https://diffgeom.subwiki.org/w/index.php?action=history&feed=atom&title=Rank_of_Jacobian_is_lower_semicontinuousRank of Jacobian is lower semicontinuous - Revision history2026-04-17T15:47:21ZRevision history for this page on the wikiMediaWiki 1.41.2https://diffgeom.subwiki.org/w/index.php?title=Rank_of_Jacobian_is_lower_semicontinuous&diff=1145&oldid=prevVipul: 1 revision2008-05-18T19:51:21Z<p>1 revision</p>
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</td></tr></table>Vipulhttps://diffgeom.subwiki.org/w/index.php?title=Rank_of_Jacobian_is_lower_semicontinuous&diff=1144&oldid=prevVipul at 23:46, 16 January 20082008-01-16T23:46:45Z<p></p>
<p><b>New page</b></p><div>==Statement==<br />
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Let <math>M</math> and <math>N</math> be [[differential manifold]]s and <math>f:M \to N</math> be a [[smooth map]]. Define the following map:<br />
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<math>rk(Df):M \to \N_0</math><br />
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which sends <math>p \in M</math> to the rank of <math>Df:T_pM \to T_{f(p)}(N)</math>.<br />
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The function function <math>rk(Df)</math> is [[lower semicontinuous function to integers|lower semicontinuous]]. In other words, given any point <math>p</math>, there exists an open set <math>U</math> containing <math>p</math> such that <math>rk(Df)(q) \ge rk(Df)(p)</math> for all <math>q \in U</math>.<br />
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Equivalently, for any <math>r \in \N_0</math>, the set of points in <math>M</math> for which <math>rk(Df) \ge r</math>, is an open subset.</div>Vipul