injective function on manifold

Question

Let $f: \mathbb{R}^n \to \mathbb{R}^n$ be a smooth function and $M$ be a smooth manifold of $\mathbb{R}^n$. Assume that $Df(x)v \neq 0$ for all $v$ being tangent to $M$ at $x$ and for all $x$ in $M$. Can we say that $f$ is locally injective on $M$?

You certainly mean $Df(x)v =0$, not $Df(x)v \neq 0$ – Georges Elencwajg Oct 16 '13 at 21:16 — Georges Elencwajg, Oct 16 '13 at 21:16

score 1 · Answer 1 · answered Oct 16 '13 at 20:35

If $M$ is a connected submanifold: Yes. (Otherwise of course no, as any disconnected open subset of $\mathbb R^n$ shows). Let $x,y \in M$ and $\gamma \colon [0,1] \to M$ a smooth map with $\gamma(0) =x$ and $\gamma(1) = y$. Then $$ (f \circ \gamma)'(t) = Df\bigl(\gamma(t)\bigr)\gamma'(t) = 0$$ as $\gamma'(t)$ is tangent to $M$ for each $t$, $\gamma$ being a path in $M$. So $f\circ \gamma$ is constant, hence $$ f(x) = (f\circ \gamma)(0) = (f\circ \gamma)(1) = f(y). $$

injective function on manifold

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