# zbMATH — the first resource for mathematics

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.
##### MSC:
 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)