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Let $X$ be a connected $n$-dimensional differentiable manifold, and $f: X \rightarrow X$ a differentiable map such that $f \circ f = f$. Now I have to show that the image $f(X)$ is a submanifold of $X$, using the following sub-results:

  • $\mathrm{rk}_p f \leq \mathrm{rk}_{f(p)} f$ for all $p \in X$

  • The rank of $f$ is constant along $f(X)$

  • The rank of $f$ is constant in an open neighbourhood of $f(X)$

I have managed to show that $f(X)$ is a submanifold when the above holds (using the constant rank theorem), but I don't see how to prove the above results.

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Recall that $d(f\circ g) = df\circ dg$. In particular, at a point $p$, we have $d_p(f\circ g) = d_{g(p)}f\circ d_pg$. Apply this to the case where $f=g$ to prove the first result, and prove the second result using this.

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