(untitled) (Annual of Sofia University St. Kliment Ohridski. Faculty of Mathematics and Informatics, 1994)
Let (M,g) be a Riemannian manifold of dimension n with curvature tensor R. If X,Y is an orthonormal pair of tangent vectors at a point p of M, we define the curvature operator λ_X,Y(u)=12(R(u,X,Y)+R(u,Y,X)). It is a symmetric operator but its eigen values depend on the base X,Y of the tangent subspace E^2(p;X,Y) spanned by X,Y. We prove: 1) the Einstein manifolds are unique for which the trace of this operator does noe depend on the base of E^2(p;X,Y); 2)the real space forms of dimension 4 are the unique for which the spectrum of this operator does not depend on the base of E^2(p;X,Y). Of course in both assertions p is an arbitrary point of M