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In this article we study almost contact manifolds admitting weakly Einstein metrics. We first prove that if a $(2n+1)$-dimensional Sasakian manifold admits a weakly Einstein metric, then its scalar curvature $s$ satisfies $-6\leqslant s \leqslant 6$ for $n=1$ and $-2n(2n+1)\frac{4n^2-4n+3}{4n^2-4n-1}\leqslant s \leqslant 2n(2n+1)$ for $n\geqslant2$. Secondly, for a $(2n+1)$-dimensional weakly Einstein contact metric $(\kappa,\mu)$-manifold with $\kappa<1$, we prove that it is flat or is locally isomorphic to the Lie group $SU(2)$, $SL(2)$, or $E(1,1)$ for $n=1$ and that for $n\geqslant2$ there are no weakly Einstein metrics on contact metric $(\kappa,\mu)$-manifolds with $0<\kappa<1$. For $\kappa<0$, we get a classification of weakly Einstein contact metric $(\kappa,\mu)$-manifolds. Finally, it is proved that a weakly Einstein almost cosymplectic $(\kappa,\mu)$-manifold with $\kappa<0$ is locally isomorphic to a solvable non-nilpotent Lie group.
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참고문헌 (14)
참고문헌 신청
H. Baltazar / Weakly Einstein critical metrics of the volume functional on compact manifolds with boundary / arXiv:1804.10706v2
M. Berger / 1970 / Quelques formules de variation pour une structure riemannienne / Ann. Sci. Ecole Norm. Sup 3:285~294
D. E. Blair / 2010 / Riemannian geometry of contact and symplectic manifolds / Progress in Mathematics 203
D. E. Blair / 1995 / Contact metric manifolds satisfying a nullity condition / Israel J. Math 91(1-3):189~214
E. Boeckx / 2000 / A full classication of contact metric(k;)-spaces / Illinois J. Math 44(1):212~219
M. Berger / 1970 / Quelques formules de variation pour une structure riemannienne / Ann. Sci. Ecole Norm. Sup 3:285~294
D. E. Blair / 2010 / Riemannian geometry of contact and symplectic manifolds / Progress in Mathematics 203
D. E. Blair / 1995 / Contact metric manifolds satisfying a nullity condition / Israel J. Math 91(1-3):189~214
E. Boeckx / 2000 / A full classication of contact metric(k;)-spaces / Illinois J. Math 44(1):212~219
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