Hi!

By a complex quasi-affine variety i mean the complement of an affine algebraic variety with respect to another algebraic variety, more precisely a quasi-affine algebraic variety is

$$V= V_{1}\setminus V_{2}$$
with
$$V_{1}:=V\left(P_{1},\ldots, P_{k}\right\)$$
$$V_{2}:=V\left(Q_{1},\ldots, Q_{l}\right\)$$
and $P_{i},Q_{j}\in \mathbb{C}\left\[x_{1},\ldots, x_{n}\right\]$. Clearly, if $V_{2}=\emptyset$, then $V=V_{1}$ is an affine variety. Suppose $V$ is smooth and connected, so it is moreover a smooth real not compact and connected differentiable manifold.
Is it possible to find a smooth embedding

$$i: V\hookrightarrow \mathbb{R}^{N}$$
for $N$ large enough s.t. $i\left(V\right)$ has an $\varepsilon$-tubular neighborhood $\mathcal{T}_{\varepsilon}V\subset \mathbb{R}^{N}$? By an $\varepsilon$-tubular neighborhood i mean
$$\mathcal{T}_{\varepsilon}V=\bigcup_{x\in i\left(V \right)}B_{\mathbb{R}^{N}}\left(x,\varepsilon\right)\qquad \varepsilon>0$$
together a smooth minimal point projection $\pi$
$$\pi: \mathcal{T}_{\varepsilon}V\rightarrow i\left( V \right)$$

that associates to any point $y\in \mathcal{T}_{\varepsilon}V$ its closest point $x\in i\left(V \right)$.

The case i have in mind $V$ is the variety of $k\times k$ complex matrices of rank exactly $r$, i don't know if this helps...

Thank you in advance!