arXiv subjects
Publications and source records attributed to Michael Klotz.
To give a criterion for the integrability of Banach-Lie triple systems, we follow the construction of the period group of a Lie algebra and define the period group of a Lie triple system as an analogous concept. We show that a Lie triple system is integrable if and only if its period group is discrete. Along the way, we see how to turn the path and the loop space of a pointed symmetric space into pointed symmetric spaces.
A Banach symmetric space in the sense of O. Loos is a smooth Banach manifold $M$ endowed with a multiplication map $μ\colon M \times M \to M$ such that each left multiplication map $μ_x := μ(x,\cdot)$ (with $x \in M$) is an involutive automorphism of $(M,μ)$ with the isolated fixed point $x$. We show that morphisms of Lie triple systems of symmetric spaces can be uniquely integrated provided the first manifold is 1-connected. The problem is attacked by showing that a continuous linear map between tangent spaces of affine Banach manifolds with parallel torsion and curvature is integrable to an affine map if it intertwines the torsion and curvature tensors provided the first manifold is 1-connected and the second one is geodesically complete. Further, we show that the automorphism group of a connected Banach symmetric space $M$ can be turned into a Banach-Lie group acting smoothly and transitively on $M$. In particular, $M$ is a Banach homogeneous space.
We derive a necessary and sufficient condition for the existence of symmetric space structures on quotients of Banach symmetric spaces. Along the way, we investigate the different kinds of reflection subspaces and their Lie triple systems.
A {1}-structure on a Banach manifold M (with model space E) is an E-valued 1-form on M that induces on each tangent space an isomorphism onto E. Given a Banach principal bundle P with connected base space and a {1}-structure on P, we show that its automorphism group can be turned into a Banach-Lie group acting smoothly on P provided the Lie algebra of infinitesimal automorphisms consists of complete vector fields. As a consequence we show that the automorphism group of a connected geodesically complete affine Banach manifold M can be turned into a Banach-Lie group acting smoothly on M.