Frontiers of Mathematics in China >
Non-embedding theorems of nilpotent Lie groups and sub-Riemannian manifolds
Received date: 13 Nov 2019
Accepted date: 18 Feb 2020
Published date: 15 Feb 2020
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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.
Yonghong HUANG , Shanzhong SUN . Non-embedding theorems of nilpotent Lie groups and sub-Riemannian manifolds[J]. Frontiers of Mathematics in China, 2020 , 15(1) : 91 -114 . DOI: 10.1007/s11464-020-0823-3
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