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Kobayashi conformal metric on manifolds, Chern-Simons and $$\eta$$- invariants. (English) Zbl 0761.53010
A canonical Riemannian smooth metric is constructed on a given uniformized conformal manifold (conformally flat manifold), a metric compatible with the conformal structure. The Kobayashi approach for the construction of a biholomorphically invariant intrinsic pseudo-metric in a complex analytic space is carried over to the conformal (Moebius) category. Infinitesimal and regularity properties for this Kobayashi conformal metric are discussed, and the Chern-Simons and $$\eta$$- invariants for conformal structures on a closed hyperbolic 3-manifold.

##### MSC:
 53A30 Conformal differential geometry (MSC2010) 58D29 Moduli problems for topological structures
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