Good afternoon, I Have problems with these exercises, I wrote the two questions because the second one needs the result of the previus one. They are:
- Let $n,m, k=n+m$ positive integers, so we can identify $\mathbb R^k\cong\mathbb R^n\times \mathbb R^m$ e $\mathbb H^k\cong\mathbb R^n\times \mathbb H^m$.
Let $W$ an open subset of $\mathbb H^k$ e $F:W\rightarrow\mathbb R^n$ a differentiable map.
Let $y\in Reg(F)\cap Reg(F|_{\partial \mathbb H^k\cap W})$, then
$\partial F^{-1}(y)=F^{-1}(y)\cap W\partial\cap \mathbb H^k$.
Let $DF=[D_1F \quad D_2F]$, according the descomposition $\mathbb R^k\cong\mathbb R^n\times \mathbb R^m$, and denote $\pi_2:\mathbb H^k\rightarrow \mathbb H^m$, the projection $\pi_2(x_1,x_2)=x_2$.
Prove that if $D_1F(x_1,x_2)$ is surjective at every point $(x_1,x_2)\in F^{-1}(y)$. then $\pi_2|_{F^{-1}(y)}:F^{-1}(y)\rightarrow\mathbb H^m$ is a local diffemorphism.
I am sorry, but I got confussed, I do not know even how to begin.
2) Let $N$ a $k-$manifold, $X$ a compact $(k+1)-$manifold in $\mathbb R^N$, and $F:X\rightarrow N$ a differentiable map. Let $y\in Reg(F)\cap Reg(F|_{\partial X})$ and let $J$ a connected component of $F^{-1}$. Suponha $\partial J\neq\emptyset$, fix a diffeomorphism $c:[0,1]\rightarrow J$, choose a base $\{w_1,\ldots,w_k\}$ de $T_yN$ and let $t_0\in[0,1]$ arbitrary. Fixe $h:U\rightarrow h(U)$ a local parametrization of $X$ in $c(t_0)$ and denote by $<.,.>$ the usual inner product of $\mathbb R^N$.
Consider $\Phi:c^{-1}(U)\times \mathbb R^{k+1}\rightarrow T_yN\times\mathbb R$ defined by
$\Phi(t,u)=(d(F\circ h)_{h^{-1}\circ c(t)}\cdot u,<dh_{h^{-1}\circ c(t)},c'(t)>).$
Show that $(w,0)\in Reg(\Phi)\cap Reg(\Phi|_{\partial (c^{-1}(U)\times \mathbb R^{k+1})})$
para todo $w\in T_yN$. Now, use the previous question to construct differentiable maps
$v_i:[0,1]\rightarrow\mathbb R^k$ only determined by $v_i(t)\in T_{c(t)}X$, $dF_{c(t)}\cdot v_i(t)=w_i$ e $<v_i(t),c'(t)>=0$. Show also that is possibel modify the $v_i$ close to $t=0,1$ to get $v_i(j)\in T_{c(j)}\partial X$, $j=0,1$.