Homogeneous Ricci solitons
Homogeneous Ricci solitons
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DOI：
10.1515/crelle-2013-0044
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发表时间：
2011-09
期刊：
影响因子：
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通讯作者：
M. Jablonski
中科院分区：
文献类型：
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作者：
M. Jablonski
在这项工作中，我们研究了齐次度量和Ricci孤子。若Ricci孤子上存在一个可迁可解等距群，则证明了它与孤子等距。此外，除非流形是平坦的，否则它必然是单连通的，并且与$\mathbb R^n$同构。在一般情况下，我们证明了齐次Ricci孤子必须是半代数Ricci孤子，在这个意义上说，他们通过膨胀和拉回的等距群的自同构下的Ricci流。在Ricci孤子上存在可迁半单群的特殊情况下，我们证明了这样的空间实际上是爱因斯坦空间。在紧致情形下，我们给出了Ricci孤子必然是Einstein孤子的新证明.最后，我们刻画了允许Ricci孤子度量的可解李群。
In this work, we study metrics which are both homogeneous and Ricci soliton. If there exists a transitive solvable group of isometries on a Ricci soliton, we show that it is isometric to a solvsoliton. Moreover, unless the manifold is flat, it is necessarily simply-connected and diffeomorphic to $\mathbb R^n$. In the general case, we prove that homogeneous Ricci solitons must be semi-algebraic Ricci solitons in the sense that they evolve under the Ricci flow by dilation and pullback by automorphisms of the isometry group. In the special case that there exists a transitive semi-simple group of isometries on a Ricci soliton, we show that such a space is in fact Einstein. In the compact case, we produce new proof that Ricci solitons are necessarily Einstein. Lastly, we characterize solvable Lie groups which admit Ricci soliton metrics.