# If $\\f:\ U\to\ R^N$ is a submersion, Prove that $g$ is $C^k$

If $$\\f:\ U\to\ R^N$$ is a submersion of class $$C^k$$ and $$g:f(U)\to\ R^M$$ is such that $$g\circ f : U\to\ R^M$$ is $$C^k$$ then $$g$$is $$C^k$$

In my attempt I know that $$D_{f(p_0)}$$ is onto, $$p_0 \in U$$ and jacobian matrix $$N\times(N+p)$$, $$J_{f(p_0)}$$ where $$N$$ columns are linearly independent in some order, take $$T:\{1,\dots,N\} \to\ \{1,\dots,N+p\}$$ injective such that the first $$N$$ columns $$C_{T_{1}},\ldots,C_{T_{N}}$$ are linearly independent and let $$L:R^{N+p}\to\ R^{N+p}$$ a linear isomorphism such that $$L(e_i)=C_{T_i}$$ for $$i:1,\ldots,N$$ the rest element in any place, where $$e_i$$ are basis for $$R^{N+p}$$. Take $$q_0$$ the only element such that $$L(q_0)=p_0$$ and $$B=L^{-1}(A)$$ where $$A$$ is neightborhood of $$p_0$$ and $$h=f\circ L: B\to\ R^{N}$$ satisfy that is $$C^k$$ and $$D_h(q_0)= D_f(p_0)L$$ is onto and its first columns of jacobian matrix linearly independent are situated in the first place. Let $$F:A=L(B)\subset R^{N+p}\to\ R^{N+p}$$ such that $$F(x,y)=(f(x,y),y)$$ . Then $$F$$ is differentiable and $$\det (D_F(p_0))\neq 0$$ and we can apply inverse theorem, hence exist open sets $$p_0\in V$$ and $$F(p_0)\in W\subset R^{N+p}$$ such that $$F:V\to\ W$$ is a diffeomorphism but $$F( x,0)=(f(x,0),0)$$ and $$g\circ F(x,0)=h(x,0)$$ is $$C^k$$, then $$g=h\circ F^{-1}$$ I am not sure about this,because is not like for all elements on the domain of $$g$$. I will appreciate any help or hints or other method for solving the problem, please and thanks.

• Thank you Bernard – weymar andres Jun 16 at 19:12