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Some basic properties of Killing spinors
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17.A basic property of Killing vectors is that their second or higher derivatives can be expressed in terms of the Killing vector and its first derivatives. Since a generalization of this property applies for KSs, the name of “Killing spinors” seems to be well justified, and we will use it without quotation marks.
18.J. Plebański, Spinors, Tetrads and Forms (Centro de Investigaciones y Estudios Avanzados del IPN, México, 1975). Our convention for normalizing Pauli matrices is ; thus, for instance, and Also, in convention: This follws the notation of Ref. 14.
19.A symbol like (AB | C | D), for instance, means that symmetrization is over all indices except C (this with obvious gen‐generalizations).
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22.Mentioned by Sommers.10
23.Petrov gives a proof of this theorem in Chapter 4 of his book21 for the particular case of a vacuum type III metric.
24.If the space is complex, one has type and in the notation of Ref. 14.
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