Manifolds with commuting Jacobi operators

Abstract

Abstract. We study the geometry of pseudo-Riemannian manifolds which are Jacobi-Tsankov, i.e. J (x)J (y) = J (y)J (x) for all x, y. We also study manifolds which are 2-step Jacobi nilpotent, i.e. J (x)J (y) = 0 for all x, y.

The global geometry of Riemannian manifolds with commuting curvature operators

“…The notation is motivated by the seminal result of Tsankov [17] who, following foundational suggestions of Stanilov, studied similar questions for hypersurfaces in R m ; Tsankov imposed an extra condition of orthogonality that we shall not impose here. Similar questions arise for the Jacobi operator; see [3] for further details.…”

Stanilov-Tsankov-Videv Theory

“…Since O is dense and λ(·) is continuous, λ(x) = 0 for all x so J (x) = 0 for all x; the usual curvature symmetries now imply the full curvature tensor R vanishes. One has the following classification result [9]; we also refer to a related result [27] if M is a hypersurface in R m+1 . (a) R = cR id has constant sectional curvature c for some c ∈ R.…”

Commuting Jacobi operators on Real hypersurfaces of Type B in the complex quadric

“…Actually, the study for the commuting property with Jacobi operators was first initiated by Brozos-Vázquex and Gilkey [4]. They gave two results for a Riemannian manifold ( M m , g), m ≥ 3, as follows: One is: if R U R V = R V R U for all tangent vector fields U, V on M , then M is flat.…”