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Consider a gradient Einstein-type metric in the setting of $K$-contact manifolds and $(\kappa,\mu)$-contact manifolds. First, it is proved that, if a complete $K$-contact manifold admits a gradient Einstein-type metric, then $M$ is compact, Einstein, Sasakian and isometric to the unit sphere $\mathbb{S}^{2n+1}$. Next, it is proved that, if a non-Sasakian $(\kappa,\mu)$-contact manifolds admits a gradient Einstein-type metric, then it is flat in dimension $3$, and for higher dimension, $M$ is locally isometric to the product of a Euclidean space $\mathbb{E}^{n+1}$ and a sphere $\mathbb{S}^n(4)$ of constant curvature $+4$.
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참고문헌 (25)
참고문헌 신청
D. E. Blair / 2010 / Riemannian geometry of contact and symplectic manifolds / Birkh¨auser Boston, Inc.
D. E. Blair / 1995 / Contact metric manifolds satisfying a nullity condition / Israel J. Math 91(1-3):189~214
E. Boeckx / 2000 / A full classification of contact metric (k, µ)-spaces / Illinois J. Math 44(1):212~219
C. P. Boyer / 2001 / Einstein manifolds and contact geometry / Proc. Amer. Math. Soc 129(8):2419~2430
C. P. Boyer / 2008 / Sasakian Geometry, Oxford Mathematical Monographs / Oxford University Press
D. E. Blair / 1995 / Contact metric manifolds satisfying a nullity condition / Israel J. Math 91(1-3):189~214
E. Boeckx / 2000 / A full classification of contact metric (k, µ)-spaces / Illinois J. Math 44(1):212~219
C. P. Boyer / 2001 / Einstein manifolds and contact geometry / Proc. Amer. Math. Soc 129(8):2419~2430
C. P. Boyer / 2008 / Sasakian Geometry, Oxford Mathematical Monographs / Oxford University Press
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