#### Problem 7

An isomorphism between vector spaces is a linear bijection. Prove that if $f:M\rightarrow N$ is a diffeomorphism between manifolds then $T_{x}f:T_{x}M\rightarrow T_{f(x)}N$ is an isomorphism for each $x\in M.$

#### Solution:

The following theorem is useful in proof.

## Theorem

Let $f$ and $g$ be a smooth maps of manifolds. If $h=g\circ f$, then

(1)If $f$ is a diffeomorphism, then its inverse $f^{-1}$ is also a smooth function. If we put $g=f^{-1}$ in above theorem, we get;

(2)Since, if $h$ is an identity map then $T_{x}h$ is also an identity map, the claim follows;

(3)Last statement proves that $T_{x}f$ has an inverse map $(T_{x}f)^{-1}=T_{f(x)}f^{-1}$, therefore $T_{x}f$ is a bijection.

Lastly, linearity can be seen from the matrix of the linear transformation definition as follows;

## Definition

For coordinates $\phi =(x^{1},...,x^{m})$ at $p\in M$, and $\psi=(y^{1},...,y^{n})$ at $f(p)\in N$, then;

(4)Thus, $T_{x}f$ is a linear bijection, i.e. isomorphism between vector spaces $T_{x}M$ and $T_{x}N.$

After correcting several typographical errors, Scott agrees!