On Conformal Transformation with Multiple Scalar Fields and Geometric
  Property of Field Space with Einstein-like Solutions
  Multiple scalar fields appear in vast modern particle physics and gravity
models. When they couple to gravity non-minimally, conformal transformation is
utilized to bring the theory into Einstein frame. However, the kinetic terms of
scalar fields are usually not canonical, which makes analytic treatment
difficult. Here we investigate under what conditions the theories can be
transformed to the quasi-canonical form, in which case the effective metric
tensor in field space is conformally flat. We solve the relevant nonlinear
partial differential equations for arbitrary number of scalar fields and
present several solutions that may be useful for future phenomenological model
building, including the $\sigma$-model with a particular non-minimal coupling.
We also find conformal flatness can always be achieved in some modified gravity
theories, for example, Starobinsky model.
