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In this paper, first we introduce the full expression of the Riemannian curvature tensor of a real hypersurface $M$ in the complex quadric~$Q^{m}$ from the equation of Gauss and some important formulas for the structure Jacobi operator ~$R_{\xi}$ and its derivatives $\nabla R_{\xi}$ under the Levi-Civita connection $\nabla$ of $M$. Next we give a complete classification of Hopf real hypersurfaces with Reeb parallel structure Jacobi operator, $\nabla_{\xi}R_{\xi}=0$, in the complex quadric $Q^{m}$ for $m \geq 3$. In addition, we also consider a new notion of $\mathcal C$-parallel structure Jacobi operator of $M$ and give a nonexistence theorem for Hopf real hypersurfaces with $\mathcal C$-parallel structure Jacobi operator in $Q^{m}$, for $m \geq 3$.
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참고문헌 (30)
참고문헌 신청
J. Berndt / 2002 / Real Hypersurfaces with Isometric Reeb Flow in Complex Two-Plane Grassmannians / Monatsh. Math 137(2):87~98
JURGEN BERNDT / 2013 / REAL HYPERSURFACES WITH ISOMETRIC REEB FLOW IN COMPLEX QUADRICS / International Journal of Mathematics 24(07):1350050
Jürgen Berndt / 2015 / Contact hypersurfaces in Kähler manifolds / Proceedings of the American Mathematical Society 143(6):2637~2649
A. L. Besse / 2008 / Einstein Manifolds / Springer-Verlag
S. Helgason / 2001 / Differential Geometry, Lie Groups, and Symmetric Spaces / American Mathematical Society
JURGEN BERNDT / 2013 / REAL HYPERSURFACES WITH ISOMETRIC REEB FLOW IN COMPLEX QUADRICS / International Journal of Mathematics 24(07):1350050
Jürgen Berndt / 2015 / Contact hypersurfaces in Kähler manifolds / Proceedings of the American Mathematical Society 143(6):2637~2649
A. L. Besse / 2008 / Einstein Manifolds / Springer-Verlag
S. Helgason / 2001 / Differential Geometry, Lie Groups, and Symmetric Spaces / American Mathematical Society
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