  We argue that field trajectories, which lead to cosmic acceleration and
feature rapid turns near the boundary of the moduli space, are in the
Swampland. We obtain this result by assuming the validity of the Swampland
Distance Conjecture (SDC) in the presence of a positive scalar potential and by
focusing on hyperbolic spaces, as prototype geometries of infinite distance
limits of Calabi-Yau compactifications. We find that, in a quasi-de Sitter
space with Hubble rate $H$ and acceleration parameter $\epsilon$, the turning
rate $\Omega$ is upper bounded such as $\Omega/H<\mathcal{O}(\sqrt{\epsilon})$.
Therefore, field trajectories consistent with the SDC can only have a
negligible deviation from geodesics. This has direct implications for the
realization and consistency of multi-field scenarios in string theory.
Moreover, it implies a tension between asymptotic accelerating expansion,
consistent with observations, and the de Sitter conjecture.
