A Tip for Landscape Riders: Multi-Field Inflation Can Fulfill the
  Swampland Distance Conjecture
  We study how both the swampland distance conjecture and the Lyth bound affect
the parameter space of multi-field models of inflation. A generic feature of
multi-field inflation is that the geodesic distance
$\left[\Delta\phi\right]_\text{G}$ separating any two points laying along the
inflationary trajectory differs from the non-geodesic distance
$\left[\Delta\phi\right]_\text{NG}$ traversed by the inflaton between those
points. These distances must respect a relation of the form
$\left[\Delta\phi\right]_\text{G} =
f\left(\left[\Delta\phi\right]_\text{NG}\right) \leq
\left[\Delta\phi\right]_\text{NG}$, where $f$ is a function determined by the
specific multi-field model under scrutiny. We show that this relation leads to
important constraints on the parameter space characterizing the multi-field
dynamics. Indeed, the swampland distance conjecture implies an upper bound on
$\left[\Delta\phi\right]_\text{G}$ set by the details of the ultraviolet
completion of inflation, whereas the Lyth bound implies a lower bound on
$\left[\Delta\phi\right]_\text{NG}$ determined by the value of the
tensor-to-scalar ratio. If future observations confirm the existence of
primordial tensor perturbations, these two bounds combined lead to tight
constraints on the possible values of the entropy mass of the isocurvature
fields orthogonal to the inflationary trajectory and the rate of turn of the
inflationary trajectory in multi-field space. We analyze the emerging
constraints in detail for the particular case of two-field inflation in
hyperbolic field spaces.
