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If $\varphi:U\subset \mathbb{R}^n \to \mathbb{R}^m$ is $C^1$, let $\mathrm{T}\varphi:\mathrm{T}U \to \mathrm{T}R^m$ be its tangent map. The inverse function theorem tells us that if $\ker(\mathrm{T}\varphi(x))$ is zero, $\varphi$ is injective in some neighborhood of $x$. If the kernel is non-zero, what can we say about $\varphi$ near $x$ provided we know the kernel? In particular, can we say anything about curves through $x$ whose tangents belong to this kernel?

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    You can consider the Constant Rank Theorem: http://en.wikipedia.org/wiki/Derivative_rule_for_inverses#Constant_rank_theorem2010-09-10
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    You should be more concrete about what you want to know. Books have been written and careers built upon the study of singularities of smooth maps, so unless you are more specific it is hard to know what you are after!2010-09-10
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    @Mariano: I don't know if mmm is in this position, but it can be difficult for someone unfamiliar with a particular field to know if their question related to it is a simple one with a concrete answer, or a fundamental problem upon which books have been written. Perhaps the best response in such a case is to leave a comment saying "Your question is a fundamental problem in [name of field]. A good reference is [citation of textbook]."2010-09-10
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    @Rahul, but s/he is familiar with what he wants to know! "What can we say about X?" is quite non-descriptive about what he wants to know.2010-09-10
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    @Pierre-Yves, I would be happy if you would add that as an answer.2010-09-11
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    @mmm Thanks for your invitation. I've just posted an answer. I hadn't done it before because I hadn't received the notification of you comment.2010-09-11

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