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Riemannian manifolds in which certain curvature operator has constant eigenvalues
along each helix

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*Yana Alexieva and Stefan Ivanov*

**Address.** University of Sofia, Faculty of Mathematics and Informatics,
Department of Geometry, bul. James Borchier 5, 1164 Sofia, BULGARIA

**E-mail:** ivanovsp@fmi.uni-sofia.bg

**Abstract.** Riemannian manifolds for which a natural skew-symmetric
curvature operator has constant eigenvalues on helices are studied. A local
classification in dimension three is given. In the three dimensional case
one gets all locally symmetric spaces and all Riemannian manifolds with
the constant principal Ricci curvatures $r_1 = r_2 = 0, r_3 \not =0$, which
are not locally homogeneous, in general.

**AMSclassification.** 53C15, 53C55, 53B35

**Keywords.** Helix, constant eigenvalues of the curvature operator,
locally symmetric spaces, curvature homogeneous spaces