low-dimensional \(\mathrm{Spin}^c\) manifolds
The energy-momentum tensor on low dimensional \(\mathrm{Spin}^c\) manifolds
The authors investigate the energy-momentum tensor on low-dimensional compact manifolds having an arbitrary \(\mathrm{Spin}^c\) structure. Their principal result is a new proof of a Bär-type inequality for the eigenvalues of the Dirac operator, and it is shown that the round sphere \(\mathbb{S}^2\) with its canonical \(\mathrm{Spin}^2\) structure satisfies this limiting case. They conclude by giving a spinorial chacterization of immersed surfaces in \(\mathbb{S}^2\times\mathbb{R}\), using solutions of the generalized Killing spinor equation associated with the induced \(\mathrm{Spin}^c\) structure on \(\mathbb{S}^2\times\mathbb{R}\).NEWLINENEWLINE Contents includes: An Introduction (containing a survey of the previous literature and a summary of their new results); Preliminaries (concerning \(\mathrm{Spin}^c\) structures and the Dirac operator); The two-dimensional case; The three-dimensional case; Characterization of surfaces in \(\mathbb S^2\times\mathbb R\).
scientific article; zbMATH DE number 6078195
scientific article