Betti numbers of nearly \(G_2\) and nearly Kähler 6-manifolds with Weyl curvature bounds

In this paper we use the Weitzenböck formulas to get information about the Betti numbers of compact nearly \(G_2\) $G_2$ 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.