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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].
MSC:
 53C27 Spin and Spin$${}^c$$ geometry 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)