Monotone functions on symmetric spaces
In [Symp. Math. 29, 183-196 (1987; Zbl 0656.43003) and J. Math. Soc. Japan 43, No. 1, 133-147 (1991; Zbl 0732.43005)] \textit{J. Faraut} raised the question whether the order intervals \([x,y]=\{z : x \preceq z\preceq y\}\) in an ordered symmetric space \(M = G/H\) of a Lie group \(G\) are compact or not. The compactness of the order intervals is necessary in order to define the algebra of Volterra kernels on such spaces. Using a result of Ol'shanskij Faraut proved compactness of the order intervals in a certain class of ordered symmetric spaces. His results also permit a generalization to general regular symmetric spaces. On the other hand there exist very general results on the structure of ordered homogeneous spaces which are not necessarily symmetric and which also contain a characterization of those spaces, where all the order intervals are compact [the author, Invent. Math. 104, No. 3, 467-496 (1991; Zbl 0762.22022)]. These spaces are called globally hyperbolic and they are characterized by the existence of a smooth monotone function \(f\) and a complete Riemannian metric \(g\) such that \(f\) increases faster than arclength along monotone curves starting in the base point. In this note we close the gap between these two branches of the theory. Namely we describe how one can construct the monotone functions and the complete Riemannian metric whose existence is guaranteed by the general characterization of the author [loc cit., Theorem 1.32] on a regular symmetric space.
scientific article; zbMATH DE number 59958