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

TL;DR

This paper establishes non-embedding results for nonabelian nilpotent Lie groups and sub-Riemannian manifolds into certain metric measure spaces, revealing fundamental geometric and analytic obstructions.

Contribution

It proves that nonabelian nilpotent Lie groups cannot be quasi-isometrically embedded into RCD(0,N) spaces and that sub-Riemannian manifolds with certain properties cannot be biLipschitzly embedded into Banach spaces with the Radon-Nikodym property, advancing understanding of their geometric rigidity.

Findings

Nonabelian nilpotent Lie groups cannot be quasi-isometrically embedded into RCD(0,N) spaces.

Sub-Riemannian manifolds with degree of nonholonomy ≥ 2 cannot be biLipschitzly embedded into Banach spaces with Radon-Nikodym property.

Regular sub-Riemannian manifolds do not satisfy CD(K,N) with N > 1.

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.