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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 $p \in M$ if and only if $M^n$ is a space of constant sectional curvature. Also we consider all hypersurfaces with $n\geq 4$ satisfying the commutation relation $ (K_{x,y} \circ K_{z,u})(u) = (K_{z,u} \circ 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 $p \in M$.