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On some properties which characterize symmetric and general \(R\)-spaces. (English) Zbl 0898.53042

A linear connection \(\nabla^c\) on a Riemannian manifold \((M,g)\) is called canonical if it satisfies \(\nabla^c g=0\) and \(\nabla^c D=0\), where \(D(X,Y)=\nabla_XY-\nabla_X^cY\) is the difference tensor of the Riemannian and canonical connections. Assume \(M\) is a compact connected Riemannian manifold isometrically and fully imbedded in a Euclidean space. The canonical covariant derivative of the second fundamental form \(\alpha\) of the imbedding is defined by \((\nabla^c_X\alpha)(Y,Z)= \nabla^\perp_X(\alpha(Y,Z))-\alpha(\nabla^c_X Y,Z)-\alpha (Y,\nabla^c_X Z).\) Earlier, E. Hulett and C. Sánchez proved that in a general \(R\)-space the second fundamental form satisfies Axiom 6; namely, for each point \(p\in M\) and every \(X,Y,Z\in T_pM\), \(\alpha_p(T(X,Y),Z)=\alpha_p(Y,D(X,Z))- \alpha_p(X,D(Y,Z)),\) where \(T(X,Y)=D(Y,X)-D(X,Y)\) is the torsion of the connection \(\nabla^c\).
In this paper, the authors prove several nice characterizations of the canonical imbeddings of general \(R\)-spaces under the assumption that the imbeddings satisfy Axiom 6.

MSC:

53C40 Global submanifolds
53C30 Differential geometry of homogeneous manifolds
53C35 Differential geometry of symmetric spaces
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