Second order parallel tensors and Ricci solitons in 3-dimensional normal paracontact geometry.(English)Zbl 1312.53049

In this paper, the authors are looking for three-dimensional Ricci-solitons with a Lorentz metric (signature $$(2, 1)$$).
They start with an almost paracontact structure $$(\varphi , \psi , \eta , g)$$ at the beginning of the second section. Here, $$\eta$$ is the almost-contact differential 1-form, $$\psi$$ is a vector field (the dual of $$\eta$$), $$\varphi$$ is a complex structure on the kernel $${\mathcal D}$$ of $$\eta$$ (actually a product structure, i.e., $$\varphi ^2 =I$$), and $$g$$ satisfies $$(\varphi \;, \varphi \;)=-g+\eta \otimes \eta$$.
$$M$$ is normal (Definition 2.1) if $$\varphi$$ extends to an integrable product structure on $$M\times {\mathbb R}$$ with the natural product of $${\mathcal D}\times {\mathbb R}^2$$. This is the same as (2.4) according to [J. Wełyczko, Result. Math. 54, No. 3–4, 377–387 (2009; Zbl 1180.53080)].
There is a serious “typo” in the page 119, line 7. “(1.4a)” should possibly be “the first equation of (2.4)”.
Definition 2.2 says that 1. $$M$$ is quasi-para-Sasakian if $$\alpha =0$$ and $$\beta \neq 0$$ (in (2.4)); and 2. para-Kenmotsu if $$\beta=0$$ and $$\alpha \neq 0$$.
One of the major result is Theorem 3.1: If $$\rho$$ is a parallel symmetric 2-tensor which is $$\varphi$$ skew-symmetric, then it is a multiple of the metric $$g$$.
By applying Theorem 3.1 to the $$\rho$$ in (4.11), the authors obtains all kind of Ricci-solitons.

MSC:

 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) 53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics

Zbl 1180.53080
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References:

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