Sasakian twistor spinors and the first Dirac eigenvalue (Q2835292)

A Sasakian spin manifold is an odd-dimensional Riemannian spin manifold \((M^{2m+1},g), m\geq 1\), such that \((M^{2m+1},\varphi,\xi,\eta,g)\) is a Sasakian structure. Denoting by \(\nabla \psi\) the Levi-Civita connection acting on sections \(\psi\) of the spinor bundle oven \(M^{2m+1}\), the Riemann Dirac operator \(D\) is given by \(D \psi = \sum_{u=1}^{2m +1} E_u \cdot \nabla _{E_u} \psi\), where \(\{E_i \}\) is a local orthonormal frame. The generalized Dirac operator \(D_{ab}\), considered by the author in [Ann. Global Anal. Geom. 45, No. 1, 67--93 (2014; Zbl 1291.53058)] corresponds to \(D\) for \(a=1, b=0\) and to the cubic Dirac operator when \(a=1, b=\frac{1}{2}\). Estimates on the first eigenvalue of \(D_{ab}\) were obtained in the above cited reference. In this paper, such estimates are improved, leading to some inequalities of which the equality cases are identified by a special class of spinors, called Sasakian duos, which appear to be a natural generalization of Sasakian Killing spinors.