A characterization of n-dimensional hypersurfaces in Rⁿ⁺¹with commuting curvature operators

Abstract

Abstract. Let M n be a hypersurface in R n+1 . We prove that two classical Jacobi curvature operators J x and J y commute on M n , n > 2, for all orthonormal pairs (x, y) and for all points p ∈ M if and only if M n is a space of constant sectional curvature. Also we consider all hypersurfaces with n ≥ 4 satisfying the commutation relation (K x,y • K z,u )(u) = (K z,u • K x,y )(u), where K x,y (u) = R(x, y, u), for all orthonormal tangent vectors x, y, z, w and for all points p ∈ M .

Stanilov-Tsankov-Videv Theory

“…We refer to [8] for the following 3-dimensional and 4-dimensional examples which generalize previous examples found in [27]. We say that M is an irreducible Riemannian manifold if there is no local product decomposition.…”

“…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.…”

“…Similarly, motivated by the seminal papers of Stanilov and Videv [26], of Tsankov [27], and of Videv [28] one studies the commutativity properties of these operators: Definition 2. Let M be a pseudo-Riemannian manifold.…”

The global geometry of Riemannian manifolds with commuting curvature operators

“…Define an embedding of (0, ∞) × N in R 4 by setting F (t, x) := tf (x). Theorem 3.1 may then be used to see the resulting hypersurface in R 4 is skew Tsankov; such hypersurfaces appear in Tsankov [17]. (2) Choose a point x ∈ N where τ N (x) = 2 and let γ x (t) := t × x.…”

The linear independence of sets of two and three canonical algebraic curvature tensors

“…Of course, there are large areas of mathematics that are concerned with the commutativity of linear operators, although recently there has been an interest in the commutativity of certain operators associated to the Riemann curvature tensor in differential geometry. Tsankov proved [16] that if (M, g) is a Riemannian hypersurface in Euclidean space, then J (x)J (y) = J (y)J (x) for x ⊥ y if and only if R g has constant sectional curvature, where J (x) is the Jacobi operator. This gave rise to a subsequent study of a study of the Tsankov condition in pseudo-Riemannian geometry, along with other ELA Linear Independence of Curvature Tensors 439 related notions.…”