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

Anton Iliashenko

Published 2023-03-31Version 1

In this paper we use the Weitzenb\"ock formulas to get information about the Betti numbers of nearly $G_2$ and nearly K\"ahler manifolds. First, we establish estimates on two curvature-type self adjoint operators on particular spaces assuming bounds on the sectional curvature. Then using the Weitzenb\"ock 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.