Centralizer of Reeb vector field in contact Lie groups (Q1784692)

Let \(G\) be a simple connected Lie group equipped with a Riemannian left invariant metric and \(\mathfrak g\) is the Lie algebra of \(G.\) Assume \(G\) is odd dimensional. \(G\) is called almost contact metric Lie group if there exist a tensor \(\phi\) of type \((1,1),\) a vector field \(\xi\) called structure vector field and \(\eta,\) the left invariant dual one-form of \(\xi\) satisfying some conditions. Moreover, if there exists a fundamental two-form \(\Phi\) on \(G\) given by \(\Phi(X,Y) = \langle X,\phi Y\rangle\) such that \(\Phi = d\eta,\) then \(G\) is said to be a contact metric Lie group. Finally let \(\mathfrak c_{\mathfrak g}(\xi) = \{X\in\mathfrak g; [X,\xi] = 0\}\) be an invariant subalgebra and \(\mathfrak n\) its orthogonal complement such that \(\mathfrak g = \mathfrak c_{\mathfrak g}(\xi) + \mathfrak n\). In the paper under review the following results for the contact Lie group \((G,\eta,\phi,\xi)\) with a left invariant Riemannian metric are proved: {\parindent=0.7cm \begin{itemize}\item[1.] \((\nabla_{X_{\mathfrak n}}\phi)Y_{\mathfrak n} = 2\langle X_{\mathfrak n},Y_{\mathfrak n}\rangle\xi\) for all \(X_{\mathfrak n}\), \(Y_{\mathfrak n}\in\mathfrak n\). \item[2.] \(\langle [X_{\mathfrak n},\xi],X_{\mathfrak n}\rangle = 0\), for any \(X_{\mathfrak n}\in\mathfrak n\). \item[3.] \([\xi,\phi X_{\mathfrak n}] = -2X_{\mathfrak n} - \phi[\xi,X_{\mathfrak n}]\), for any \(X_{\mathfrak n}\in\mathfrak n\). \end{itemize}} Furthermore, in the last section a special automorphism \(\psi\in\mathrm{aut}\,\mathfrak g\) is defined and studied which maintains the contact form. This means \(\eta(X) = \eta(\psi X)\) for any \(X\in\mathfrak g\).