On 3-dimensional Riemannian manifolds with prescribed Ricci eigenvalues

Stana Nikcevic. Universidade de Belgrado

Resumo:

For three-dimensional Riemannian manifold (M, g) we have the following simple criterion: (M, g) is curvature homogeneous if and only if all principle Ricci curvatures of (M, g) are constant. Thus the problem of classifying all three-dimensional Riemannian manifold with prescribed three constant Ricci eigenvalues is of considerable interest. For every prescribed constant r₁, r₂, r₃ there exist a three-dimensional Riemannian manifold with contant Ricci eigenvalues r₁, r₂, r₃. If the numbers r_i are not equal, then there always exists a corresponding Riemannian manifold (M, g) which is not locally homogeneous. On the other hand, it is not always possible to find a locally homogeneous Riemannian 3-manifold with arbitrary prescribed Ricci eigenvalues. We determine precisely the range of all triplets from R³ which can occur as corresponding Ricci eigenvalues of locally homogeneous Riemannian 3-manifolds. This is the join work with Oldrich Kowalski.

© Stana Nikcevic.