Cross ratios on boundaries of higher rank symmetric spaces. (arXiv:1701.09096v2 [math.DG] UPDATED)

We construct (generalized) cross ratios on Furstenberg boundaries (or flag manifolds) of higher rank symmetric spaces of non-compact type. We show several basic properties of it; including continuity, the connection to translation lengths of hyperbolic elements, and the behavior under products. Moreover, we motivate that these are suitable generalizations of cross ratios on ideal boundaries of rank one symmetric spaces, by proving that every continuous cross ratio-preserving map on the maximal Furstenberg boundary (or full flag manifold) is induced by an isometry, after multiplying metrics on de Rham factors by positive constants. If the symmetric space is irreducible this yields a one-to-one correspondence of isometries and continuous cross ratio-preserving maps. 查看全文>>