Hijazi, Oussama Spectral properties of the Dirac operator and geometrical structures. (English) Zbl 1017.58021 Ocampo, Hernan (ed.) et al., Geometric methods for quantum field theory. Proceedings of the summer school, Villa de Leyva, Colombia, July 12-30, 1999. Singapore: World Scientific. 116-169 (2001). Summary: These lectures aim to give an elementary exposition on basic results about the first eigenvalue of the Dirac operator, on compact Riemannian spin manifolds with positive scalar curvature. For this, we select some key ingredients which illustrate the basic objects and some of their properties as Clifford algebras, spin groups, connections, covariant derivatives, Dirac and twistor operators. We end by pointing out how the size of the gap around zero in the spectrum of the Dirac operator, increases when the geometrical structure is Kähler or quaternion-Kähler.For the entire collection see [Zbl 0964.00053]. Cited in 1 ReviewCited in 25 Documents MSC: 58J50 Spectral problems; spectral geometry; scattering theory on manifolds 53C27 Spin and Spin\({}^c\) geometry Keywords:conformal spin geometry; compact Riemannian spin manifolds with positive scalar curvature; Clifford algebras; spin groups; connections; Dirac and twistor operators; Kähler or quaternion-Kähler PDFBibTeX XMLCite \textit{O. Hijazi}, in: Geometric methods for quantum field theory. Proceedings of the summer school, Villa de Leyva, Colombia, July 12--30, 1999. Singapore: World Scientific. 116--169 (2001; Zbl 1017.58021)