Non-embedding theorems of nilpotent Lie groups and sub-Riemannian manifolds

doi:10.1007/s11464-020-0823-3

Front. Math. China

2020, Vol. 15

Issue (1) : 91-114 https://doi.org/10.1007/s11464-020-0823-3

RESEARCH ARTICLE

Non-embedding theorems of nilpotent Lie groups and sub-Riemannian manifolds

Yonghong HUANG^1,³(

), Shanzhong SUN^1,²

¹. Department of Mathematics, Capital Normal University, Beijing 100048, China
². Academy for Multidisciplinary Studies, Capital Normal University, Beijing 100048, China
³. Qiannan Preschool Education College for Nationalities, Guiding 558000, China

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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 curvaturedimension condition RCD(0;N) with N 2 R and N>1: In fact, we can prove that a sub-Riemannian manifold whose generic degree of nonholonomy is not smaller than 2 cannot be bi-Lipschitzly embedded in any Banach space with the Radon-Nikodym property. We also get that every regular sub-Riemannian manifold do not satisfy the curvature-dimension condition CD(K;N); where K;N 2 R and N>1: Along the way to the proofs, we show that the minimal weak upper gradient and the horizontal gradient coincide on the Carnot-Carathéodory spaces which may have independent interests.

Keywords Nilpotent Lie group curvature-dimension condition bi-Lipschitz embedding sub-Riemannian manifold

Corresponding Author(s): Yonghong HUANG

Issue Date: 09 March 2020

Cite this article:

Yonghong HUANG,Shanzhong SUN. Non-embedding theorems of nilpotent Lie groups and sub-Riemannian manifolds[J]. Front. Math. China, 2020, 15(1): 91-114.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-020-0823-3
https://academic.hep.com.cn/fmc/EN/Y2020/V15/I1/91

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