  Model reduction of large nonlinear systems often involves the projection of
the governing equations onto linear subspaces spanned by carefully-selected
modes. The criteria to select the modes relevant for reduction are usually
problem-specific and heuristic. In this work, we propose a rigorous
mode-selection criterion based on the recent theory of Spectral Submanifolds
(SSM), which facilitates a reliable projection of the governing nonlinear
equations onto modal subspaces. SSMs are exact invariant manifolds in the phase
space that act as nonlinear continuations of linear normal modes. Our criterion
identifies critical linear normal modes whose associated SSMs have locally the
largest curvature. These modes should then be included in any projection-based
model reduction as they are the most sensitive to nonlinearities. To make this
mode selection automatic, we develop explicit formulas for the scalar curvature
of an SSM and provide an open-source numerical implementation of our
mode-selection procedure. We illustrate the power of this procedure by
accurately reproducing the forced-response curves on three examples of varying
complexity, including high-dimensional finite element models.
