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arXiv · 1801.05626

Non-Embedding Theorems of Nilpotent Lie groups and Sub-Riemannian Manifolds

Yonghong Huang·Shanzhong Sun
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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.

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Yonghong Huang, Shanzhong Sun. 2018-01-17. Non-Embedding Theorems of Nilpotent Lie groups and Sub-Riemannian Manifolds. https://arxiv.org/abs/1801.05626

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