New estimates for eigenvalues of the basic Dirac operator (Q961442)

For Riemannian foliations with transverse spin structure, there exists the notion of a basic Dirac operator \(D_b\) acting on transverse spinor fields. There are lower bounds known for the square of the first eigenvalue of \(D_b\) in terms of the transversal scalar curvature, the mean curvature and the first eigenvalue of the basic Yamabe operator [cf. \textit{S. D. Jung}, J. Geom. Phys. 39, No.~3, 253--264 (2001; Zbl 1024.53019) and \textit{S. D. Jung, B. H. Kim} and \textit{J. S. Pak}, J. Geom. Phys. 51, No.~2, 166--182 (2004; Zbl 1076.58022)]. These bounds generalise known results due to Th. Friedrich and O. Hijazi for the first eigenvalue of the Dirac operator on a Riemannian spin manifold. In the current work, the authors improve (the mean curvature part of) these lower bounds by using a modified transverse spinor connection. In the limiting case, the underlying foliation is transversally Einstein with minimal leaves.