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Recently I am studying Ricci flow and its self-similar solution called Ricci soliton. In this respect I found some papers which focuses Ricci soliton in the setting of various kind of contact manifolds. My question is what is the significance of studying ricci soliton in contact geometry setting rather than general Riemannian setting? What kind of geometric idea is behind this scene?

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  • $\begingroup$ Sasakian manifolds in contact geometry are odd dimensional counterpart of Kaehler geometry, Ricci solitons in Sasakian manifolds are very important. You can find a lot of results on Sasaki-Ricci soliton here arxiv.org/pdf/0806.0373.pdf $\endgroup$
    – user21574
    Commented Jan 30, 2016 at 5:21
  • $\begingroup$ ya this is a good paper on sasakian geometry. But my question is not answered there. $\endgroup$ Commented Feb 1, 2016 at 3:51

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Ricci solitons have been studied in the context of Kaehler geometry. Kaehler manifolds are complex and contact Riemannian manifolds are real. So, it makes more sense to study Ricci solitons within the frame-work of contact geometry than complex geometry. Moreover, there are well-known examples of non-gradient expanding Ricci solitons that carry contact structures, for example the 3-dimensional Heisenberg group and solvable Lie groups with certain left-invariant metrics. It has been shown that a non-trivial Ricci soliton as a Sasakian metric is transversally Calabi-Yau. Further, if one considers a contact metric as Ricci soliton with Reeb vector field as the soliton vector field, then it becomes K-contact (for which the Reeb vector field is Killing). Thus, it is interesting to study contact metrics as Ricci solitons, and the current developments indicate that there is ample scope for further advancements in this direction.

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