Sasakian twistor spinors and the first Dirac eigenvalue
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On a closed eta-Einstein Sasakian spin manifold of dimension $2m+1 \geq 5, \, m \, \equiv 0 \, \mbox{mod} \, 2,$ we prove a new eigenvalue estimate for the Dirac operator. In dimension 5, the estimate is valid without the eta-Einstein condition. Moreover, we show that the limiting case of the estimate is attained if and only if there exists such a pair $(\varphi_{\frac{m}{2} -1} , \, \varphi_{\frac{m}{2}} )$ of spinor fields (called {\it Sasakian duo}, see Definition 2.1) that solves a special system of two differential equations.
Abstract
On a closed eta-Einstein Sasakian spin manifold of dimension $2m+1 \geq 5, \, m \, \equiv 0 \, \mbox{mod} \, 2,$ we prove a new eigenvalue estimate for the Dirac operator. In dimension 5, the estimate is valid without the eta-Einstein condition. Moreover, we show that the limiting case of the estimate is attained if and only if there exists such a pair $(\varphi_{\frac{m}{2} -1} , \, \varphi_{\frac{m}{2}} )$ of spinor fields (called {\it Sasakian duo}, see Definition 2.1) that solves a special system of two differential equations.
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