Geometry of Selberg's bisectors in the symmetric space $SL(n,\mathbb{R})/SO(n,\mathbb{R})$

Abstract

We study some properties of a family of symmetric spaces, namely $SL(n,\mathbb{R})/SO(n,\mathbb{R})=:\mathcal{P}(n)$, where we replace the Riemannian metric on $\mathcal{P}(n)$ with a premetric suggested by Selberg. These include criteria for intersections of Selberg's bisectors, the shape of Dirichlet-Selberg domains, and angle-like functions between Selberg's bisectors. These properties are generalizations of properties on the hyperbolic spaces ${\mathbf{H}}^n$ related to Poincaré's fundamental polyhedron theorem.

arXiv commentThe most recent revision introduced Lancaster and Rodman's congruence normal form of symmetric matrix pencils and used it to improve the main theorems in Sections 3 and 4. I also added a brief discussion of the isometry equivalent classes of $2$-subspaces of $\mathcal{P}(3)$ to the appendix. Comments are welcome

References in corpus (1)

• Simultaneous Diagonalization of Matrices and its Applications in Quadratically Constrained Quadratic Programming