论文标题
几乎是riemannian歧管不满足$ \ mathsf {cd} $条件
Almost-Riemannian manifolds do not satisfy the $\mathsf{CD}$ condition
论文作者
论文摘要
Lott-Sturm-Villani曲率维度条件$ \ MATHSF {CD}（K，N）$为度量计量空间提供了一个合成的概念，可以使曲率从下方界限为$ K $，并且从上面限制的尺寸为$ n $。 Juillet证明，对于任何$ k \ in \ Mathbb r $ in \ Mathbb r $和$ n \ in（1，\ infty）$，一大批\ sr歧管不满足$ \ mathsf {cd}（k，n）$条件。但是，他的结果并不能涵盖几乎是里曼尼的歧管的情况。在本文中，我们解决了在此设置中反驳$ \ mathsf {cd} $条件的问题，从而提供了一种新策略，该策略使我们能够与$ \ mathsf {cd} $条件的$ 1 $二维版本相矛盾。特别是，我们证明，对于任何$ k，n \ in（1，\ iffty）$，我们证明了$ 2 $几乎是差不多的差异歧管和强烈常规的几乎经常的歧管歧管，并不满足$ \ m athsf {cd}（k，n）$条件。
The Lott-Sturm-Villani curvature-dimension condition $\mathsf{CD}(K,N)$ provides a synthetic notion for a metric-measure space to have curvature bounded from below by $K$ and dimension bounded from above by $N$. It was proved by Juillet that a large class of \sr manifolds do not satisfy the $\mathsf{CD}(K,N)$ condition, for any $K\in\mathbb R$ and $N\in(1,\infty)$. However, his result does not cover the case of almost-Riemannian manifolds. In this paper, we address the problem of disproving the $\mathsf{CD}$ condition in this setting, providing a new strategy which allows us to contradict the $1$-dimensional version of the $\mathsf{CD}$ condition. In particular, we prove that $2$-dimensional almost-Riemannian manifolds and strongly regular almost-Riemannian manifolds do not satisfy the $\mathsf{CD}(K,N)$ condition for any $K\in\mathbb R$ and $N\in(1,\infty)$.
