[proofplan] Existence is proved by constructing connections locally on trivialising charts, then gluing them via a partition of unity. The key observation is that the Leibniz rule — the sole defining property of a connection — is preserved under convex combinations, so averaging local connections produces a global one. The affine structure follows from computing the difference of two connections: the $C^\infty(M)$-linearity of the Leibniz remainder forces the difference to be a tensor in $\mathrm{End}(E) \otimes T^*M$, and conversely any such tensor can be added to a connection to produce a new connection. Together these two facts identify $\mathcal{A}_E$ as a non-empty affine space modelled on $\Omega^1(\mathrm{End}(E))$. [/proofplan] [step:Construct a local connection in each trivialising chart via a choice of connection matrix] Since $E \to M$ is a smooth vector bundle, there exists a trivialising open cover $\{U_\alpha\}_{\alpha \in I}$ of $M$ together with local frames $e^\alpha = (e^\alpha_1, \ldots, e^\alpha_r)$ for $E|_{U_\alpha}$, where $r = \operatorname{rank}(E)$. On each $U_\alpha$, choose any matrix $\theta_\alpha \in \Omega^1(U_\alpha; \mathrm{End}(\mathbb{R}^r))$ of 1-forms — for instance $\theta_\alpha = 0$ — and define \begin{align*} d_{A_\alpha}: \Gamma(E|_{U_\alpha}) &\to \Gamma(E|_{U_\alpha} \otimes T^*U_\alpha) \\ s &\mapsto ds + \theta_\alpha \cdot s, \end{align*} where $s = \sum_{k=1}^r s_k e^\alpha_k$ and $ds := \sum_{k=1}^r ds_k \otimes e^\alpha_k$. A direct computation shows that $d_{A_\alpha}(f s) = df \otimes s + f\, d_{A_\alpha} s$ for every $f \in C^\infty(U_\alpha)$, so $d_{A_\alpha}$ is a connection on $E|_{U_\alpha}$. [guided] To build a global connection we first need local connections to glue. Over a trivialising open $U_\alpha$ the bundle $E|_{U_\alpha}$ is isomorphic to $U_\alpha \times \mathbb{R}^r$ via a local frame $e^\alpha = (e^\alpha_1, \ldots, e^\alpha_r)$, and every section $s \in \Gamma(E|_{U_\alpha})$ has a unique expansion $s = \sum_k s_k e^\alpha_k$ with $s_k \in C^\infty(U_\alpha)$. The simplest connection on a trivial bundle is the flat connection $s \mapsto ds := \sum_k ds_k \otimes e^\alpha_k$. More generally, any matrix $\theta_\alpha \in \Omega^1(U_\alpha; \mathrm{End}(\mathbb{R}^r))$ of 1-forms defines \begin{align*} d_{A_\alpha}: \Gamma(E|_{U_\alpha}) &\to \Gamma(E|_{U_\alpha} \otimes T^*U_\alpha) \\ s &\mapsto ds + \theta_\alpha \cdot s. \end{align*} Why does this satisfy the Leibniz rule? For $f \in C^\infty(U_\alpha)$ and $s$ as above, \begin{align*} d_{A_\alpha}(fs) = d(fs) + \theta_\alpha \cdot (fs) = df \otimes s + f\, ds + f\, \theta_\alpha \cdot s = df \otimes s + f\, d_{A_\alpha} s, \end{align*} so $d_{A_\alpha}$ is a connection on $E|_{U_\alpha}$. Any choice of $\theta_\alpha$ will do — the simplest is $\theta_\alpha = 0$. [/guided] [/step] [step:Glue the local connections via a partition of unity using the Leibniz rule's convex stability] Choose a smooth partition of unity $\{\lambda_\alpha\}_{\alpha \in I}$ subordinate to $\{U_\alpha\}$, so $\lambda_\alpha \in C^\infty(M)$, $\operatorname{supp} \lambda_\alpha \subseteq U_\alpha$, $\sum_\alpha \lambda_\alpha \equiv 1$, and the sum is locally finite. Define \begin{align*} d_A: \Gamma(E) &\to \Gamma(E \otimes T^*M) \\ s &\mapsto \sum_{\alpha \in I} \lambda_\alpha \cdot d_{A_\alpha}(s|_{U_\alpha}), \end{align*} with the convention that $\lambda_\alpha \cdot d_{A_\alpha}(s|_{U_\alpha})$ is extended by zero outside $U_\alpha$. Local finiteness of the partition makes the sum well-defined in $\Gamma(E \otimes T^*M)$. For $f \in C^\infty(M)$, \begin{align*} d_A(fs) = \sum_\alpha \lambda_\alpha \big(df \otimes s + f\, d_{A_\alpha} s\big) = \Big(\sum_\alpha \lambda_\alpha\Big) df \otimes s + f \sum_\alpha \lambda_\alpha\, d_{A_\alpha} s = df \otimes s + f\, d_A s, \end{align*} so $d_A$ is a connection on $E$. This proves (i). [guided] We now glue the local connections into a global connection on $E$. Paracompactness of $M$ guarantees a smooth partition of unity $\{\lambda_\alpha\}_{\alpha \in I}$ subordinate to $\{U_\alpha\}$: $\lambda_\alpha \in C^\infty(M)$, $\operatorname{supp} \lambda_\alpha \subseteq U_\alpha$, $\sum_\alpha \lambda_\alpha \equiv 1$ on $M$, and the sum is locally finite. Define \begin{align*} d_A: \Gamma(E) &\to \Gamma(E \otimes T^*M) \\ s &\mapsto \sum_{\alpha \in I} \lambda_\alpha \cdot d_{A_\alpha}(s|_{U_\alpha}), \end{align*} where each summand is extended by zero outside $U_\alpha$; the extension is smooth because $\operatorname{supp} \lambda_\alpha \subseteq U_\alpha$. Local finiteness makes the sum well-defined pointwise. Why does $d_A$ satisfy the Leibniz rule? Take $f \in C^\infty(M)$. Each $d_{A_\alpha}$ is a connection, so it satisfies the Leibniz rule $d_{A_\alpha}(fs) = df \otimes s + f\, d_{A_\alpha} s$ on $U_\alpha$. Then \begin{align*} d_A(fs) = \sum_\alpha \lambda_\alpha \big(df \otimes s + f\, d_{A_\alpha} s\big) = \Big(\sum_\alpha \lambda_\alpha\Big) df \otimes s + f \sum_\alpha \lambda_\alpha\, d_{A_\alpha} s = df \otimes s + f\, d_A s, \end{align*} using $\sum_\alpha \lambda_\alpha = 1$ at the end. The key structural fact here is that the Leibniz rule is affine in the connection: any convex (indeed any partition-of-unity) combination of connections is again a connection, because the non-linear "$df \otimes s$" term only involves $f$ and $s$, not the connection, so it passes through the sum when weighted by a partition of unity. This proves (i): $E$ admits a global connection. [/guided] [/step] [step:Compute the difference of two connections and verify $C^\infty(M)$-linearity] Let $d_A, d_B \in \mathcal{A}_E$ and define \begin{align*} D: \Gamma(E) &\to \Gamma(E \otimes T^*M) \\ s &\mapsto d_A s - d_B s. \end{align*} $D$ is $\mathbb{R}$-linear because $d_A$ and $d_B$ are. For $f \in C^\infty(M)$ and $s \in \Gamma(E)$, applying the Leibniz rule to each connection gives \begin{align*} D(fs) = d_A(fs) - d_B(fs) = \big(df \otimes s + f\, d_A s\big) - \big(df \otimes s + f\, d_B s\big) = f\, D(s), \end{align*} so $D$ is $C^\infty(M)$-linear. [/step] [step:Identify $C^\infty(M)$-linear maps with sections of $\mathrm{End}(E) \otimes T^*M$ via the tensor--hom correspondence] By the tensor--hom correspondence for vector bundles, the space of $C^\infty(M)$-linear maps \begin{align*} \operatorname{Hom}_{C^\infty(M)}\big(\Gamma(E), \Gamma(E \otimes T^*M)\big) \end{align*} is canonically isomorphic to $\Gamma(\mathrm{Hom}(E, E \otimes T^*M))$. Using the canonical identifications \begin{align*} \mathrm{Hom}(E, E \otimes T^*M) \cong E^* \otimes E \otimes T^*M \cong \mathrm{End}(E) \otimes T^*M, \end{align*} the space of global sections is $\Omega^1(\mathrm{End}(E))$. Applying this to $D$ from the previous step, there exists a unique $\eta \in \Omega^1(\mathrm{End}(E))$ with $D(s) = \eta \cdot s$ for all $s \in \Gamma(E)$. Hence $d_A - d_B \in \Omega^1(\mathrm{End}(E))$. [guided] The step "$C^\infty(M)$-linear $\Longrightarrow$ tensor" is a recurring theme in differential geometry, and here we deploy it to identify $D$ with a section of an endomorphism bundle. Precisely: for any two smooth vector bundles $F_1, F_2 \to M$, the tensor--hom correspondence gives a natural isomorphism \begin{align*} \operatorname{Hom}_{C^\infty(M)}(\Gamma(F_1), \Gamma(F_2)) \cong \Gamma(\mathrm{Hom}(F_1, F_2)). \end{align*} The content of this statement is that a $C^\infty(M)$-linear map on sections is determined pointwise: it arises from a smoothly varying family of linear maps $F_{1,p} \to F_{2,p}$. Apply this with $F_1 = E$ and $F_2 = E \otimes T^*M$. Then \begin{align*} \mathrm{Hom}(E, E \otimes T^*M) \cong E^* \otimes (E \otimes T^*M) \cong (E^* \otimes E) \otimes T^*M \cong \mathrm{End}(E) \otimes T^*M, \end{align*} where the first isomorphism is $\mathrm{Hom}(V, W) \cong V^* \otimes W$ applied fibrewise, and the last uses $\mathrm{End}(E) = E^* \otimes E$. Taking sections, \begin{align*} \Gamma(\mathrm{Hom}(E, E \otimes T^*M)) \cong \Gamma(\mathrm{End}(E) \otimes T^*M) = \Omega^1(\mathrm{End}(E)). \end{align*} Since $D$ is $C^\infty(M)$-linear by the previous step, it corresponds under this isomorphism to a unique $\eta \in \Omega^1(\mathrm{End}(E))$ acting as $D(s) = \eta \cdot s$. This is the sense in which "$d_A - d_B$ is a tensor": the Leibniz non-linearity cancels, and what remains is a genuine section of an endomorphism-valued 1-form bundle. [/guided] [/step] [step:Verify that adding any $\eta \in \Omega^1(\mathrm{End}(E))$ to a connection yields a connection] Let $d_A \in \mathcal{A}_E$ and $\eta \in \Omega^1(\mathrm{End}(E))$. Define \begin{align*} d_{A + \eta}: \Gamma(E) &\to \Gamma(E \otimes T^*M) \\ s &\mapsto d_A s + \eta \cdot s. \end{align*} For $f \in C^\infty(M)$, using the Leibniz rule for $d_A$ and the $C^\infty(M)$-linearity of $\eta \cdot (-)$, \begin{align*} d_{A+\eta}(fs) = d_A(fs) + \eta \cdot (fs) = df \otimes s + f\, d_A s + f\, \eta \cdot s = df \otimes s + f\, d_{A+\eta} s, \end{align*} so $d_{A+\eta}$ is a connection. Hence the map $\Omega^1(\mathrm{End}(E)) \to \mathcal{A}_E$, $\eta \mapsto d_A + \eta$, is a well-defined action of $\Omega^1(\mathrm{End}(E))$ on $\mathcal{A}_E$ by translations. [/step] [step:Conclude that $\mathcal{A}_E$ is an affine space over $\Omega^1(\mathrm{End}(E))$] Step 1--2 established $\mathcal{A}_E \ne \varnothing$, proving (i). Steps 3--4 show that for any $d_A, d_B \in \mathcal{A}_E$, the difference $d_A - d_B$ is a unique element of $\Omega^1(\mathrm{End}(E))$, and conversely every $\eta \in \Omega^1(\mathrm{End}(E))$ defines a translation $d_A \mapsto d_A + \eta$ on $\mathcal{A}_E$. These data — a non-empty set, a free and transitive action by a vector space — are precisely the definition of an affine space. Hence $\mathcal{A}_E$ is an affine space for $\Omega^1(\mathrm{End}(E))$, proving (ii). [/step]