Eigenvalue estimates for the basic Dirac operator on a Riemannian foliation admitting a basic harmonic 1-form
Assume that \(M\) is a compact Riemannian manifold with a transverse spin foliation \({\mathcal F}\) of codimension \(q\geq3\) and a bundle-like metric \(g_M\) such that \(\Delta_B\kappa=0\), where \(\Delta_B\) is the basic Laplacian acting on the basic forms \(\Omega_B^*({\mathcal F})\) and \(\kappa\in\Omega^1({\mathcal F})\) is the mean curvature form for the tangent bundle of \({\mathcal F}\). Denote by \(\sigma^\Delta\) the transversal scalar curvature of \({\mathcal F}\) and assume that the function \(K^\sigma=\sigma^\Delta+|\kappa|^2\) satisfies \(K^\sigma>0\). If \((M,g_M{\mathcal F})\) admits a non-trivial basic harmonic 1-form \(\omega\) of constant length, then every eigenvalue \(\lambda\) of the basic Dirac operator has the property \(\lambda^2\geq\frac{q-1}{4(q-2)}\inf\limits_M K^\sigma\), according to one of the main theorems of the paper under review (Theorem~3.4). If this equality is satisfied for some eigenvalue, then the foliation \({\mathcal F}\) is minimal and the 1-form \(\omega\) is parallel (Theorem~4.1).
scientific article; zbMATH DE number 5134197