  We develop an approach for linking the power spectra, bispectrum, and
trispectrum to the geometric and kinematical features of multifield
inflationary Lagrangians. Our geometric approach can also be useful in
determining when a complicated multifield model can be well approximated by a
model with one, two, or a handful of fields. To arrive at these results, we
focus on the mode interactions in the kinematical basis, starting with the case
of no sourcing and showing that there is a series of mode conservation laws
analogous to the conservation law for the adiabatic mode in single-field
inflation. We then treat the special case of a quadratic potential with
canonical kinetic terms, showing that it produces a series of mode sourcing
relations identical in form to that for the adiabatic mode. We build on this
result to show that the mode sourcing relations for general multifield
inflation are extension of this special case but contain higher-order covariant
derivatives of the potential and corrections from the field metric. In
parallel, we show how these interactions depend on the geometry of the
inflationary Lagrangian and on the kinematics of the associated field
trajectory. Finally, we consider how the mode interactions and effective number
of fields active during inflation are reflected in the spectra and introduce a
multifield consistency relation, as well as a multifield observable that can
potentially distinguish two-field scenarios from scenarios involving three or
more effective fields.
