scenarios and to deduce the existence of parallel complex structures in the nonpositive ICC case. The paper also provides constructive examples—via convex hypersurfaces and ellipsoids—that satisfy the pinching condition, illustrating sharpness and giving a broad family of sphere-type submanifolds.

Abstract

We investigate the topology and geometry of compact submanifolds in space forms of nonnegative curvature that satisfy a lower bound on the sectional curvature, depending only on the length of the mean curvature vector of the immersion. We show that this condition imposes strong constraints on either the topology or geometry of the submanifold. Additionally, we provide examples that demonstrate the sharpness of our result.

Topological and geometric rigidity of nonnegatively curved submanifolds

TL;DR

This work derives a sharp sectional-curvature pinching bound K_{ ext{min}} ≥ b(n,H,c) (with b(n,H,c)= rac{1}{2}igl(c+ rac{n}{4}H^2igr) for n≥5 and rac{1}{3}(c+H^2) for n=4) for compact submanifolds in space forms, tying intrinsic curvature to the mean curvature length H. For n≥5, Lawson–Simons type vanishing theorems yield that M^n is topologically a sphere, with diffeomorphism results in some dimensions; for n=4, the bound forces a sharp dichotomy: either M^4 ≅ S^4 or, when equality occurs, M^4 is isometric to CP_r^2 and immersed via an umbilical inclusion, with the immersion structure explicitly described. The 4D case is analyzed using isotropic curvature and Bochner techniques, leveraging the decomposition of 2-forms and harmonic forms to classify equality scenarios and to deduce the existence of parallel complex structures in the nonpositive ICC case. The paper also provides constructive examples—via convex hypersurfaces and ellipsoids—that satisfy the pinching condition, illustrating sharpness and giving a broad family of sphere-type submanifolds.

Abstract

We investigate the topology and geometry of compact submanifolds in space forms of nonnegative curvature that satisfy a lower bound on the sectional curvature, depending only on the length of the mean curvature vector of the immersion. We show that this condition imposes strong constraints on either the topology or geometry of the submanifold. Additionally, we provide examples that demonstrate the sharpness of our result.

Paper Structure

This paper contains 7 sections, 8 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. The case of dimensions $n\geq5$
  3. Background
  4. Geometry of 4-dimensional manifolds
  5. Isotropic curvature
  6. Proof of Theorem \ref{['thm1']} for the case $n=4$
  7. Examples of submanifolds satisfying condition \ref{['*']}

Key Result

Theorem 1

Let $f\colon M^n\to\mathbb{Q}_c^{n+m}$, $n\geq 4, c\geq 0$, be an isometric immersion of a compact, oriented Riemannian manifold. Assume that at any point of $M^n$, the sectional curvature satisfies Then the following assertions hold: $(i)$ If $n\geq 5$, then $M^n$ is homeomorphic to the sphere $\mathbb{S}^n$. $(ii)$ If $n=4$, then either $M^4$ is diffeomorphic to $\mathbb{S}^4$, or equality hold

Theorems & Definitions (8)

  • Theorem 1
  • Theorem 2
  • Proposition 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Proposition 7
  • Proposition 8