  We formulate scalar field theories coupled non-conformally to gravity in a
manifestly frame-independent fashion. Physical quantities such as the $S$
matrix should be invariant under field redefinitions, and hence can be
represented by the geometry of the target space. This elegant geometric
formulation, however, is obscured when considering the coupling to gravity
because of the redundancy associated with the Weyl transformation. The
well-known example is the Higgs inflation, where the target space of the Higgs
fields is flat in the Jordan frame but is curved in the Einstein frame.
Furthermore, one can even show that any geometry of O$(N)$ nonlinear $\sigma$
models can be flattened by an appropriate Weyl transformation. In this Letter,
we extend the notion of the target space by including the conformal mode of the
metric, and show that the extended geometry provides a compact formulation that
is manifestly Weyl-transformation/field-redefinition invariant. We identify the
cutoff scale with the inverse of square root of the extended target space
curvature and confirm that it coincides with that obtained from two-to-two
scattering amplitudes based on our formalism.
