Four-dimensional Riemannian manifolds with commuting higher order Jacobi operators

Abstract

We consider four-dimensional Riemannian manifolds with commuting higher order Jacobi operators defined on two-dimensional orthogonal subspaces (polygons) and on their orthogonal subspaces. More precisely, we discuss higher order Jacobi operator $\mathcal{J}(X)$ and its commuting associated operator $\mathcal{J}(X^{\perp })$ induced by the orthogonal complement $X^{\perp }$ of the vector $X$, i. e. $\mathcal{J}(X)\circ \mathcal{J}(X^{\perp })=\mathcal{J}(X^{\perp })\circ \mathcal{J}(X)$. At the end some new central theorems have been cited. The latter are due to P. Gilkey, E. Puffini and V. Videv, and have been recently obtained.

10 pages

Cited by in corpus (1)

• Stanilov-Tsankov-Videv Theory