  We study multifield inflation in scenarios where the fields are coupled
non-minimally to gravity via $\xi_I(\phi^I)^n g^{\mu\nu}R_{\mu\nu}$, where
$\xi_I$ are coupling constants, $\phi^I$ the fields driving inflation,
$g_{\mu\nu}$ the space-time metric, $R_{\mu\nu}$ the Ricci tensor, and $n>0$.
We consider the so-called $\alpha$-attractor models in two formulations of
gravity: in the usual metric case where $R_{\mu\nu}=R_{\mu\nu}(g_{\mu\nu})$,
and in the Palatini formulation where $R_{\mu\nu}$ is an independent variable.
As the main result, we show that, regardless of the underlying theory of
gravity, the field-space curvature in the Einstein frame has no influence on
the inflationary dynamics at the limit of large $\xi_I$, and one effectively
retains the single-field case. However, the gravity formulation does play an
important role: in the metric case the result means that multifield models
approach the single-field $\alpha$-attractor limit, whereas in the Palatini
case the attractor behaviour is lost also in the case of multifield inflation.
We discuss what this means for distinguishing between different models of
inflation.
