"Symplectic leaves"의 두 판 사이의 차이
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| − | + | ‹william›The symplectic leaves are equivalence relations <math>x \tilde y</math> if and only if <math>x</math> can be connected to <math>y</math> be a piece-wise Hamiltonian path | |
| + | 27/02/201123:12:31‹william›<math>x \sim y</math>27/02/201123:14:18‹william›Let <math>D</math> be a degenerate distribution27/02/201123:14:49‹william›this means that for every point <math>x \in M</math>, <math>D_x</math> is a subset of <math>T_x M</math>27/02/201123:14:58‹william›subset = subspace27/02/201123:15:18‹william›distribution normally means that <math>D_x</math> is constant rank27/02/201123:15:25‹william›and <math>D_x</math> is spanned by vector fields27/02/201123:15:58‹william›which means that for every <math>x</math> there is vector fields <math>X_1,\ldots,X_r</math> locally defined around <math>x</math> such that <math>X_1(x),\ldots,X_r(x)</math> span <math>D_x</math>27/02/201123:16:19‹william›and <math>X_1(y),\ldots,X_r(y)</math> lie in <math>D_y</math> for all <math>y</math> where they are defined27/02/201123:18:02‹william›a foliation of <math>D</math> is an immersed manifold <math>A</math> of <math>M</math> with <math>TA = D</math>27/02/201123:19:10‹william›Let <math>M^{dis}</math> be the manifold with underlying set <math>M</math> and the discrete topology27/02/201123:20:09‹william›<math>M^{dis}</math> is an immersed manifold for <math>D = M \times 0</math>27/02/201123:20:36‹william›<math>M = \R^2</math>27/02/201123:20:46‹william›<math>D = \R^2 \times \R</math>27/02/201123:24:17‹william›the foliation is the map <math>\bigcup_{\R} \R \arr \R^2</math>27/02/201123:24:25‹william›the foliation is the map <math>\bigcup_{\R} \R \rightarrow \R^2</math> | ||
2011년 2월 28일 (월) 15:01 판
‹william›The symplectic leaves are equivalence relations \(x \tilde y\) if and only if \(x\) can be connected to \(y\) be a piece-wise Hamiltonian path 27/02/201123:12:31‹william›\(x \sim y\)27/02/201123:14:18‹william›Let \(D\) be a degenerate distribution27/02/201123:14:49‹william›this means that for every point \(x \in M\), \(D_x\) is a subset of \(T_x M\)27/02/201123:14:58‹william›subset = subspace27/02/201123:15:18‹william›distribution normally means that \(D_x\) is constant rank27/02/201123:15:25‹william›and \(D_x\) is spanned by vector fields27/02/201123:15:58‹william›which means that for every \(x\) there is vector fields \(X_1,\ldots,X_r\) locally defined around \(x\) such that \(X_1(x),\ldots,X_r(x)\) span \(D_x\)27/02/201123:16:19‹william›and \(X_1(y),\ldots,X_r(y)\) lie in \(D_y\) for all \(y\) where they are defined27/02/201123:18:02‹william›a foliation of \(D\) is an immersed manifold \(A\) of \(M\) with \(TA = D\)27/02/201123:19:10‹william›Let \(M^{dis}\) be the manifold with underlying set \(M\) and the discrete topology27/02/201123:20:09‹william›\(M^{dis}\) is an immersed manifold for \(D = M \times 0\)27/02/201123:20:36‹william›\(M = \R^2\)27/02/201123:20:46‹william›\(D = \R^2 \times \R\)27/02/201123:24:17‹william›the foliation is the map \(\bigcup_{\R} \R \arr \R^2\)27/02/201123:24:25‹william›the foliation is the map \(\bigcup_{\R} \R \rightarrow \R^2\)