  This paper explores the relationship that exists between nonlinear normal
modes (NNMs) defined as invariant manifolds in phase space and the spectral
expansion of the Koopman operator. Specifically, we demonstrate that NNMs
correspond to zero level sets of specific eigenfunctions of the Koopman
operator. Thanks to this direct connection, a new, global parametrization of
the invariant manifolds is established. Unlike the classical parametrization
using a pair of state-space variables, this parametrization remains valid
whenever the invariant manifold undergoes folding, which extends the
computation of NNMs to regimes of greater energy. The proposed ideas are
illustrated using a two-degree-of-freedom system with cubic nonlinearity.
