Complex Lipschitz structures and bundles of complex Clifford modules

Let (M,g) be a pseudo-Riemannian manifold of signature (p,q). We construct mutually quasi-inverse equivalences between the groupoid of bundles of weakly-faithful complex Clifford modules on (M,g) and the groupoid of reduced complex Lipschitz structures on (M,g). As an application, we show that (M,g)...

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Bibliographic Details
Published in:Differential geometry and its applications 2018-12, Vol.61, p.147-169
Main Authors: Lazaroiu, C.I., Shahbazi, C.S.
Format: Article
Language:English
Subjects:
Clifford bundles
Lipschitz structures
Spin geometry
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Summary:Let (M,g) be a pseudo-Riemannian manifold of signature (p,q). We construct mutually quasi-inverse equivalences between the groupoid of bundles of weakly-faithful complex Clifford modules on (M,g) and the groupoid of reduced complex Lipschitz structures on (M,g). As an application, we show that (M,g) admits a bundle of irreducible complex Clifford modules if and only if it admits either a Spinc(p,q) structure (when p+q is odd) or a Pinc(p,q) structure (when p+q is even). When p−q≡83,4,6,7, we compare with the classification of bundles of irreducible real Clifford modules which we obtained in previous work. The results obtained in this note form a counterpart of the classification of bundles of faithful complex Clifford modules which was previously given by T. Friedrich and A. Trautman.
ISSN:0926-2245
1872-6984
DOI:10.1016/j.difgeo.2018.08.006