As it well known, Riemannian manifolds are very useful tools for practical problems. To give some examples, we underline their utilization in economic theory, in system modeling and optimization or to statistical theory. That is why they are intenively studied by famous scientist in the world.

Contenents

1. Hessian type of structure on pseudo-Reimannian manifolds

2. Hessian structures via invariant calculus

3. Self-concordant functions on Reimannian manifolds

4. Hessian Reimannian manifolds

5. Geometrical characteristics of special functions

6. Role of Hessian type metrics in the interior point methods

7. Affine connections and affine hypersurfaces

8. Ricci solitons and gradient ricci solitons

9. Classes of gradient ricci solitons