n维流形,拓扑空间<math>\mathcal</math>在满足以下条件时,称<math>\mathcal</math>为<math>m</math>维流形,即 <math>\mathcal</math>为豪斯多夫空间,.流形定义 拓扑空间mathmathcalmath在满足以下条件时,称mathmathcalmath为mathmmath维流形,即 mathmathcalmath为豪斯多夫空间, 对于任意一点mathpinmathcalmath,存在包含mathpmath的mathmmath维坐标邻域math(U,varphi)math. 可微流形定义 设mathrgeq 1math的自然数或者为mathinftymath,拓扑空间mathmathcalmath被称为是mathmmath维mathmathbf^rmath可微流形,如果, mathmathcalmath为豪斯多夫空间 mathmathcalmath被m维坐标邻域所覆盖,换句话说,存在mathmathcalmath的mathmmath维坐标邻域族mathleft{(U_alpha,varphi_alpha)right}_{alphain A}math,使得mathmathcal=cup_{alpha in A} U_alphamath 满足mathU_alpha cap U_{beta}neq phimath的任意mathalpha,betain Amath,坐标转换 mathvarphi_betacdot varphi_alpha^: varphi_alpha (U_alphacap U_beta) to varphi_beta (U_alphacap U_beta)math 为mathmathbf^rmath映射. 当mathr=0math时,mathmathbf^0math流形称为是拓扑流形;当mathr=inftymath时,mathmathbf^inftymath流形称为是光滑流形.