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GR: Gravitation und Relativitätstheorie

GR 5: Grundlegende Probleme und alternative Ans
ätze

GR 5.4: Fachvortrag

Tuesday, March 25, 2003, 17:30–17:50, A310

The Cotton 2-form in arbitrary dimensions — •Christian Heinicke — Institut für Theoretische Physik, Universität zu Köln, 50923 Köln

In spaces of dimension greater than 3, the vanishing of the Weyl 2-form is the necessary and sufficient condition for conformal flatness. In 3 dimensions, the Weyl 2-form vanishes identically and the Cotton 2-form [1,2] plays the role of the Weyl 2-form.

We will discuss the meaning of the Cotton 2-form in arbitrary dimensions. We derive in a systematic way how its specific role in three dimension comes about. The argument is based on the Bianchi identities. We use the formalism of exterior calculus in which the arguments become very transparent [3]. The number of independent components of the Cotton 2-form are determined, and a classification is presented which is based on its number of independent eigenvectors.

[1] E. Cotton: Ann. Fac. d. Sc. Toulouse (II) 1 (1899) 385

[2] J. A. Schouten: Ricci Calculus (Springer, Berlin, 1954)

[3] C. Heinicke: Gen. Rel. Grav. 33 (2001) 1115

[4] A. Garcia, F. W. Hehl, C. Heinicke, A. Macias: The Cotton tensor in Riemannian spacetimes, to be published.