On a generalization of the Jacobi operator in the Riemannian geometry

Abstract

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 ${\lambda }_{X,Y}(u)=\frac{1}{2}(R(u,X,Y)+R(u,Y,X))$. 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 $M$