# Smooth bijection and tangent spaces

Let $$f:M\to N$$ be a smooth bijection between manifolds with same dimension. Do we necessarily have

$$df_p(T_pM)=T_{f(p)}N.$$

I think it is probably not true. But I can't give a counterexample...

• Is this question the same as "Are smooth bijections of manifolds of the same dimension submersions?" ? – Selene Auckland Jul 24 at 7:54

## 2 Answers

The map $$f:\mathbb{R}\to\mathbb{R}$$ given by $$f:x\mapsto x^3$$ is a smooth bijection but $$df_0(T_0\mathbb{R}) = 0\neq T_{f(0)}\mathbb{R}$$.

• Is this question the same as "Are smooth bijections of manifolds of the same dimension submersions?" please? – Selene Auckland Jul 24 at 7:52
• @SeleneAuckland Let me answer your question with a question: is the function $f$ in my answer a smooth bijection between two manifolds of the same dimension that is not a submersion? :) – Neal Jul 24 at 12:59
• Thanks. I guess, yes that $f$ is such. I was just wondering why OP didn't use the word "submersion". So it is the same question then? – Selene Auckland Jul 25 at 2:00

How about $$f: \mathbb{R} \to \mathbb{R}$$ by $$f(x)=x^3$$? Certainly smooth, certainly bijective. However, $$df_0$$ will map the $$1$$-dimensional tangent space to a point. The root issue is that a smooth bijection can still have critical points.