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Five-dimensional \(\varphi\)-symmetric spaces. (English) Zbl 0889.53031

It is well-known that all known examples of D’Atri spaces, harmonic spaces, ball-homogeneous spaces and \(C\)-spaces are locally homogeneous, and it is an interesting open problem to determine whether all these spaces are indeed locally homogeneous or not. The authors study this problem in the special framework of Sasakian geometry, and they provide some partial positive answers. In particular, they prove that a five-dimensional Sasakian manifold is ball-homogeneous and \(\eta\)-parallel if and only if it is locally \(\varphi\)-symmetric. They also show that a five-dimensional Sasakian manifold is a D’Atri space or a \(C\)-space if and only if it is locally \(\varphi\)-symmetric, and that a \(K\)-contact metric manifold is harmonic if and only if it has constant curvature 1.

MSC:

53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C30 Differential geometry of homogeneous manifolds
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