A generalized Jacobi operator in the 4-dimensional Riemannian geometry

Abstract

We consider a Riemannina manifold $(M,g)$ of dimension $n$ with curvature tensor $R$. At a point $p\in M$ we take a 2-dimensional tangent plane $E^2$ of the tangent space $M_p$, spanned by an orthonormal pair of vectors $X,Y$. We define the lienar operator ${\lambda }_{X,Y}:M_p\to M_p$ by ${\lambda }_{X,Y}(u)=R(u,X,X)+R(u,Y,Y)$. It is a symmetric operator and a very important fact is namely the assertion it is invariant operator under the orthogonal transformations in $E^2$. This gives us the possibility to define an operator in respect to any $E^2$ in $M_p:{\lambda }_{E^2}={\lambda }_{X,Y}$. In the general case its eigen values depend of $p$ and of $E^2$ also. Then we investigate the class of $S$-Riemannian manifolds of dimension 4 with the property the eigen values $c_i(p;E^2),i=1,2,3,4$ of the operators ${\lambda }_{E^2}$ are independent of $E^2$