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Linear connections and geodesics on Fréchet manifolds. (English. Russian original) Zbl 0864.58003
Russ. Math. 39, No. 7, 74-86 (1995); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1995, No. 7, 78-90 (1995).
This article generalizes the method of projection of linear connections on principal bundles, developed by K. M. Yegiazaryan, to the case of Fréchet manifolds: Let \(\gamma\) be a connection in a Fréchet principal bundle \((\mathcal{E},\pi,\mathcal{B},\mathcal{G})\) and \(\nabla\) be a \(\mathcal{G}\)-invariant linear connection on \(\mathcal{E}\), then \(\gamma\) and \(\nabla\) induce a linear connection on \(\mathcal{B}\).
This method is applied to the case where \(\mathcal{E}\) is an open subset of a Fréchet vector space. As an example, a connection on the space of \(C^\infty\) conformal structures is defined and its curvature and its geodesics are calculated.
58B20 Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds
53C05 Connections, general theory
58C20 Differentiation theory (Gateaux, Fréchet, etc.) on manifolds
58D17 Manifolds of metrics (especially Riemannian)