{'fraction_non_alphanumeric': 0.05732484076433121, 'fraction_numerical': 0.010422698320787493, 'mean_word_length': 3.818688981868898, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Ricci solitons on Sasakian manifolds

We show that a Sasakian metric which also satisfies the gradient Ricci soliton equation is necessarily Einstein.

Question: Does there exist a shrinking Ricci soliton on Sasakian manifolds which is not Einstein?
In the contrast to Einstein manifolds, we have the following non-existence result on gradient Ricci solitons.
Theorem 1.1. Suppose (M, g) is a Sasakian manifold and satisfies the gradient Ricci soliton equation, then f is a constant function and (M, g) is an Einstein manifold.
Remark 1.2. Note that in our paper a Ricci soliton with Sasakian metric is different from the Sasaki-Ricci soliton in the context of transverse Kähler structure in a Sasakian manifold, see for example [FOW].
Remark 1.3. Since all compact Ricci solitons are gradient ones from [Pe], Theorem 1.1 implies that there is no compact non-Einstein Ricci soliton in Sasakian manifolds.
Remark 1.4. On the other hand, there exist non-gradient expanding Ricci solitons on noncompact Sasakian manifolds which are not Einstein. They were first discovered by P. Baird and L. Danielo in [BD], and independently by J. Lott in [Lo]. They are left invariant metrics on some solvable Lie groups and appear as type III singularities models of Ricci flow. The one with the lowest dimension is the three dimensional Heisenberg group with a left invariant metric. One can easily check that the Reeb vector field is given by F 3 in Example 1.54 in [Ch].
Remark 1.5. After the first version of our paper appeared in the arXiv, we learned that similar results have been obtained in K-contact manifolds by R. Sharma, see [Sh].
Acknowledgment: We thank Huai-Dong Cao for bringing this problem to our attention and useful discussions. We also want to thank Xiaofeng Sun for useful conservations, and Mircea Crasmareanu for informing us the reference [Sh].

Proof of Theorem 1.1
In this section we prove Theorem 1.1. The basic properties of Sasakian manifolds can be found in [BG] or [YK, Chapter V].
Proof. Recall that a Sasakian manifold (M 2m+1 , g, ξ, η, Φ) is a contact manifold on which there exists a unit Killing vector field ξ(called the Reeb vector field), η is the dual one-form of ξ and Φ is a (1, 1)-tensor defined by Φ(Y ) = ∇ Y ξ. The metric (M, g) is called Sasakian if and only if for any vector fields Y, Z ∈ T M . Let D ⊂ T M be the distribution defined by η(Y ) = g(Y, ξ) = 0. Then D is nowhere integrable as η is a contact 1-form. Since ξ is a unit Killing vector field we have Φ(Y ) = ∇ Y ξ ∈ D for any Y ∈ T M . From the curvature condition (2) for any Y ∈ D and Z ∈ T M we have R(Y, ξ, Z, Y ) = −g(Y, Z)g(ξ, Y ) + g(Z, ξ)g(Y, Y ) = g(Z, ξ) |Y | 2 and thus ξ is an eigenvector field of Ric with Ric(ξ) = 2mξ.
In the following, we do the calculation on a general Ricci soliton. At the end, we use the fact that X = ∇f to draw the conclusion. Since ξ is a Killing vector field, taking the Lie derivative to the soliton equation (1) yields Thus, for any vector fields Y, Z ∈ T M we have We let Y be orthogonal to ξ and Z = ξ and then the above equation with the curvature condition (2) implies that where the vector field ∇ ξ ξ vanishes as ξ is a unit Killing vector field.
Note that from the soliton equation, for any W ∈ T M we have Let W = ξ then from the above equation we have 2(λ − 2m) = 2g(∇ ξ X, ξ), i.e, g(∇ ξ X, ξ) = λ − 2m, or equivalently, The middle term in the equation (3) vanishes and we have for any vector field Y ⊥ ξ.