Ricci soliton homogeneous nilmanifolds
Ricci soliton homogeneous nilmanifolds
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DOI：
10.1007/pl00004456
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发表时间：
2001-04
影响因子：
1.4
通讯作者：
J. Lauret
中科院分区：
文献类型：
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作者：
J. Lauret
研究了幂零李群N上左不变黎曼度量的爱因斯坦条件的一个弱化概念.我们考虑N的李代数的某些导子D满足Rici的度量，其中Rici表示的Ricci算子。这个条件等价于度量是Ricci孤子。证明了N上的Ricci孤子左不变度量在等距和标度下是唯一的.给出了如下刻划：（N，g）是Ricci孤子当且仅当（N，g）有一个度量标准可解扩张，其对应的标准可解流形是Einstein.这给了几个家庭的新的例子里奇孤子。通过变分方法，我们进一步证明了Ricci孤子齐次流形（N，g）正是定义在向量空间上的自然泛函的临界点，该向量空间包含所有给定维数的齐次流形作为一个真实的代数集.
We study a notion weakening the Einstein condition on a left invariant Riemannian metricgon a nilpotent Lie groupN. We consider those metrics satisfying Ricfor someand some derivationDof the Lie algebraofN, where Ricdenotes the Ricci operator of. This condition is equivalent to the metricgto be a Ricci soliton. We prove that a Ricci soliton left invariant metric onNis unique up to isometry and scaling. The following characterization is also given: (N,g) is a Ricci soliton if and only if (N,g) admits a metric standard solvable extension whose corresponding standard solvmanifoldis Einstein. This gives several families of new examples of Ricci solitons. By a variational approach, we furthermore show that the Ricci soliton homogeneous nilmanifolds (N,g) are precisely the critical points of a natural functional defined on a vector space which contains all the homogeneous nilmanifolds of a given dimension as a real algebraic set.