Four-dimensional Riemannian manifolds with commuting higher order Jacobi operators

Details Four-dimensional Riemannian manifolds with commuting higher order Jacobi operators Four-dimensional Riemannian manifolds with commuting higher order Jacobi operators

2007-01-03

arxiv.org/abs/math/0701090v3

Mathematics

Differential Geometry

10 pages

Scientific paper

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.

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