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Partial Scalar Curvatures and Topological Obstructions for Submanifolds

C. -R. Onti, K. Polymerakis, Th. Vlachos

TL;DR

This work introduces the $k$-th scalar curvature $\rho_k$ as an interpolation between $\rho_1$ and the normalized scalar curvature $\rho$ and leverages the DDVV framework to obtain topological obstructions for compact submanifolds $M^n$ immersed in space forms with $c\ge0$ by lower bounding the $L^{n/2}$-norm of $c+H^2-\rho^\perp-\rho_k$ in terms of Betti numbers. A key contribution is the auxiliary algebraic inequality that yields a uniform bound $\varphi_{V,W}(\beta)\ge\delta(n,m)(\psi_p(\beta))^{d/n}$, enabling global integral estimates that connect intrinsic curvature data to topology. The main results cover $1\le k\le n-1$, providing rigidity when the integral is small and, for Einstein submanifolds, showing the obstruction strengthens to a sphere or Eells–Kuiper manifold under tuned thresholds; the methodology also extends to a conformally invariant form (Theorem 2). The paper also demonstrates the limitation of the $k=n$ case by constructing new compact 3-dimensional Wintgen-ideal submanifolds in even spheres with arbitrarily large first Betti number, highlighting both the power and boundary of the theory.

Abstract

We investigate specific intrinsic curvatures $ρ_k$ (where $1\leq k\leq n$) that interpolate between the minimum Ricci curvature $ρ_1$ and the normalized scalar curvature $ρ_n=ρ$ of $n$-dimensional Riemannian manifolds. For $n$-dimensional submanifolds in space forms, these curvatures satisfy an inequality involving the mean curvature $H$ and the normal scalar curvature $ρ^\perp$, which reduces to the well-known DDVV inequality when $k=n$. We derive topological obstructions for compact $n$-dimensional submanifolds based on universal lower bounds of the $L^{n/2}$-norms of certain functions involving $ρ_k,H$ and $ρ^\perp$. These obstructions are expressed in terms of the Betti numbers. Our main result applies for any $1\leq k \leq n-1$, but it generally fails for $k=n$, where the involved norm vanishes precisely for Wintgen ideal submanifolds. We demonstrate this by providing a method of constructing new compact 3-dimensional minimal Wintgen ideal submanifolds in even-dimensional spheres. Specifically, we prove that such submanifolds exist in $\mathbb{S}^6$ with arbitrarily large first Betti number.

Partial Scalar Curvatures and Topological Obstructions for Submanifolds

TL;DR

This work introduces the -th scalar curvature as an interpolation between and the normalized scalar curvature and leverages the DDVV framework to obtain topological obstructions for compact submanifolds immersed in space forms with by lower bounding the -norm of in terms of Betti numbers. A key contribution is the auxiliary algebraic inequality that yields a uniform bound , enabling global integral estimates that connect intrinsic curvature data to topology. The main results cover , providing rigidity when the integral is small and, for Einstein submanifolds, showing the obstruction strengthens to a sphere or Eells–Kuiper manifold under tuned thresholds; the methodology also extends to a conformally invariant form (Theorem 2). The paper also demonstrates the limitation of the case by constructing new compact 3-dimensional Wintgen-ideal submanifolds in even spheres with arbitrarily large first Betti number, highlighting both the power and boundary of the theory.

Abstract

We investigate specific intrinsic curvatures (where ) that interpolate between the minimum Ricci curvature and the normalized scalar curvature of -dimensional Riemannian manifolds. For -dimensional submanifolds in space forms, these curvatures satisfy an inequality involving the mean curvature and the normal scalar curvature , which reduces to the well-known DDVV inequality when . We derive topological obstructions for compact -dimensional submanifolds based on universal lower bounds of the -norms of certain functions involving and . These obstructions are expressed in terms of the Betti numbers. Our main result applies for any , but it generally fails for , where the involved norm vanishes precisely for Wintgen ideal submanifolds. We demonstrate this by providing a method of constructing new compact 3-dimensional minimal Wintgen ideal submanifolds in even-dimensional spheres. Specifically, we prove that such submanifolds exist in with arbitrarily large first Betti number.
Paper Structure (7 sections, 11 theorems, 82 equations)

This paper contains 7 sections, 11 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. An auxiliary lemma
  3. Algebraic preliminaries
  4. Proofs of the Results
  5. Counterexamples to Theorem \ref{['main1']} for $k=n$
  6. Minimal surfaces in spheres
  7. A class of compact 3-dimensional Wintgen submanifolds

Key Result

Theorem 1

Given integers $n\geq 3$ and $m\geq 1$, there exists for every $k\in\{1,\dots,n-1\}$ a positive constant $\varepsilon_k(n,m)$, depending only on $n$ and $m$, such that if $M^n$ is a compact Riemannian manifold that admits an isometric immersion into $\mathbb{Q}_c^{n+m}$ with $c\geq 0$, then for any coefficient field $\mathbb{F}$. Moreover, if for some $k\in\{1,\dots,n-1\}$, then $M^n$ is either

Theorems & Definitions (19)

  • Theorem 1
  • Corollary 2
  • Corollary 3
  • Theorem 4
  • Lemma 5
  • proof
  • Lemma 7
  • proof
  • Proposition 8
  • proof
  • ...and 9 more