The prescribed Ricci curvature problem for naturally reductive metrics on non-compact simple Lie groups (Q6652123)

The title and abstract of the paper give a good quick summary, as well as the short introductory section provides a nice presentation of the subject. We next provide some technicalities to state some of its main results, which are not shown in the introduction.\N\NLet \(G\) be a non-compact simple Lie group and let \(K\) be a maximal compact subgroup of \(G\), which is unique up to conjugation. For a given Lie algebra \(\mathfrak h\), \(\operatorname{B}_{\mathfrak h}\) stands for its Killing form. We write \(\mathfrak k=\mathfrak k_0\oplus \dots\oplus \mathfrak k_{r}\), where \(\mathfrak k_1,\dots,\mathfrak k_r\) are the simple ideals of \([\mathfrak k,\mathfrak k]\) and \(\mathfrak k_{0}=\mathfrak z(\mathfrak k)\). For each \(i\), there is \(\kappa_i\geq0\) such that \(\operatorname{B}_{\mathfrak k_i}=\kappa_i\, \operatorname{B_{\mathfrak g}}|_{\mathfrak k_i}\).\N\NLet \(\mathcal M_K\) denote the space of left-invariant metrics on \(G\) that are naturally reductive with respect to the action of \(G\times K\) on \(G\) given by \((x,k)\cdot y=xyk^{-1}\), for \(x,y\in G\) and \(k\in K\).\N\NWe are now in position to state some of the main results. Let \(T\) be a left-invariant \((0,2)\)-tensor field \(T\) on \(G\) written as \(T=-T_{\mathfrak p}Q|_{{\mathfrak p}}+T_0Q|_{\mathfrak k_0}+\dots+T_0Q|_{\mathfrak k_r}\) with \(T_{\mathfrak p},T_0,\dots,T_r>0\).\N\begin{itemize}\N\item[Theorem 4.1] There is \(g\in\mathcal M_K\) satisfying \(\operatorname{Ric}(g)=T\) if and only if \(4T_i-\kappa_i>0\) for all \(i=1,\dots,r\) and \[T_{\mathfrak p}=\sum_{i=0}^{r} \frac{2\dim\mathfrak k_i(1-\kappa_i)+\dim\mathfrak k_i\sqrt{(4T_i-\kappa_i)(1-\kappa_i)}}{2n}.\] Furthermore, there is at most one such \(g\), up to scaling.\N\item[Theorem 5.10(1)] There exists at least one pair \((g,c)\in\mathcal M_K\times\mathbb R_{>0}\) satisfying \(\operatorname{Ric}(g)=cT\) if \N\[ \sum_{i=0}^{r} \frac{\dim \mathfrak p\, \kappa_i T_{\mathfrak p}^2-(\dim\mathfrak k_i)^2(1-\kappa_i)T_i^2}{nT_{\mathfrak p}T_i}-2n<0. \]\N\item[Theorem 5.10(2)] If additionally to the condition from the previous item, one has that \[\frac{\kappa_m \operatorname{tr}_Q(T)}{T_m}<\dim\mathfrak k+\dim\mathfrak k_m(1-\kappa_m)-3\dim\mathfrak p\] for some index \(m\) satisfying \(\frac{\kappa_m}{T_m}=\max_{1\leq i\leq r}\frac{\kappa_i}{T_i}\), then there are at least two pairs \((g,c)\in\mathcal M_K\times\mathbb R_{>0}\) of non-homothetic metrics \(g\) satisfying \(\operatorname{Ric}(g)=cT\).\N\end{itemize}\N\NAmong other things, the paper also considers the special case when \(K\) is simple, and provides three particular examples in Section 7. Of independent interest are the formulas for the Ricci tensor of \(g\) in \(\mathcal M_K\) (Theorem 3.6).\N\NThe article can be seen as a continuation of [\textit{R. M. Arroyo} et al., Differ. Geom. Appl. 78, Article ID 101794, 14 p. (2021; Zbl 1477.53056)], which deals with the case \(G\) compact.