Suppose we have a diffeomorphism $f:{\mathbb{S}}^n\_{+}\to\mathbb{S}^n$ of class $C^1$ of the closed upper hemisphere onto a submanifold of $\mathbb{S}^n$ with boundary.



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> **Question.** Is it possible to extend it to a diffeomorphism $F:\mathbb{S}^n\to\mathbb{S}^n$?
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I believe that when $n=2$ this should follow from the smooth Schoenflies theorem, but still some work is necessary. I think I know how to do, but I did not write a rigorous proof.


From what I understand when $n\geq 3$, the generalized Schoenflies theorem allows us to extend $f$ to a homeomorphism, but that is much less than extending to a diffeomorphism.


I expect that the answer might depend on $n$ ($n\geq 3$, $n=4$, $n\geq 5$).


I would greatly appreciate it if you could provide references where I cold find relevant theorems (if there are any), including the case $n=2$. I need references for a proper citation in my paper.


**Edit:** This question is strictly related to another post: [Gluing two diffeomorphisms together](https://mathoverflow.net/q/38498/121665)



Extending diffeomorphisms
The answer is positive and follows from Corollary 2 in


*Palais, Richard S.*, [**Extending diffeomorphisms**](http://dx.doi.org/10.2307/2032968), Proc. Am. Math. Soc. 11, 274-277 (1960). [ZBL0095.16502](https://zbmath.org/?q=an:0095.16502).


(A caveat: Palais is not entirely clear about the degree of smoothness he allows, he just says "differentiable." However, I think, it works for $C^k$-smooth map for every $k>0$.)


Applying Corollary 2 in the case of maps of closed $n$-balls $B^n$ to $S^n$, one obtains that if $\phi, \psi: B^n \to S^n$ are smooth embeddings, then there exists a diffeomorphism $F: S^n\to S^n$ such that $\phi=F\circ \psi$ on $B^n$. Now, take $\psi$ to be the identity embedding $B^n\to S^n$ (where $B^n$ is a hemisphere). Then it follows that $F$ is the desired extension of $\phi$.


PS: For some reason I had Palais' paper open in my browser for a week before you posted the question. :)


