Linear connections for reproducing kernels on vector bundles

Abstract

We construct a canonical correspondence from a wide class of reproducing kernels on infinite-dimensional Hermitian vector bundles to linear connections on these bundles. The linear connection in question is obtained through a pull-back operation involving the tautological universal bundle and the classifying morphism of the input kernel. The aforementioned correspondence turns out to be a canonical functor between categories of kernels and linear connections. A number of examples of linear connections including the ones associated to classical kernels, homogeneous reproducing kernels and kernels occurring in the dilation theory for completely positive maps are given, together with their covariant derivatives.

Appendix: On linear connections and their pull-backs

For the reader’s convenience, we record here some general facts on connections on Banach fiber bundles that are needed in the present paper. We use [16] and [17] as the main references, but we will also provide proofs for some results where we were unable to find convenient references in the literature.

1.1 Connections on fiber bundles

Definition 6.1

Let $\varphi :M\rightarrow Z$ a fiber bundle and consider both vector bundle structures of the tangent space $TM$:

• $\tau _M:TM\rightarrow M$, the tangent bundle of the total space $M$.

• $T\varphi :TM\rightarrow TZ$, the tangent map of $\varphi$.

A connection on the bundle $\varphi :M\rightarrow Z$ is a smooth map $\Phi :TM\rightarrow TM$ with the following properties:

• (i) $\Phi \circ \Phi =\Phi$;

• (ii) the pair $(\Phi ,\mathrm{id}_M)$ is an endomorphism of the bundle $\tau _M:TM\rightarrow M$;

• (iii) for every $x\in M$, if we denote $\Phi _x:=\Phi |_{T_xM}:T_xM\rightarrow T_xM$, then we have $\mathrm{Ran}\,(\Phi _x)=\mathrm{Ker}\,(T_x\varphi )$, so that we get an exact sequence

  $$\begin{aligned} 0\rightarrow H_xM\hookrightarrow T_xM\mathop {\longrightarrow }\limits ^{\Phi _x} T_xM \mathop {\longrightarrow }\limits ^{T_x\varphi } T_{\varphi (x)}Z \rightarrow 0. \end{aligned}$$

Here $H_xM:=\mathrm{Ker}\,(\Phi _x)$ is a closed linear subspace of $T_xM$ called the space of horizontal vectors at $x\in M$. Similarly, the space of vertical vectors at $x\in M$ is ${\mathcal {V}}_xM:=\mathrm{Ker}\,(T_x\varphi )$. Then we have the direct sum decomposition $T_xM=H_xM\oplus {\mathcal {V}}_xM$, for every $x\in M$ (cf. [16, subsect. 37.2]).

We consider in this paper two special types of connections.

• (1) If $\varphi :M\rightarrow Z$ is a principal bundle with structure group $G$ acting to the right on $M$ by

  $$\begin{aligned} (x,g)\mapsto \mu _g(x)=\mu (x,g),\quad M\times G\rightarrow M \end{aligned}$$

  then a connection $\Phi$ on $\varphi :M\rightarrow Z$ is called principal whenever it is $G$-equivariant, that is,

  $$\begin{aligned} T( \mu _g)\circ \Phi =\Phi \circ T( \mu _g) \end{aligned}$$

  for all $g\in G$ (cf. [16, subsect. 37.19]).

• (2) If $\varphi :M\rightarrow Z$ is a vector bundle then a connection $\Phi$ on $\varphi :M\rightarrow Z$ is called linear if the pair $(\Phi , \mathrm{id}_{TZ})$ is an endomorphism of the vector bundle $T\varphi :TM\rightarrow TZ$ (i.e., if $\Phi$ is linear on the fibers of the bundle $T\varphi$); see [16, subsect. 37.27].

We are interested in particular in vector bundles constructed out of principal ones. Recall how they appear: Let $\pi :{\mathcal {P}}\rightarrow Z$ be a principal Banach bundle with the structure Banach-Lie group $G$ and the action $\mu :{\mathcal {P}}\times G\rightarrow {\mathcal {P}}$. Assume that $\rho :G\rightarrow {\mathcal {B}}(\mathbf{E})$ is a smooth representation of $G$ by linear operators on a Banach space $\mathbf{E}$, and denote by

$$\begin{aligned} {[(p,e)]}\mapsto \pi (p),\quad \Pi :D={\mathcal {P}}\times _G\mathbf{E}\rightarrow Z \end{aligned}$$

the associated vector bundle (see [7, subsect. 6.5] and [16, subsect. 37.12]). Here ${\mathcal {P}}\times _G\mathbf{E}$ denotes the quotient of ${\mathcal {P}}\times \mathbf{E}$ with respect to the equivalence relation defined by

$$\begin{aligned} (\forall g\in G)\quad (p,e)\sim (\mu (p,g),\rho (g^{-1})e)=:\bar{\mu }(g)(p,e) \end{aligned}$$

whenever $(p,e)\in {\mathcal {P}}\times \mathbf{E}$, and we denote by $[(p,e)]$ the equivalence class of any pair $(p,e)$.

In this way, $\Pi :{\mathcal {P}}\times _G\mathbf{E}\rightarrow Z$ is a vector $G$-bundle.

Remark 6.2

Every connection on a principal bundle $\pi$ induces a linear connection on any vector bundle associated to $\pi$. A good reference for that induction procedure in infinite dimensions is Kriegl and Michor [16]. We will recall here the corresponding construction since we need it in order to describe specific induced connections (see for instance the comment prior to Theorem 2.2 above).

For a Banach-Lie group $G$ with the Lie algebra ${\mathfrak {g}}=T_{\mathbf{1}}G$ let $\lambda _g:G\rightarrow G,\,\lambda _g(h)=gh$ for all $g,h\in G$. Then the mapping $(g,X)\mapsto T_{\mathbf{1}}(\lambda _g)X$ is a diffeomorphism $G\times {\mathfrak {g}}\rightarrow TG$, and thus the tangent manifold $TG$ is endowed with structure of a semidirect product of groups $TG\equiv G \ltimes _{\mathrm{Ad}_{G}}{\mathfrak {g}}$ defined by the adjoint action of $G$ on ${\mathfrak {g}}$; see [16, Cor. 38.10]. The multiplication in the group $TG$ is given by

$$\begin{aligned} (g_1,X_1)(g_2,X_2)=(g_1 g_2,\mathrm{Ad}_{G}(g_2^{-1})X_1+X_2), \quad (g_1,g_2\in G; X_1,X_2\in {\mathfrak {g}}). \end{aligned}$$

Let $\pi :{\mathcal {P}}\rightarrow Z$ be a principal bundle with the structure group $G$ acting to the right by $\mu :{\mathcal {P}}\times G\rightarrow {\mathcal {P}}$. If $\rho :G\rightarrow {\mathcal {B}}(\mathbf E)$ is a smooth representation as above, then we can form the associated vector bundle $\Pi :D={\mathcal {P}}\times _{G}\mathbf{E}\rightarrow Z$.

For describing a connection induced on $\Pi$, one needs a specific description of the tangent space of the total space ${\mathcal {P}}\times _{G}\mathbf{E}$, and to this end one uses the fact that the tangent functor commutes with the construction of associated bundles. In fact, the tangent bundle $T\pi :T{\mathcal {P}}\rightarrow TZ$ is a principal bundle with the structure group $TG=G\ltimes _{\mathrm{Ad}_{G}}{\mathfrak {g}}$ and right action $T\mu :T{\mathcal {P}}\times TG\rightarrow T{\mathcal {P}}$ ([16, Th. 37.18(1)]). The representation $\rho$ gives a linear action $G\times \mathbf{E}\rightarrow \mathbf{E}$, and by computing the tangent map of that action it follows that the tangent map of the above representation can be viewed as the smooth representation $T\rho :G\ltimes _{\mathrm{Ad}_{G}}{\mathfrak {g}}\rightarrow {\mathcal {B}}(\mathbf{E}\oplus \mathbf{E})$, which is easily computed as

where the resulting matrix is to be understood as acting on vectors of $\mathbf{E}\oplus \mathbf{E}$ written in column form. Using the representation $T\rho$, the tangent bundle of the vector bundle $\Pi :D={\mathcal {P}}\times _{G}\mathbf{E}\rightarrow Z$ can be described as the vector bundle

$$\begin{aligned} \tau _D:TD=T{\mathcal {P}}\times _{G\ltimes _{\mathrm{Ad}_{G}}{\mathfrak {g}}}(\mathbf{E}\oplus \mathbf{E}) \rightarrow {\mathcal {P}}\times _{G}\mathbf{E}=D, \end{aligned}$$

which is associated to the principal bundle $T\pi :T{\mathcal {P}}\rightarrow TZ$ and is defined by

$$\begin{aligned} \tau _D:[(v_p,(f,h))]\mapsto [(p,f)] \qquad (v_p\in T_p{\mathcal {P}}; f,h\in \mathbf E) \end{aligned}$$

([16, Th. 37.18(4)]).

If now $\Phi :T{\mathcal {P}}\rightarrow T{\mathcal {P}}$ is a principal connection on the principal bundle $\pi :{\mathcal {P}}\rightarrow Z$, then the mapping $\Phi \times \mathrm{id}_{T\mathbf{E}}:T{\mathcal {P}}\times T\mathbf{E}\rightarrow T{\mathcal {P}}\times T\mathbf{E}$ is $TG$-equivariant and factorizes through a map $\bar{\Phi }:T({\mathcal {P}}\times _G\mathbf{E})\rightarrow T({\mathcal {P}}\times _G\mathbf{E}) =T{\mathcal {P}}\times _{TG}T\mathbf{E}$. That is, there exists the commutative diagram

and $\bar{\Phi }$ is the connection induced by $\Phi$ on $T({\mathcal {P}}\times _G\mathbf{E})$; see [16, subsect. 37.24].

We now briefly recall the covariant derivatives (as in Vilms [34]) and we then provide a proposition needed in the specific computations carried out in the present paper.

Let $\Pi :D\rightarrow Z$ be a vector bundle with a linear connection $\Phi :TD\rightarrow TD$. Let ${\mathcal {V}}D=\mathrm{Ker}\,(T\Pi )$ ($\subseteq TD$) be the vertical part of the tangent bundle $\tau _D:TD\rightarrow D$. A useful description of ${\mathcal {V}}D$ can be obtained by considering the fibered product $D\mathop {\times }\limits _Z D:=\{(x_1,x_2)\in D\times D\mid \Pi (x_1)=\Pi (x_2)\}$ along with the natural maps $r_j:D\mathop {\times }\limits _Z D\rightarrow D,\,r_j(x_1,x_2)=x_j$ for $j=1,2$. Define for every $(x_1,x_2)\in D\mathop {\times }\limits _Z D$ the path $c_{x_1,x_2}:\mathbb {R}\rightarrow D,\,c_{x_1,x_2}(t)=x_1+tx_2$. Then it is easily seen that we have a well-defined diffeomorphism $\varepsilon :D\mathop {\times }\limits _Z D\rightarrow {\mathcal {V}}D,\quad \varepsilon (x_1,x_2)=\dot{c}_{x_1,x_2}(0)\in T_{x_1} D$, which is in fact an isomorphism between the vector bundles $r_1:D\mathop {\times }\limits _Z D\rightarrow D$ and $\tau _D\vert _{{\mathcal {V}}D}:{\mathcal {V}}D\rightarrow D$. We then get a natural mapping $r:=r_2\circ \varepsilon ^{-1} :{\mathcal {V}}D\rightarrow D$ and the pair $(r,\Pi )$ is a homomorphism of vector bundles from $\tau _D\vert _{{\mathcal {V}}D}:{\mathcal {V}}D\rightarrow D$ to $\Pi :D\rightarrow Z$.

Next let $\Omega ^1(Z,D)$ the space of locally defined smooth differential 1-forms on $Z$ with values in the bundle $\Pi :D\rightarrow Z$, hence the set of smooth mappings $\eta :\tau _Z^{-1}(Z_\eta )\rightarrow D$, where $\tau _Z:TZ\rightarrow Z$ is the tangent bundle and $Z_\eta$ is a suitable open subset of $Z$, such that for every $z\in Z_\eta$ we have a bounded linear operator $\eta _z:=\eta \vert _{T_zZ}:T_zZ\rightarrow D_z=\Pi ^{-1}(z)$. (So the pair $(\eta ,\mathrm{id}_Z)$ is a homomorphism of vector bundles from the tangent bundle $\tau _D\vert _{Z_\eta }$ to the bundle $\Pi$.) For the sake of simplicity we actually omit the subscript $\eta$ in $Z_\eta$, as if the forms were always defined throughout $Z$; in fact, the algebraic operations are performed on the intersections of the domains, and so on. Similarly, we let $\Omega ^0(Z,D)$ be the space of locally defined smooth sections of the vector bundle $\Pi$.

Definition 6.3

The covariant derivative for the linear connection $\Phi$ is the linear mapping $\nabla :\Omega ^0(Z,D)\rightarrow \Omega ^1(Z,D)$, defined for every $\sigma \in \Omega ^0(Z,D)$ by the composition

$$\begin{aligned} \nabla \sigma :TZ\mathop {\longrightarrow }\limits ^{T\sigma } TD\mathop {\longrightarrow }\limits ^{\Phi } {\mathcal {V}}D\mathop {\longrightarrow }\limits ^{r}D \end{aligned}$$

that is, $\nabla \sigma =(r\circ \Phi )\circ T\sigma$. (The composition $r\circ \Phi$ is the so-called connection map.)

Proposition 6.4

Let $\Pi :D\rightarrow Z$ and $\widetilde{\Pi }:\widetilde{D}\rightarrow \widetilde{Z}$ be vector bundles endowed with the linear connections $\Phi$ and $\widetilde{\Phi }$, with the corresponding covariant derivatives $\nabla$ and $\widetilde{\nabla }$, respectively. Assume that $\Theta =(\delta ,\zeta )$ is a homomorphism of vector bundles from $\Pi$ into $\widetilde{\Pi }$ such that $T\delta \circ \Phi =\widetilde{\Phi }\circ T\delta$. If $\sigma \in \Omega ^0(Z,D)$ and $\widetilde{\sigma }\in \Omega ^0(\widetilde{Z},\widetilde{D})$ are such that $\delta \circ \sigma =\widetilde{\sigma }\circ \zeta$, then $\delta \circ \nabla \sigma =\widetilde{\nabla }\widetilde{\sigma }\circ T\zeta$.

Proof

First let $r:{\mathcal {V}}D\rightarrow D$ and $\widetilde{r}:{\mathcal {V}}\widetilde{D}\rightarrow \widetilde{D}$ be the natural mappings and note that

$$\begin{aligned} \delta \circ r=\widetilde{r}\circ T\delta . \end{aligned}$$

(6.1)

In order to see why this equality holds true we need the mapping $\delta \mathop {\times }\limits _Z\delta :D\mathop {\times }\limits _Z D\rightarrow \widetilde{D}\mathop {\times }\limits _{\widetilde{Z}}\widetilde{D}$ given by $(\delta \mathop {\times }\limits _Z\delta )(x_1,x_2)=(\delta (x_1),\delta (x_2))$, which is well defined since $\widetilde{\Pi }\circ \delta =\zeta \circ \Pi$. Since $\delta$ is fiberwise linear, it follows that with the notation of Definition 6.3 we have $\delta \circ c_{x_1,x_2}=c_{\delta (x_1),\delta (x_2)}:\mathbb {R}\rightarrow \widetilde{D}$ for all $(x_1,x_2)\in D\mathop {\times }\limits _Z D$. By taking the velocity vectors at $0\in \mathbb {R}$ for these paths we get $T\delta \circ \varepsilon =\widetilde{\varepsilon }\circ (\delta \mathop {\times }\limits _Z\delta ):D\mathop {\times }\limits _Z D\rightarrow T\widetilde{D}$. Therefore $\widetilde{\varepsilon }^{-1}\circ T\delta =(\delta \mathop {\times }\limits _Z\delta )\circ \varepsilon ^{-1}$ and then, by using the obvious equality $\widetilde{r}_2\circ (\delta \mathop {\times }\limits _Z\delta )=\delta \circ r_2:D\mathop {\times }\limits _Z D\rightarrow \widetilde{D}$, we get

$$\begin{aligned} \widetilde{r}\circ T\delta =\widetilde{r}_2\circ \widetilde{\varepsilon }^{-1}\circ T\delta =\widetilde{r}_2\circ (\delta \mathop {\times }\limits _Z\delta )\circ \varepsilon ^{-1} =\delta \circ r_2\circ \varepsilon ^{-1} =\delta \circ r \end{aligned}$$

hence (6.1) holds true.

We now come back to the proof of the assertion. By using (6.1) and the equality $T\delta \circ \Phi =\widetilde{\Phi }\circ T\delta$ we get

$$\begin{aligned} \delta \circ (r\circ \Phi )=\widetilde{r}\circ T\delta \circ \Phi =(\widetilde{r}\circ \widetilde{\Phi })\circ T\delta . \end{aligned}$$

(6.2)

On the other hand we have $\delta \circ \sigma =\widetilde{\sigma }\circ \zeta$, and therefore $T\delta \circ T\sigma =T\widetilde{\sigma }\circ T\zeta$. We then get

$$\begin{aligned} \widetilde{\nabla }\widetilde{\sigma }\circ T\zeta =(\widetilde{r}\circ \widetilde{\Phi })\circ T\widetilde{\sigma }\circ T\zeta =(\widetilde{r}\circ \widetilde{\Phi })\circ T\delta \circ T\sigma =\delta \circ (r\circ \Phi )\circ T\sigma =\delta \circ \nabla \sigma \end{aligned}$$

where the next-to-last equality follows by (6.1), and this completes the proof. $\square$

1.2 Pull-backs of connections

Pull-backs of connections on various types of finite-dimensional bundles have been studied in several papers; see for instance [18, 22, 23, 31, 32]. We now establish a result (Proposition 6.6) that belongs to that circle of ideas and is appropriate for the applications we want to make in infinite dimensions. Unlike the descriptions of the pull-backs of connections that we were able to find in the literature, the method provided here is more direct in the sense that it requires neither the connection map, nor any connection forms, nor the covariant derivative, but rather the connection itself. The intertwining property of the covariant derivatives follows at once (Corollary 6.8).

We will need the following simple lemma.

Lemma 6.5

Let $T:{{\mathcal {E}}}\rightarrow \widetilde{{\mathcal {E}}}$ be a continuous (conjugate-)linear operator between two Banach spaces ${\mathcal {E}}$ and $\widetilde{{\mathcal {E}}}$. Let us assume that there are two closed linear subspaces ${\mathcal {F}}\subset {\mathcal {E}}$ and $\widetilde{{\mathcal {F}}}\subset \widetilde{{\mathcal {E}}}$ such that:

• (i) the operator $T$ induces a (conjugate-)linear isomorphism $T|_{{\mathcal {F}}}:{\mathcal {F}}\rightarrow \widetilde{{\mathcal {F}}}$;

• (ii) $\mathrm{Ran}\,\widetilde{P}=\widetilde{{\mathcal {F}}}$, for some projection $\widetilde{P}:\widetilde{{\mathcal {E}}}\rightarrow \widetilde{{\mathcal {E}}}$.

Then there exists a unique projection $P\in \mathrm{End}\,({\mathcal {E}})$ such that $\mathrm{Ran}\,P={\mathcal {F}}$ and ${\widetilde{P}}\circ T=T\circ P$.

Proof

Existence: Define

$$\begin{aligned} P:=(T|_{{\mathcal {F}}})^{-1}\circ \widetilde{P}\circ T\in \mathrm{End}\,({\mathcal {E}}) \end{aligned}$$

(6.3)

It is clear that $\mathrm{Ran}\,P={\mathcal {F}}$ and moreover $P|_{{\mathcal {F}}}=\hbox {id}_{{\mathcal {F}}}$, hence $P\circ P=P$. Then the commutativity of the diagram is satisfied by the construction of $P$.

Uniqueness: Assume that $P_1\in \mathrm{End}\,({\mathcal {E}})$ is another operator satisfying the properties of the statement. Then for arbitrary $x\in {\mathcal {E}}$ we have $T(P_1x)=\widetilde{P} Tx=T(Px)$. Since $P_1x,Px\in {\mathcal {F}}$ and $T|_{{\mathcal {F}}}:{\mathcal {F}}\rightarrow \widetilde{{\mathcal {F}}}$ is an isomorphism, it then follows that $P_1x=Px$. Thus $P_1=P$ and we are done. $\square$

Proposition 6.6

Let $\varphi :M\rightarrow Z$ and $\widetilde{\varphi }:\widetilde{M}\rightarrow \widetilde{Z}$ be fiber bundles modeled on Banach spaces, and let $\Theta =(\delta ,\zeta )$ be a bundle homomorphism, that is, the diagram

is commutative and both $\delta$ and $\zeta$ are smooth. In addition, assume that for every $s\in Z$ the mapping $\delta$ induces a diffeomorphism of the fiber $M_s:=\varphi ^{-1}(\{s\})$ onto the fiber $\widetilde{M}_{\zeta (s)} :=\widetilde{\varphi }^{-1}(\zeta (s))$.

Then for every connection $\widetilde{\Phi }$ on the bundle $\widetilde{\varphi }:\widetilde{M}\rightarrow \widetilde{Z}$ there exists a unique connection $\Phi$ on the bundle $\varphi :M\rightarrow Z$ such that the diagram

is commutative.

Moreover, if both $\varphi :M\rightarrow Z$ and $\widetilde{\varphi }:\widetilde{M}\rightarrow \widetilde{Z}$ are principal (vector) bundles, the pair $\Theta =(\delta ,\zeta )$ is a homomorphism of principal bundles (or of vector bundles, and in this case $\delta$ can be linear) bundles, and $\widetilde{\Phi }$ is a principal (linear or conjugate-linear) connection, then so is $\Phi$.

Proof

We have for every $x\in M$ the continuous operator $T_x\delta :T_xM\rightarrow T_{\delta (x)}\widetilde{M}$ (which is either linear or conjugate-linear), and also the relations $T_x(M_{\varphi (x)}) = {\mathcal {V}}_x\hookrightarrow T_xM$ and

$$\begin{aligned} T_{\delta (x)}(\widetilde{M}_{\zeta (\varphi (x))})=T_{\delta (x)}(\widetilde{M}_{\widetilde{\varphi }(\delta (x))}) ={\mathcal {V}}_{\delta (x)}\widetilde{M}\hookrightarrow T_{\delta (x)}\widetilde{M}. \end{aligned}$$

Since $\delta |_{M_{\Pi (x)}}:M_{\Pi (x)}\rightarrow \widetilde{M}_{\zeta (\Pi (x))}$ is a diffeomorphism by hypothesis, it thus follows that the operator $T_x\delta$ induces a (conjugate-)linear isomorphism ${\mathcal {V}}_xM\rightarrow {\mathcal {V}}_{\delta (x)}\widetilde{M}$. Now Lemma 6.5 shows that there exists a unique idempotent operator $\Phi _x:T_xM\rightarrow T_xM$ such that $\mathrm{Ran}\,\Phi _x={\mathcal {V}}_xM$ and $(T_x\delta )\circ \Phi _x=\widetilde{\Phi }_{\delta (x)}\circ (T_x\delta )$. In fact it is defined by

$$\begin{aligned} \Phi _x:=(T_x\delta |_{{\mathcal {V}}_x M})^{-1}\circ \widetilde{\Phi }_{\delta (x)}\circ T_x\delta \qquad (x\in M). \end{aligned}$$

If we put together the operators $\Phi _x$ with $x\in M$, we get the map $\Phi :TM\rightarrow TM$ we were looking for. What still remains to be done is to check that $\Phi$ is smooth. Since this is a local property, we may assume that both bundles $\Pi$ and $\widetilde{\Pi }$ are trivial. Let $S$ and $\widetilde{S}$ be their typical fibers, respectively. Then $M=Z\times S$ and $\widetilde{M}=\widetilde{Z}\times \widetilde{S}$, hence $TM=TM\times TS$ and $T\widetilde{M}=T\widetilde{Z}\times T\widetilde{S}$. The fact that $\widetilde{\Phi }$ is a connection means that for every $(\widetilde{z},\widetilde{k})\in \widetilde{Z}\times \widetilde{S}$ we have an idempotent operator $\widetilde{\Phi }_{(\widetilde{z}, \widetilde{k})}$ on $T_{\widetilde{z}}\widetilde{Z}\times T_{\widetilde{k}}\widetilde{S}$ with $\mathrm{Ran}\,\widetilde{\Phi }_{(\widetilde{z}, \widetilde{k})}=\{0\}\times T_{\widetilde{k}}\widetilde{S}$.

Moreover, we have the smooth map $\delta :Z\times S\rightarrow \widetilde{Z}\times \widetilde{S}$ for which there exists a smooth map $d:Z\times S\rightarrow \widetilde{S}$ such that $\delta (z, k)=(\zeta (z),d(z,k))$ for all $z\in Z$ and $k\in S$. The hypothesis that $\delta$ is a fiberwise diffeomorphism is equivalent to the fact that for every $z\in Z$ we have the diffeomorphism $d(z,\ \cdot \ ):S\rightarrow \widetilde{S}$. It follows by (6.3) that, for arbitrary $(z,k)\in Z\times S$,

$$\begin{aligned} \Phi _{(z,k)}=T_k(d(z,\ \cdot \ ))^{-1} \circ \widetilde{\Phi }_{\delta (z,k)} \circ T_{(z,k)}\delta \in \mathrm{End}\,(T_z Z\times T_k S) \end{aligned}$$

which clearly shows that $\Phi :TZ\times TS\rightarrow TZ\times TS$ is smooth. (Note that the smoothness of the mapping $(z,k)\mapsto T_k(d(z,\ \cdot \ ))^{-1}$ is ensured by the fact that we are working with Banach manifolds.)

The remainder of the proof is straightforward. $\square$

Definition 6.7

In the setting of Proposition 6.6 we say that the connection $\Phi$ is the pull-back of the connection $\widetilde{\Phi }$ and we denote $\Phi =\Theta ^*(\widetilde{\Phi })$.

Corollary 6.8

Let $\Pi :D\rightarrow Z$ and $\widetilde{\Pi }:\widetilde{D}\rightarrow \widetilde{Z}$ be vector bundles. Assume that $\Theta =(\delta ,\zeta )$ is a homomorphism of vector bundles from $\Pi$ into $\widetilde{\Pi }$ such that for every $s\in Z$ the mapping $\delta$ induces an isomorphism of the fiber $D_s:=\Pi ^{-1}(\{s\})$ onto the fiber $\widetilde{D}_{\zeta (s)}:=\widetilde{\Pi }^{-1}(\zeta (s))$. Consider any linear connection $\widetilde{\Phi }$ on the vector bundle $\Pi$ and its pull-back $\Phi =\Theta ^*(\widetilde{\Phi })$ on the vector bundle $\widetilde{\Pi }$, with the corresponding covariant derivatives $\nabla$ and $\widetilde{\nabla }$, respectively. If we have $\sigma \in \Omega ^0(Z,D)$ and $\widetilde{\sigma }\in \Omega ^0(\widetilde{Z},\widetilde{D})$ such that $\delta \circ \sigma =\widetilde{\sigma }\circ \zeta$, then $\delta \circ \nabla \sigma =\widetilde{\nabla }\widetilde{\sigma }\circ T\zeta$.

Proof

Use Propositions 6.6 and 6.4. $\square$