Is an injective smooth map an immersion? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T07:35:12Z http://mathoverflow.net/feeds/question/15317 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/15317/is-an-injective-smooth-map-an-immersion Is an injective smooth map an immersion? Daniel Barter 2010-02-15T05:36:36Z 2011-02-06T18:38:21Z <p>Suppose $M$ and $N$ are smooth manifolds. An immersion is a smooth map $f: M \rightarrow N$ whose pushforward is injective at each point.</p> <p>Is a smooth injective map an immersion? </p> <p>We can actually simplify the question further. </p> <p>Suppose $f : M \rightarrow N$ is a smooth injective map. Suppose $(U, \phi)$ and $(V, \psi)$ are smooth charts for $M$ and $N$ respectively. Fix $p \in U$. Then</p> <p><code>$$f_\ast = ( \psi^{-1}\circ \psi \circ f \circ \phi^{-1} \circ \phi)_{\ast} = (\psi^{-1})_\ast \circ (\psi \circ f \circ \phi^{-1})_\ast \circ \phi_\ast$$</code></p> <p>As <code>$\phi$</code> and <code>$\psi$</code> are diffeomorphisms, $\phi_\ast$ and <code>$(\psi^{-1})_\ast$</code> are linear isomorphisms. </p> <p>Therefore, if <code>$(\psi \circ f \circ \phi^{-1})_\ast$</code> is injective then $f_\ast$ is injective.</p> <p>This shows that if every smooth injective map between open subsets of euclidean space is an immersion, then every smooth injective map between smooth manifolds is an immersion.</p> http://mathoverflow.net/questions/15317/is-an-injective-smooth-map-an-immersion/15318#15318 Answer by Kevin Lin for Is an injective smooth map an immersion? Kevin Lin 2010-02-15T05:55:36Z 2010-02-15T10:17:50Z <p>The answer is no. </p> <p>Here's an example --- which is based on an example I gave in my multivariable calculus discussion section recently!! :-)</p> <p>Let M be (0,pi) with coordinate t. Let N be R^2 with coordinates x and y. Define a map x(t)=sin(t), y(t)=sin(t) for t in (0,pi/2] and x(t)=2-cos(t-pi/2), y(t)=2-cos(t-pi/2) for t in (pi/2,pi). This map is smooth and injective on points but the derivative is zero at t=pi/2.</p> <p>Edit: Ok, I got a bit too excited about using multivariable calculus. As algori mentions, t^3 is a simpler example. Or just taking one of the coordinate functions in my original example works too. There's no need for multivariable calculus. </p> http://mathoverflow.net/questions/15317/is-an-injective-smooth-map-an-immersion/54528#54528 Answer by Giuseppe Tortorella for Is an injective smooth map an immersion? Giuseppe Tortorella 2011-02-06T15:38:30Z 2011-02-06T18:38:21Z <p>For a smooth injective map $f:M\rightarrow N$, there is only an obstruction to be $f$ an immersion, and it is that its rank is not constant.</p>