https://diffgeom.subwiki.org/w/index.php?title=Submanifold_(differential_sense)&feed=atom&action=history Submanifold (differential sense) - Revision history 2019-11-15T02:36:12Z Revision history for this page on the wiki MediaWiki 1.29.2 https://diffgeom.subwiki.org/w/index.php?title=Submanifold_(differential_sense)&diff=1463&oldid=prev Vipul: 1 revision 2008-05-18T20:10:16Z <p>1 revision</p> <table class="diff diff-contentalign-left" data-mw="interface"> <tr style='vertical-align: top;' lang='en'> <td colspan='1' style="background-color: white; color:black; text-align: center;">← Older revision</td> <td colspan='1' style="background-color: white; color:black; text-align: center;">Revision as of 20:10, 18 May 2008</td> </tr><tr><td colspan='2' style='text-align: center;' lang='en'><div class="mw-diff-empty">(No difference)</div> </td></tr></table> Vipul https://diffgeom.subwiki.org/w/index.php?title=Submanifold_(differential_sense)&diff=1462&oldid=prev Vipul at 20:47, 13 January 2008 2008-01-13T20:47:33Z <p></p> <p><b>New page</b></p><div>==Definition==<br /> <br /> Let &lt;math&gt;M&lt;/math&gt; be a [[differential manifold]]. A '''submanifold''' of &lt;math&gt;M&lt;/math&gt; can be viewed as the following data: An abstract [[differential manifold]] &lt;math&gt;N&lt;/math&gt; along with a [[smooth map]] &lt;math&gt;f&lt;/math&gt; from &lt;math&gt;N&lt;/math&gt; to &lt;math&gt;M&lt;/math&gt; such that:<br /> <br /> # The map &lt;math&gt;f&lt;/math&gt; is an [[immersion]]; in other words, the induced map &lt;math&gt;(Df)_p&lt;/math&gt; on the tangent space at any point &lt;math&gt;p \in M&lt;/math&gt; is injective<br /> # The map is injective i.e. &lt;math&gt;f(p) = f(q) \implies p = q&lt;/math&gt;<br /> # The map is a homeomorphism to its image<br /> <br /> Note that when &lt;math&gt;N&lt;/math&gt; is compact, the third condition is redundant, because [[tps:Injection from compact to Hausdorff implies embedding|any injective map from a compact space to a Hausdorff space is an embedding]].<br /> <br /> To complete the definition, we need to observe that a submanifold is completely determined, upto diffeomorphism, by its set-theoretic image in the manifold. In other words, if &lt;math&gt;f_1:N_1 \to M&lt;/matH&gt; and &lt;math&gt;f_2:N_2 \to M&lt;/math&gt; are submanifolds with the same set-theoretic image, then there is a diffeomorphism &lt;math&gt;g:N_1 \to N_2&lt;/math&gt; such that &lt;math&gt;f_1 = f_2 \circ g&lt;/math&gt;.</div> Vipul