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I came across this exercise in Warner's Foundations of Differential Manifolds and Lie Groups,

Let $N \in M$ be a submanifold. Let $\gamma \colon (a,b) \to M$ be a $C^{\infty}$ curve such that $\gamma(a,b) \subset N$. Show that it is not necessarily true that $\dot \gamma (t) \in N_{\gamma (t)}$ for each $t \in (a,b)$.

I'm having trouble trying to find an example of such a curve. Could someone give me an example or show me the way? Either way thanks!

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what does $\dot{\gamma}(t)\in N_{\gamma(t)}$ mean? The thing on the left is a tangent vector. –  Michael Jul 11 '12 at 18:58
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Try with a figure eight in $\mathbb{R}^2$. Not every author would call that a "submanifold", though. –  Giuseppe Negro Jul 11 '12 at 19:10
    
Oh right, sorry... $N_{\gamma(t)} = T_{\gamma(t)}N$ the tangent space of $\gamma(t) \in N$, and $\dot \gamma(t)=d\gamma(\frac{d}{dt})$ is the tangent vector at $t$ to the curve $\gamma(t)$. By sub manifold the author means an injective immersion. @GiuseppeNegro Thanks, I'll look into the eight figure. –  Bruce Wayne Jul 11 '12 at 19:28
    
@Michael, this is exercise 16 on page 51. $M_m$ is the tangent space at $m \in M,$ defined as linear operators on germs of functions, page 12 Def. 1.14. On page 17, 1.23 (e) he defines $\dot{\gamma}.$ In the figures on page 23 he distinguishes three similar immersions, the one in the middle, a submanifold but not an imbedding. Warner would not call an immersed figure 8 a submanifold. On page 25, and again on page 29, he shows a noncompact submanifold figure 8, say $\gamma : (-\infty, \infty) \rightarrow \mathbb R^2,$ with $\gamma(0) = (0,0)$ and –  Will Jagy Jul 11 '12 at 19:29
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It is exactly as Giuseppe said. Suppose $\psi:M\to \mathbb R^2$ is an injective immersion with image a figure eight with singularity at the origin and two branches along the axes. Then $\psi(t)$ will run, say, tangent to the $x$- axis at time zero. Now there exists a curve $\gamma$ tangent to the $y$-axis, entirely contained in the figure eight, but whose velocity $\gamma'(0)\neq 0$ at time zero is along the $y$-axis.Then that velocity is not in $N_{\gamma(0)}=N_{(0,0)}=x$-axis. (To tell the truth, I don't like this definition of submanifold at all!) –  Georges Elencwajg Jul 11 '12 at 21:22
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