Parallelity and extrinsic homogeneity
It is well known that the parallelity of torsion and curvature tensor of a canonical connection on a Riemannian manifold, implies local homogeneity [\textit{W. Ambrose} and \textit{I. M. Singer}, Duke Math. J. 25, 647-669 (1958; Zbl 0134.17802)]. An analogous result for submanifolds of Euclidean spaces, with the second fundamental form playing the rôle of the curvature tensor, was stated in [\textit{C. Olmos}, J. Differ. Geom. 38, No. 2, 225-234 (1993; Zbl 0791.53051)] and homogeneous submanifolds of Euclidean spaces were studied in [\textit{C. Console}, Boll. Unione Mat. Ital., VII. Ser., B 10, No. 1, 129-148 (1996; Zbl 0860.53031)]. In the present paper, the theory is extended to submanifolds of homogeneous spaces \(G/H\), giving a characterization of orbits \(M\) of closed subgroups of \(G\), in terms of the existence of a canonical connection on the restriction of the tangent bundle of \(G/H\) to \(M\), which is a \(G\)-connection, i.e., the parallel transport \(\tau_c\) along any piecewise differentiable curve \(c: [ a, b] \to M\) is given by the action of some element \(g\) of \(G\), namely \(\tau_c (v)=g \cdot v\). An application to (extrinsic) symmetric submanifolds of Riemannian symmetric spaces is given.
scientific article; zbMATH DE number 1268070