Busemann functions on \(\Re_{\text RI}(m,n)\) and \(\Re_{\text{IV}}(n)\) (Q1425359)

For a complete Riemannian manifold \(M\), let \(r(t)\) be a geodesic ray, where \(t\) is the arc length parameter, and \(d\) the Riemannian distance on \(M\). The function \(\beta_r\) defined by \(\beta_r(x):=\lim_{t\to\infty}(d(x,r(t))-t)\), \(x\in M\), is called the Busemann function determined by \(r\). \textit{J. Q. Zhong} [Acta Math. Sin. 33, No. 5, 577--591 (1990; Zbl 0716.53046)] has obtained the Busemann function on the so-called Cartan domain of the first type explicitly and also announced that his method is suitable for the Cartan domains of the second and third type [see also \textit{M. Kou} and \textit{Z. Zhao}, Adv. Math., Beijing 30, No. 5, 414--426 (2001; Zbl 1004.32008)]. Let \(\mathbb{R}^{m\times n}\) be the set of all real \(m\times n\) matrices. In this paper, an explicit formula for the Busemann function on \(\{X\in \mathbb{R}^{m\times n}\;| \;I-XX'>0\}\), \(m\leq n\) [resp. \(\{z=(z_1,\dots, z_n)\in \mathbb{C}^n\;| \;1+| zz'| ^2-2| z| ^2>0,\;1-| zz'| >0\}\), that is the Cartan domain of the fourth type] determined by any geodesic ray through \(0\) [resp. the geodesic ray joining \(0\) to \(\lambda=(\lambda_1, i\lambda_2, 0,\dots,0)\), \(\lambda_1\geq\lambda_2\geq 0\) and \(1>\lambda_1+\lambda_2\)] is established.