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Conformal holonomy of bi-invariant metrics. (English) Zbl 1138.53041
In conformal geometry (in dimension \(\geq 3\)) there exists an invariant notion of holonomy which can naturally be defined by means of the canonical Cartan connection of conformal geometry. The conformal holonomy algebra and group are useful invariants for a space with conformal structure, because (for example) they describe invariant substructures and indicate solutions of certain overdetermined conformally covariant partial differential equations.
The aim of this paper is to develop an invariant Cartan calculus and apply it to the conformal geometry of bi-invariant metrics on compact semisimple Lie groups, in order to compute explicitely the conformal holonomy algebras and groups. In two sections, the author recalls briefly the basic concepts of conformal geometry and, respectively, the basic notions of bi-invariant metrics. In the next section, the author develops a conformal Cartan calculus adapted to bi-invariant metrics. The canonical connection and its curvature are described by certain linear maps \(\gamma_{\text{nor}}\) and \(\kappa\). After discussing the properties of these maps the author derives a formula for the conformal holonomy algebra. In the last section, he makes explicit calculations for the bi-invariant metrics on SO(3) and SO(4). It is proved that, up to constant scales, the bi-invariant metric is the only Einstein metric in its conformal class on SO(4). Moreover, SO(4) is not (locally) conformally equivalent to a product of Einstein metrics in such a manner that the Schouten tensor of the product equals the product of the Schouten tensors of the factors.

MSC:
53C29 Issues of holonomy in differential geometry
53B15 Other connections
53C20 Global Riemannian geometry, including pinching
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