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Let $M$ be a curvature homogeneous or ball-homogeneous non-coK\"{a}hler almost coK\"{a}hler $3$-manifold. In this paper, we prove that $M$ is locally isometric to a unimodular Lie group if and only if the Reeb vector field $\xi$ is an eigenvector field of the Ricci operator. To extend this result, we prove that $M$ is homogeneous if and only if it satisfies $\nabla_\xi h=2f\phi h$, $f\in\mathbb{R}$.
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참고문헌 (21)
참고문헌 신청
D. E. Blair / 1967 / The theory of quasi-Sasakian structures / J. Dierential Geometry 1:331~345
D. E. Blair / 2010 / Riemannian Geometry of Contact and Symplectic Manifolds / Birkhauser Boston, Inc.
W. M. Boothby / 1958 / On contact manifolds / Ann. of Math. (2) 68:721~734
G. Calvaruso / 2000 / Torsion and homogeneity on contact metric threemanifolds / Ann. Mat. Pura Appl. (4) 178:271~285
G. Calvaruso / 2016 / Natural almost contact structures and their 3D homogeneous models / Math. Nachr 289(11-12):1370~1385
D. E. Blair / 2010 / Riemannian Geometry of Contact and Symplectic Manifolds / Birkhauser Boston, Inc.
W. M. Boothby / 1958 / On contact manifolds / Ann. of Math. (2) 68:721~734
G. Calvaruso / 2000 / Torsion and homogeneity on contact metric threemanifolds / Ann. Mat. Pura Appl. (4) 178:271~285
G. Calvaruso / 2016 / Natural almost contact structures and their 3D homogeneous models / Math. Nachr 289(11-12):1370~1385
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