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Betti numbers of nearly $G_2$ and nearly Kähler manifolds with Weyl curvature bounds

Anton Iliashenko

Abstract

In this paper we use the Weitzenböck formulas to get information about the Betti numbers of compact nearly $G_2$ and compact nearly Kähler $6$-manifolds. First, we establish estimates on two curvature-type self adjoint operators on particular spaces assuming bounds on the sectional curvature. Then using the Weitzenböck formulas on harmonic forms, we get results of the form: if certain lower bounds hold for these curvature operators then certain Betti numbers are zero. Finally, we combine both steps above to get sufficient conditions of vanishing of certain Betti numbers based on the bounds on the sectional curvature.

Betti numbers of nearly $G_2$ and nearly Kähler manifolds with Weyl curvature bounds

Abstract

In this paper we use the Weitzenböck formulas to get information about the Betti numbers of compact nearly and compact nearly Kähler -manifolds. First, we establish estimates on two curvature-type self adjoint operators on particular spaces assuming bounds on the sectional curvature. Then using the Weitzenböck formulas on harmonic forms, we get results of the form: if certain lower bounds hold for these curvature operators then certain Betti numbers are zero. Finally, we combine both steps above to get sufficient conditions of vanishing of certain Betti numbers based on the bounds on the sectional curvature.
Paper Structure (30 sections, 44 theorems, 322 equations)

This paper contains 30 sections, 44 theorems, 322 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Organization of the paper and main results
  4. Acknowledgements
  5. Notation
  6. Curvature estimates
  7. Estimates for $\hat{R}$
  8. Estimates for $\mathring{R}$
  9. General Weitzenböck formulas
  10. Nearly $G_2$ manifolds
  11. Preliminaries
  12. Curvature identities
  13. Weitzenböck formulas
  14. $2$-forms
  15. $3$-forms
  16. ...and 15 more sections

Key Result

Lemma 2.5

The following identities hold: where by $g \bar{\mathbin{\bigcirc\mspace{-16mu}\wedge\mspace{3mu}}} g$ we mean that we apply the $\bar{\space}$ operator to $g \mathbin{\bigcirc\mspace{-16mu}\wedge\mspace{3mu}} g \in \mathcal{R}.$ Similarly, for the $g \hat{\mathbin{\bigcirc\mspace{-16mu}\wedge\mspace{3mu}}} g$ and $g \mathring{\m

Theorems & Definitions (105)

  • Remark 1.1
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Lemma 2.5
  • proof
  • Definition 2.6
  • Lemma 2.7
  • proof
  • ...and 95 more