Studying analysis on manifolds, I have found, in the proof of the existence of tubular neighborhoods, a reference to theorem 3.1.2 in "Topologie algebrique et theorie des faisceaux" of Godement.
Without going through the machinery of the sheaves, at least now, is it possible to bypass the Godement's result?
And, if yes, what is an accessible (not sheaf-theoretic) route?
This is the initial setting:
$J:N\rightarrow M$ is a smooth embedding.
$\pi:E\rightarrow N$ is a vector bundle, and $s_0:N\rightarrow E$ is the zero section of $\pi$.
$\psi:U\rightarrow M$ is a smooth map from a neighborhood $U$ of $s_0(N)$ in $E.$
$\psi$ is a local diffeomorphism in each point of $s_0(N),$ and $\psi\circ s_0=j$.
At this point there is a reference to the argument of Godement in order to prove that:
(*)There exists a neighborhood $V$ of $s_0(N)$ in $U$ such that $\psi|_V$ is a diffeomorphism.
What is the argument (differential geometric, not sheaf-theoretic) in order to conclude (*)?
Added by Mariano: I now have a copy in my hands. Theorem 3.1.2 reads (my translation):
Let $$0\to\mathscr L'\to \mathscr L\to\mathscr L''\to0$$ be a short exact sequence of sheaves of abelian groups. If $\mathscr L'$ is flasque, then for all open sets $U$ there is a short exact sequence $$0\to\mathscr L'(U)\to \mathscr L(U)\to\mathscr L''(U)\to0$$
He remarks that we therefore have a short exact sequence of pre-sheaves.
Best Answer
In the finite-dimensional setting, it's possible to construct tubular neighborhoods without anything like Godement's lemma. Many sources simply rely on a point-set topology argument that's based on the same idea as Godement's lemma (to be precise, I'm talking about the argument on p. 109 of Lang's book Differential and Riemannian Manifolds, which he says follows Godement). I'll explain another approach.
The idea is to use a Riemannian metric on the manifold $M$, which also induces a Riemannian metric on $TM$ (viewed as a manifold in its own right). The geodesic distance then gives a (topological) metric on $TM$. If $Y\subset M$ is a (not necessarily closed) submanifold, then a simple metric geometry argument can then be used to find a neighborhood of the zero section of $N(Y)$ (thought of as the perpendicular complement of $TY$ inside $TM$) on which the exponential map is injective. The key fact about the exponential map $f$ is that every point in the zero section of $N(Y)$ has a neighborhood on which $f$ is a diffeomorphism onto an open subset of $M$. (Edit: Note that in the finite dimensional setting, $N(Y)$ is automatically a locally trivial vector bundle. This does not seem to be the case for arbitrary infinite dimensional Riemannian manifolds, as discussed here: Orthogonal complements in Hilbert bundles. Hence the discussion that follows does not work in as great generality as arguments based on Godement's lemma.)
The general metric geometry fact is this:
Proof. For each $y\in Y$, $f(B_{\epsilon (y)/2} (y, D))$ is open in $X$, hence contains $B_{\epsilon'(y)} (y, X)$, for some $\epsilon'_y < \epsilon_y/4$ (remember that $f(y) = y$). Now consider the inverse image $Z_y$ of $B_{\epsilon'_y} (y, X)$ under the restriction of $f$ to $B_{\epsilon (y)/2} (y, D)$. Since $f$ is a homeomorphism when restricted to this ball, $Z_y$ is open as a subset of $D$. Now I claim that $f$ is injective on $D' = \bigcup_{y\in Y} Z_y$. Say $f(z_1) = f(z_2) = y_0$ with $z_1 \in Z_{y_1}$ and $z_2\in Z_{y_2}$, and assume $\epsilon_{y_1} \geq \epsilon_{y_2}$. Then we have $$d(z_2, y_1) \leq d(z_2, y_2) + d(y_2, y_0) + d(y_0, y_1) < \epsilon_{y_2}/2 + \epsilon'_{y_2} + \epsilon'_{y_1} $$ $$< \epsilon_{y_1}/2 + \epsilon_{y_2}/4 + \epsilon_{y_1}/4 \leq \epsilon_{y_1}$$ (for the second inequality, note that by definition, $y_0 = f(z_i) \in f(Z_{y_i}) \subset B_{\epsilon'_{y_i}} (y_i, X)$ for $i=1, 2$). So $z_2$ and $z_1$ both lie in $B_{\epsilon_{y_1}} (y_1, D)$, and since $f$ is injective on this ball we have $z_1 = z_2$.
This argument is useful for the construction of equivariant tubular neighborhoods in certain infinite-dimensional settings. See http://arxiv.org/abs/1006.0063; I just updated it so I guess the new version will show up tomorrow. The equivariant version of the above argument is in Proposition 2.3 or 2.4, depending on the version.