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Spinor fields on Riemannian manifolds. (English) Zbl 0847.53006
Bureš, J. (ed.) et al., Proceedings of the winter school on geometry and physics, Zdíkov, Czech Republic, January 1993. Palermo: Circolo Matematico di Palermo, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 37, 201-205 (1994).
Let \((M,g)\) be a connected Riemannian manifold of dimension \(n\), let \(S\) be a spinor bundle on \(M\) and \(\Gamma (S)\) the space of all smooth sections of \(S\). Let a spinor field \(\psi \in \Gamma (S)\) satisfy \[ \nabla^S_X \psi + {1 \over n} X \cdot D \psi = 0, \] where \(D\) denotes the Dirac operator, for all vector fields \(X\) on \(M\). Such a field \(\psi\) is called \(E\)-spinor if \[ \nabla^S_X (D \psi) + {R \over 4 (n - 1)} X \cdot \psi = 0, \] where \(R\) is the scalar curvature on \(M\).
\(E\)-spinors are constructed on the noncompact Riemannian manifolds \(S^2 \times \mathbb{R}^1\) and \(H^2 \times \mathbb{R}^1\).
For the entire collection see [Zbl 0823.00015].
53C27 Spin and Spin\({}^c\) geometry
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)