The unit tangent sphere bundle whose characteristic Jacobi operator is pseudo-parallel
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\noindent We study the characteristic Jacobi operator $\ell=\bar R(\cdot,\xi)\xi$ (along the Reeb flow $\xi$) on the unit tangent sphere bundle $T_1 M$ over a Riemannian manifold $(M^n,g)$. We prove that if $\ell$ is pseudo-parallel, i.e., $\bar R\cdot \ell=L \mathcal{Q}(\bar g,\ell)$, by a non-positive function $L$, then $M$ is locally flat. Moreover, when $L$ is a constant and $n\neq 16$, $M$ is of constant curvature $0$ or $1$.
Abstract
\noindent We study the characteristic Jacobi operator $\ell=\bar R(\cdot,\xi)\xi$ (along the Reeb flow $\xi$) on the unit tangent sphere bundle $T_1 M$ over a Riemannian manifold $(M^n,g)$. We prove that if $\ell$ is pseudo-parallel, i.e., $\bar R\cdot \ell=L \mathcal{Q}(\bar g,\ell)$, by a non-positive function $L$, then $M$ is locally flat. Moreover, when $L$ is a constant and $n\neq 16$, $M$ is of constant curvature $0$ or $1$.
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