Let $S$ be a smooth compact $k$-dimensional manifold.
Suppose we are given a $W^{2,2}$ immersion $F:S \to \mathbb{R}^d$, that is $F \in W^{2,2}(S,\mathbb{R}^d)$ and $dF$ is an immersion a.e.
Is there any sense in considering $dF(TS)$ as a topological vector bundle over $S$?
(we endow $dF(TS)$ with the subspace topology induced by $T\mathbb{R}^d$).
The point is that while the fibers $p \to T_pS$ change continuously (even smoothly), $p \to dF_p$ does not...
I wonder if there is anything intelligible one can say of this "weak bundle" (whose fibers are defined only almost everywhere).
Probably there is no reasonable way to consider this weak bundle as something similar to a true smooth bundle. I guess this is part of the reason why, when phrasing regularity theory (e.g weak harmonicity), we usually work directly with the smooth "ambient bundle" $T\mathbb{R}^d|_{S}$, and not with $dF(TS) \subseteq T\mathbb{R}^d|_{S}$.
From aesthetic point of view, I would prefer a more intrinsic version of regularity theory, but maybe this is too much to ask for.
Admittedly, I did not write here anything mindblowing, but I thought some answer is needed, so we could take away this question from the "unanswered list".