通过不可压缩欧拉方程得到的黎曼流形光滑等距嵌入的动力学-拓扑障碍
微分几何
2023-09-14 v4 偏微分方程分析
摘要
我们获得了带边黎曼流形存在等距嵌入的一个动力学-拓扑障碍：若的第一实同调非平凡，若其基本群的中心平凡，且若等距嵌入到维数至少为的欧氏空间中，则该等距嵌入必违反某一动力学、与动能相关的条件（定义1.1中的“刚性同伦扩张性质”）。论证的动机来自流体力学中具有指定初态与终态构型的不可压缩欧拉方程。
引用
@article{arxiv.2006.15423,
title = {A dynamical--topological obstruction for smooth isometric embeddings of Riemannian manifolds via incompressible Euler equations},
author = {Siran Li},
journal= {arXiv preprint arXiv:2006.15423},
year = {2023}
}
备注
The main Theorem 1.4 of this manuscript is flawed --- in fact, it is vacuously true, for the "rigid isotopy extension property'' can never hold for any isometric embedding. We shall correct this issue in a new preprint entitled "A smooth isotopy of volume-preserving diffeomorphisms on unit cube saving energy through extra dimensions'' (to be uploaded to ArXiv shortly)