On the curvature groups of a Riemannian foliation (Q2907036)

In [Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 22, 331--341 (1968; Zbl 0155.49803)], \textit{I. Vaisman} introduces curvature groups and curvature numbers associated to a linear connection \(\Gamma\) on a C\(^{\infty}\) manifold \(M\). In [J. Differ. Geom. 9, 547--555 (1974; Zbl 0287.53031)], \textit{S. I Goldberg} and \textit{N. C. Petridis} study this type of curvature groups for the general case of a semi-Riemannian manifold. The authors of this paper generalize these notions by considering basic curvature groups in the case of a transverse Levi-Civita connection \(\Gamma\) of a foliated Riemannian manifold \((M, \mathcal F, g)\), although their construction could be generalized to the case of an arbitrary projectable connection on a foliated principal bundle over a foliated manifold. They prove some interesting results among which the one stating that, if the transverse Ricci tensor is everywhere non-degenerate, the basic curvature groups of \((M, \mathcal F, g)\) vanish. Therefore, an Einstein foliation with at least one non-zero basic curvature group is transversally Ricci flat. Interesting examples of curvature groups of Riemannian manifolds foliated by level hypersurfaces and of curvature groups of foliations defined by suspensions are also presented.