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A canonical lift of linear connection in Fréchet principal bundles. (English. Russian original) Zbl 0873.53013
Russ. Math. 40, No. 2, 69-77 (1996); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1996, No. 2(405), 73-81 (1996).
Let, in the paper’s notations, $$\lambda=({\mathcal E},\pi,{\mathcal B},{\mathcal G})$$ be a principal Fréchet bundle whose structure group $$\mathcal G$$ admits an exponential map. Then, given an infinitesimal connection on $$\lambda$$ and a linear connection $$\overset{*}\nabla$$ on $$\mathcal B$$, the author constructs a covariant derivation $$\nabla$$ on $$\mathcal E$$, corresponding to an appropriate linear condition. Generalizing K. M. Egiazaryan [Tr. Geom. Semin. 12, 27-37 (1980; Zbl 0493.53013)], $$\nabla$$ is called a canonical lift of $$\overset{*}\nabla$$. It is shown that $$\nabla$$ is $${\mathcal G}$$-invariant.
The techniques of this construction are used to obtain a linear connection on the space of smooth Riemannian metrics $${\mathfrak M}$$ of an oriented closed smooth manifold $$M$$. It is proved that $${\mathfrak M}$$ is locally symmetric with respect to the previous linear connection if and only if $$\dim M=2$$. In the latter case, the explicit form of the corresponding geodesics is given.
##### MSC:
 53C05 Connections, general theory 58B20 Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds
##### Keywords:
space of Riemannian metrics; linear connection