The parallel transport map over reductive homogeneous space
Masahiro Morimoto
TL;DR
The paper extends the theory of parallel transport maps to reductive homogeneous spaces using a natural torsion-free connection, proving that the lifted parallel-transport map is an affine submersion with a well-behaved horizontal distribution and an alternating fundamental tensor. It establishes compactness of the fibers’ shape operators and develops two notions of regularized mean curvature for affine Fredholm submanifolds in Hilbert spaces, showing that fibers are minimal in both senses and that lifted submanifolds preserve base-curvature properties. The work generalizes prior results for affine symmetric spaces and connects infinite-dimensional submanifold theory with affine differential geometry, offering tools to study equifocal-type structures in this broader setting. It also provides foundational links between horizontal-mean curvature behavior and the geometry of the parallel transport map, with potential implications for isoparametric-like phenomena in infinite dimensions.
Abstract
We show that the parallel transport map over a reductive homogeneous space with natural torsion-free connection becomes an affine submersion with horizontal distribution. This generalizes one of the main results in the author's previous paper in the case of affine symmetric spaces. We also prove the compactness of the shape operators of the submanifold lifted by the parallel transport map. This improves a previous result by the author and generalizes some results of Terng-Thorbergsson and of Koike. Furthermore we propose two definitions for the regularized mean curvatures of affine Fredholm submanifolds in Hilbertable spaces and discuss their relations to the parallel transport map. In particular, each fiber of the parallel transport map over a reductive homogeneous space is shown to be minimal in both senses.