Canonical connection and the canonical conformal group on a Riemannian almost-product manifold. (English) Zbl 0709.53024

Mihova, Vesselka

Serdica 15, No. 4, 351-358 (1989).

Let \(M_{2n}\) be a manifold with an almost-product structure (P,g) and let \(\tilde g\) be the associated metric. The author defines an action of the fundamental group of the structure (P,g) on the bundle \({\mathcal T}\) of tensors of type (0,3) which are skew-symmetric in the first two arguments and determines a decomposition of the bundle \({\mathcal T}\) in the direct sum of four invariant and orthogonal subbundles. Then he shows that there exists a unique linear connection \(\nabla\) compatible with the structure (P,g) whose torsion tensor field T belongs to \({\mathcal T}_ 1\oplus {\mathcal T}_ 3\). Finally the author proves that the group of the general conformal transformations of the metric g generates a group of conformal transformations of the canonical connection \(\nabla\).

Reviewer: V.Cruceanu

MSC:

53C15	General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C10	\(G\)-structures

Keywords:

almost-product structure; fundamental group; linear connection; conformal transformations