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Euler Characteristic of Closed Manifolds with Almost Nonnegative Curvature Operator

Jing-Bin Cai

Abstract

We study closed manifolds with almost nonnegative curvature operator and address a question of Herrmann--Sebastian--Tuschmann concerning the sign of their Euler characteristic. Our main result shows that if a closed $2n$-dimensional manifold admits an almost nonnegative curvature operator together with a uniform upper bound on the curvature operator, then its Euler characteristic is nonnegative. In addition, under an ANCO-type condition and assuming that the fundamental group is infinite, we prove vanishing results for the Euler characteristic, the signature, and, in the spin case, the $\widehat{A}$-genus, extending recent work of Chen--Ge--Han from almost nonnegative Ricci curvature to the curvature-operator setting.

Euler Characteristic of Closed Manifolds with Almost Nonnegative Curvature Operator

Abstract

We study closed manifolds with almost nonnegative curvature operator and address a question of Herrmann--Sebastian--Tuschmann concerning the sign of their Euler characteristic. Our main result shows that if a closed -dimensional manifold admits an almost nonnegative curvature operator together with a uniform upper bound on the curvature operator, then its Euler characteristic is nonnegative. In addition, under an ANCO-type condition and assuming that the fundamental group is infinite, we prove vanishing results for the Euler characteristic, the signature, and, in the spin case, the -genus, extending recent work of Chen--Ge--Han from almost nonnegative Ricci curvature to the curvature-operator setting.
Paper Structure (3 sections, 12 theorems, 56 equations)

This paper contains 3 sections, 12 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Curvature operator and Weitzenböck formulas
  3. Eigenvalues of curvature operators with a uniform upper bound

Key Result

Theorem 1.1

Let $(X,g)$ be a closed smooth $2n$-dimensional Riemannian manifold with $b_1(X) > 0$. There exists a uniformly positive constant $c = c(n) > 0$ such that if the eigenvalues of the curvature operator satisfy then $\chi(X) = 0$.

Theorems & Definitions (18)

  • Theorem 1.1: HT23
  • Corollary 1.2
  • Theorem 1.3
  • Proposition 2.1: PW21, Lemma 2.1
  • Remark 2.2
  • Theorem 2.3: PW21, Theorems A and B
  • Theorem 2.4
  • proof
  • Theorem 3.1: Bha97, Theorem VI.2.1
  • Theorem 3.2: Kas89
  • ...and 8 more