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Given a submanifold $(S,\phi)$ of a manifold $M$, how do we find the subspace of $T_pM$ that is equal to $T_pS$ for $p\in \phi(S)$. I know how to do it for level sets. Is there a way for general submanifolds?

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  • $\begingroup$ $T_pS$ is the subspace consisting of all $X \in T_pM$ such that whenever $f$ is a function which vanishes on $S$, then $Xf = 0$. $\endgroup$ – Santiago Canez Feb 6 '16 at 4:23
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For a point $p \in \phi(S)$, put local coordinates $(x_1,\dots,x_n)$ on $S$ around the point $\phi^{-1}(p)$. Then the tangent space $T_{\phi^{-1}(p)}S$ is spanned by $\frac{\partial}{\partial x_1}, \dots, \frac{\partial}{\partial x_n}$. The tangent space $T_p\phi(S)$ as a subset of $T_pM$ should then be the image of $T_{\phi^{-1}(p)}S$ under the pushforward map $\phi_*$. That is, the space you are looking for should be the linear span of $\phi_* \frac{\partial}{\partial x_1}, \dots, \phi_* \frac{\partial}{\partial x_n}$ in $T_pM$.

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  • $\begingroup$ This looks good. Thanks. Do you think this is reason why we define submanifold map to be injective and an immersion? $\endgroup$ – Mathew George Feb 12 '16 at 8:31
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    $\begingroup$ @Matt I think it is! $\endgroup$ – Gustavo Mar 19 '16 at 13:25

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