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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 $1$.
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참고문헌 (14)
참고문헌 신청
D. E. Blair / 2010 / Riemannian Geometry of Contact and Symplectic Manifolds / Birkhauser Boston, Inc.
E. Boeckx / 2007 / Flow-invariant structures on unit tangent bundles / Publ. Math. Debrecen 70(1-2):167~178
E. Boeckx / 1997 / Characteristic reflections on unit tangent sphere bundles / Houston J. Math 23(3):427~448
J. T. Cho / 2003 / On the classification of contact Riemannian manifolds satisfying the condition (C) / Glasg. Math. J 45(3):475~492
조종택 / 2005 / Pseudo-symmetric contact $3$-manifolds / 대한수학회지 42(5):913~932
E. Boeckx / 2007 / Flow-invariant structures on unit tangent bundles / Publ. Math. Debrecen 70(1-2):167~178
E. Boeckx / 1997 / Characteristic reflections on unit tangent sphere bundles / Houston J. Math 23(3):427~448
J. T. Cho / 2003 / On the classification of contact Riemannian manifolds satisfying the condition (C) / Glasg. Math. J 45(3):475~492
조종택 / 2005 / Pseudo-symmetric contact $3$-manifolds / 대한수학회지 42(5):913~932
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