Let $p:E \to B$ be a smooth vector bundle of rank $n$ over a manifold $B$ and we identify $B$ with the image of the corresponding zero section. For $b\in B$ denote by $E_b = p^{-1}(b)$ the fiber over $b$.
Let also $\mathrm{End}_{B}(E)$ be space of smooth vector bundle morphisms $h:E \to E$ "fixed on $B$", i.e. such that $p\circ h = p$, and $\mathrm{Aut}_{B}(E) \subset \mathrm{End}_{B}(E)$ be the subset consisting of vector bundle automorphisms.
Associating to each $h\in\mathrm{End}(E)$ the collection of linear maps $\{h|_{E_b}: E_b \to E_b\}_{b\in B}$, one can identify $\mathrm{End}(E)$ with the space of smooth section of the vector bundle $q:E^{*} \otimes E = \mathrm{Hom}(E,E) \to B$.
Moreover, let $\mathrm{Iso}(E,E) \subset \mathrm{Hom}(E,E)$ be the subset consisting of all "non-degenerate linear self-maps of all fibers". Then the restriction of the projection $q:\mathrm{Iso}(E,E) \to B$ is a $GL(\mathbb{R},n)$-bundle. Then under above correspondence $\mathrm{Aut}_{B}(E)$ corresponds to the sections of $q$ whose image is contained in $\mathrm{Iso}(E,E)$.
Now, as in the case when $B$ is just a point, one might define a "fiberwise exponential map" $$\exp: \mathrm{Hom}(E,E) \to \mathrm{Iso}(E,E).$$
(A) I am looking for the paper describing the construction of such exponential map.
Moreover, let $s_0: B \to \mathrm{Hom}(E,E)$ be the zero zection of $q$ and $s_1:B \to \mathrm{Iso}(E,E)$ be the section of $q$ corresponding to the identity automorphism $\mathrm{id}_{E}$ of $E$.
(B) I also need the fact that $\exp$ yields a diffeomorphism of some neighborhood the image $s_0(B)$ of $s_0$ onto some neighborhood the image $s_1(B)$ of $s_1$.
I can define this construction explicitly and prove what I need (see below just for completeness), but I feel that it must be written somewhere.
Construction of the fiberwise exponential map.
Consider the Whitney sum of $2n$ copies of $p$, so take a product of $2n$ times of $p$: $$p^{2n}: E^{2n} \to B^{2n},$$ let $\Delta_B = \{ (b,\ldots,b) \mid b\in B\} \subset B^{2n}$ be the diagonal, and $\oplus_{2n} E := (p^{2n})^{-1}(\Delta_B)$. Then the projection $p^{2n}: \oplus_{2n} E \to \Delta_B \equiv B$ is the mentioned above Whitney sum of $2n$ copies of $p$.
Thus each element of $\oplus_{2n} E$ is an ordered $2n$-tuple $(v_1,\ldots,v_n, w_1,\ldots,w_{2n})$ of elements of $E$ belonging to the same fiber of $p$. We can write that $2n$-tuple as $(v,w)$, where $v=(v_1,\ldots,v_n)$ and $w=(w_1,\ldots,w_n)$
Then there is a natural right action of $GL(\mathbb{R},n)$ on $\oplus_{2n} E$, by $(v,w)A = (vA, wA)$. This action is also fiberwise. Moreover, the following subsets $$ X = \{ (v,w) \in \oplus_{2n} E \mid \text{$v$ is linearly independent} \}, $$ $$ Y = \{ (v,w) \in \oplus_{2n} E \mid \text{both $v$ and $w$ are linearly independent} \} $$ are invariant with respect to that action. Then $\mathrm{Hom}(E,E):=X/GL(\mathbb{R},n)$ and $\mathrm{Iso}(E,E):=Y/GL(\mathbb{R},n)$ are the correponding quotients.
Notice that for each $(v,w)\in X$ there exists a unique matrix $A_{v,w}$ such that $w=vA_{v,w}$ (since $v$ is linearly independent $n$-tuple). Moreover, $(v,w)\in Y$ iff $A_{v,w}$ is non-degenerate.
Also $A_{vB, wB} = B^{-1} A_{v,w}B$. Indeed, $(vB, wB) = (vB, vA_{v,w} B) = (vB, v B B^{-1} A_{v,w} B)$.
Define now the map $\exp': X \to Y$ by $\exp'(v,w) = (v, v e^{A_{v,w}})$. This map commutes with the action of $GL(\mathbb{R},n)$. Indeed, \begin{align} \exp'(v,w) B &= (v, v e^{A_{v,w}})B = (vB, v e^{A_{v,w}}B) = (vB, v B B^{-1} e^{A_{v,w}}B) = \\ &= (vB, v B e^{B A_{v,w} B^{-1}}) = (vB, v B e^{A_{vB,wB}}) = \exp'(vB,wB). \end{align}
Hence it yields a well-defined map of the quotients $\exp:\mathrm{Hom}(E,E)\to \mathrm{Iso}(E,E)$ which is the required fiberwise exponential map.
Could you please provide a reference concerning that map. Thank you in advance.
