  We consider the Calder\'on problem for systems with unknown zeroth and first
order terms, and improve on previously known results. More precisely, let $(M,
g)$ be a compact Riemannian manifold with boundary, let $A$ be a connection
matrix on $E = M \times \mathbb{C}^r$ and let $Q$ be a matrix potential. Let
$\Lambda_{A, Q}$ be the Dirichlet-to-Neumann map of the associated connection
Laplacian with potential $\mathcal{L}_{A, Q}$. Under the assumption that $(M,
g)$ is isometrically contained in the interior of $(\mathbb{R}^2 \times M_0,
c(e \oplus g_0))$, where $(M_0, g_0)$ is an arbitrary compact Riemannian
manifold with boundary, $e$ is the Euclidean metric on $\mathbb{R}^2$, and $c >
0$, we show that $\Lambda_{A, Q}$ uniquely determines $(A, Q)$ up to natural
gauge invariances. Moreover, we introduce a new complex ray transform and a new
complex parallel transport problem, and study their fundamental properties and
relations to the Calder\'on problem.

Analysis of PDEs;Differential Geometry