On the curvature of affine homogeneous convex domains
Let \(\Omega\) be a convex domain in \({\mathbb{R}}^ n\) not containing affine lines and let G(\(\Omega)\) be the group of all affine automorphisms of \(\Omega\). \(\Omega\) is homogeneous when G(\(\Omega)\) acts transitively. It is known that any homogeneous convex domain is affinely equivalent to \(\Omega\) (U) where \(U=\sum_{1\leq i,j\leq r}\quad U_{ij}\) is a T- algebra of rank r \((r>1)\), * is an involutive automorphism, \[ T=\{(t_{ij});\quad t_{ii}>0\quad (1\leq i\leq r-1),\quad t_{rr}=1,t_{ij}=0\quad (1\leq j<i\leq r)\},\quad \Omega (U)=\{tt^*;\quad t\in T\}. \] Moreover, since the rank of the T-algebra \(U_ 1r_ kU\), is an affine invariant of \(\Omega\), the rank of \(\Omega\) is defined to be \(r_ kU-1\). Also, when [t,t] is Abelian, t the Lie algebra of T, \(\Omega\) (U) is said to be metabelian. Throughout this paper the author considers homogeneous convex domains given by T-algebras and determines those of non-positive sectional curvature (with respect to the canonical metric) among homogeneous convex domains of low rank (section 3), or of metabelian type (sections 2,3). Furthermore, in section 4, homogeneous convex domains over cones of square or dual type of nonpositive curvature are characterized completely.
scientific article; zbMATH DE number 4002983