A diffeomorphism between two smooth manifolds \(M\) and \(N\) is a smooth bijective map \(F\colon M\to N\) that has a smooth inverse.
We say \(M\) and \(N\) are diffeomorphic if there exists a diffeomorphism between them.
Examples
- Smooth coordinate maps are diffeomorphisms.
Remark
- By the chain rule , if \(F\colon M\to N\) is diffeomorph, then \(dF_p\) is an isomorphism, and \[ (dF_p)^{-1}=d(F^{-1})_{F(p)}. \]
- If \(F\) is a diffeomorphism , then \(dF\colon TM\to TN\) is also a diffeomorphism, and \((dF)^{-1}=d(F^{-1})\).