Abstract
Starting from the infinitesimal holonomy groupH i of aV 4, (+++−) the spinholonomy group\(\tilde H_i \equiv \bar \sigma ^1 (H_i )\) defined by the covering isomorphism\(\sigma :G \to L_ + ^ \uparrow \) is introduced. In Einstein-spaces we may replace its real Lie-algebra by a complex one. With the complex calculus we may reproduce the results ofSchell, Goldberg andKerr with very much simplified proofs. A theorem on non-empty Einstein-spaces is given.
In part 4 we prove a theorem on the connection between theH i -behaviour of a vector (spinor) and its covariant derivative in aV 4. With its help we get in a simple manner the metiics of aV 4 with givenH i and Dim (H i ) <6; our results agree with those given byGoldberg andKerr, Cahen andDebever. Finally we make some new statements on imperfect holonomy groups.
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Der Verfasser dankt Herrn Prof. P.Jordan für die Unterstützung dieser Arbeit und den Mitgliedern des Hamburger Seminars für Allgemeine Relativitätstheorie für wertvolle Diskussionen. Besonderer Dank gebührt Dr. J.Ehlers für eine detaillierte Kritik dieser Veröffentlichung. Ich danke auch dem Bundesministerium für Wissenschaftliche Forschung für Unterstützung.
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Beiglböck, W. Zur Theorie der infinitesimalen Holonomiegruppe in der Allgemeinen Relativitätstheorie. Z. Physik 179, 148–160 (1964). https://doi.org/10.1007/BF01381251
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DOI: https://doi.org/10.1007/BF01381251