Smoothing continuous families of formal immersions

Encode the data as $a(p,x)=e(f_p(x))$ and $B(p,x)=d{e}_{f_p(x)}{F}_{p,x}\in \mathrm{Hom}(T_{x}M,{\mathbb{R}}^b)$. They have jointly continuous $x$-derivatives by [F1]. All their source fibres remain $T_{x}M$. In the vector bundle $V={\mathbb{R}}^b\oplus \mathrm{Hom}(TM,{\mathbb{R}}^b)$ over $M$, the set $\mathcal{O}$ of pairs $(a,B)$ with $a\in T$ and $d{r}_{a}B$ injective is open: in local frames injectivity is the nonvanishing of an appropriate minor. The original pairs lie in $\mathcal{O}$ because $d{r}_{a}B=B$. Equip $V$ with a smooth fibre norm using [L2].

Write $P=P_0\times [0,1{]}^d$. Embed the compact boundaryless $P_0$ in Euclidean space and use its smooth tubular projection; in the interval factors use coordinate clamps. Together these give a continuous projection $\pi$ from a Euclidean neighbourhood of $P$ to $P$, independent of $x$, and fix $P$ pointwise. Extend $a,B$ by evaluation at $\pi (p)$. On every compact source set, the extension retains all jointly continuous $x$-derivatives. Convolution in the Euclidean parameter coordinates therefore gives smooth data $a_{\delta },B_{\delta }$ on a neighbourhood of $P$ times $M$; $B_{\delta }(p,x)$ is the integral in the single vector space $\mathrm{Hom}(T_{x}M,{\mathbb{R}}^b)$, so it is intrinsically defined and is smooth in any local source frame.

Choose a countable locally finite smooth partition $({\lambda }_j)$ on $M$ with compact supports $K_j$ inside chart domains, using [L2]. Compactness of $P\times K_j$ and openness of $\mathcal{O}$ give a positive constant $c_j$ such that perturbations of norm less than $c_j$ of the original pair over this product remain in $\mathcal{O}$. Put $\epsilon (x)=\frac{1}{2}{\sum }_j{\lambda }_j(x)c_j>0$. At each $x$ this is at most half the largest active $c_j$; that $c_j$ is valid at $x$, so the fibre ball of radius $\epsilon (x)$ about every original pair at $x$ lies in $\mathcal{O}$. Choose ${\delta }_j>0$ so that the error of $(a_{{\delta }_j},B_{{\delta }_j})$ on $P\times K_j$ is less than $\frac{1}{2}{\min }_{K_j}\epsilon$, which is positive by compactness. These are independent countably many choices. Define $a^{\mathrm{\prime }}={\sum }_j{\lambda }_j{a}_{{\delta }_j}$ and $B^{\mathrm{\prime }}={\sum }_j{\lambda }_j{B}_{{\delta }_j}$. They are smooth by local finiteness, and their combined error at $(p,x)$ is less than $\epsilon (x)\mathrm{/}2$. Empty supports are omitted.

Choose a smooth $\chi :P\to [0,1]$ equal to one near $Q$ and supported in $W$, using [L2] on $P_0\times {\mathbb{R}}^d$ and restricting to $P$; for $Q=\mathrm{\varnothing }$ take $\chi =0$. Replace $(a^{\mathrm{\prime }},B^{\mathrm{\prime }})$ by $(\bar{a},\bar{B})=\chi (a,B)+(1-\chi )(a^{\mathrm{\prime }},B^{\mathrm{\prime }})$. This pair is smooth: the original data are smooth where $\chi$ is supported, and the other summand is smooth everywhere. For $s\in [0,1]$ set $(a_s,B_s)=(1-s)(a,B)+s(\bar{a},\bar{B})$. Its error from the original pair is still less than $\epsilon (x)\mathrm{/}2$, so every pair lies in $\mathcal{O}$.

Set $f_s=e^{-1}r(a_s)$ and $F_s=d(e^{-1}{)}_{r(a_s)}d{r}_{a_s}{B}_s$. Each $F_s$ is a linear injection from the original $T_{x}M$ to $T_{f_s(p,x)}N$; no approximation has moved $x$. At $s=0$ these recover $(f,F)$, and at $s=1$ they are jointly smooth in $(p,x)$. Every source derivative is jointly continuous in $(s,p,x)$, because the sums are locally finite and the retraction is smooth. By [F1] this is a continuous homotopy through formal immersions. It is fixed where $\chi =1$, proving the relative assertion. If $P$ or $M$ is empty the unique family already suffices.