A canonical lift of linear connection in Fréchet principal bundles (Q677781)

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 \textit{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.