## On 3-dimensional Riemannian manifolds with prescribed Ricci eigenvalues

### Stana Nikcevic. Universidade de Belgrado

Luns 1 de decembro de 2008 ás 17 horas na aula 9.

**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 *r*_{1}, r_{2}, r_{3} there exist a three-dimensional Riemannian manifold with contant Ricci eigenvalues *r*_{1}, r_{2}, r_{3}. 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 **R**^{3} which can occur as corresponding Ricci eigenvalues of locally homogeneous Riemannian 3-manifolds. This is the join work with Oldrich Kowalski.

©
Stana Nikcevic.