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

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