The zero set of a twistor spinor in any metric signature. (English) Zbl 1327.53017
In Chapter 2, “Basic facts about twistor spinors and conformal Cartan geometry”, the reader finds the statement that “[…] there is a natural double covering from \(\mathrm{Spin}(p,q)\) onto \(\mathrm{SO}(p,q)\)”. According to common denotations \(\mathrm{SO}(p,q)\) is the special pseudo-orthogonal group of a quadratic form with signature \((p,q)\) on a real vector space, and \(\mathrm{Spin}(p,q)\) is the reduced Clifford group which is defined by means of the real Clifford algebra belonging to this quadratic form. If the quadratic form is definite, then the statement above is correct. If it is indefinite, however, this statement is generally wrong: in case of signature \((2,1)\) for instance there can be easily found an element of \(\mathrm{SO}(2,1)\) which is not an image of an element of \(\mathrm{Spin}(2,1)\).
The paper will need a revision.
The paper will need a revision.
Reviewer: Peter Kraut (Erlabrunn)
MSC:
| 53B30 | Local differential geometry of Lorentz metrics, indefinite metrics |
| 53C27 | Spin and Spin\({}^c\) geometry |
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