Non-Embedding Theorems of Nilpotent Lie groups and Sub-Riemannian Manifolds
Differential Geometry
2020-02-20 v3
Abstract
We prove that there do not exist quasi-isometric embeddings of connected nonabelian nilpotent Lie groups equipped with left invariant Riemannian metrics into a metric measure space satisfying the RCD(0,N), with N > 1. In fact, we can prove that a subRiemannian manifold whose generic degree of nonholonomy is not smaller than 2 can not be biLipschitzly embedded in any Banach space with the Radon-Nikodym property. We also get that every regular sub-Riemannian manifold do not satisfy the CD(K,N) with N > 1. We also prove that the subRiemannian manifold is infinitesimally Hilbert space.
Keywords
Cite
@article{arxiv.1801.05626,
title = {Non-Embedding Theorems of Nilpotent Lie groups and Sub-Riemannian Manifolds},
author = {Yonghong Huang and Shanzhong Sun},
journal= {arXiv preprint arXiv:1801.05626},
year = {2020}
}
Comments
To appear in Frontiers of Mathematics in China