zbMATH — the first resource for mathematics

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.
53C05 Connections, general theory
58B20 Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds