Homogeneous Einstein metrics on \(SU(n)\)
The author obtains explicitly left-invariant Einstein metrics on the compact Lie group \(SU(n)\) which are not bi-invariant, by imposing two kinds of ansatzes (named schemes 1 and 2). However, it seems that the metrics obtained are known by previous works of Jensen and D'Atri--Ziller.NEWLINENEWLINEMore precisely, in scheme 1 the left-invariant metrics depend on three parameters \(x_1, x_2, x_3\) and the Einstein metrics obtained are (up to scale) either bi-invariant \(x_1=x_2=x_3\) or \(x_1=x_3=\frac{3n+2}{n-2}\) \((n\geq 3)\), \(x_2=1\), which is known by \textit{G. R. Jensen}, [``Left invariant metrics on Lie groups'', J. Differ. Geom. 8, 599--614 (1973; Zbl 0284.53038)] and \textit{A. Besse}, Einstein Manifolds (Chapter 9). In scheme 2 the left-invariant metrics depend on four parameters \(x_1, x_2, x_3, x_4\) and are associated to the decomposition \(S(U(p)\times U(q))\subset SU(p+q)\), for any \(n\geq 2\), \(0\leq p\leq n\) and \(q=m-p\). The Einstein metrics obtained are (up to scale) \(x_1=x_2=x_3=1\), \(x_4=\frac2{pq(p+1)}\) and \(x_2=\frac{q}{p}x_1\), \(x_1=f(p,q)\), \(x_4=g(p,q)x_1\), where \(f(p,q), g(p,q)\) are rational functions of \(p,q\). These metrics are discussed in [\textit{J. E. D'Atri} and \textit{W. Ziller}, ``Naturally reductive metrics and Einstein metrics on compact Lie groups'', Mem. Am. Math. Soc. 215, 72 p. (1979; Zbl 0404.53044)].NEWLINENEWLINEIn both schemes the Einstein metrics obtained are naturally reductive.
scientific article; zbMATH DE number 6031295
scientific article