  For the study of highly nonlinear, conservative dynamic systems, finding
special periodic solutions which can be seen as generalization of the
well-known normal modes of linear systems is very attractive. However, the
study of low-dimensional invariant manifolds in the form of nonlinear normal
modes is rather a niche topic, treated mainly in the context of structural
mechanics for systems with Euclidean metrics, i.e., for point masses connected
by nonlinear springs. Newest results emphasize, however, that a very rich
structure of periodic and low-dimensional solutions exist also within nonlinear
systems such as elastic multi-body systems encountered in the biomechanics of
humans and animals or of humanoid and quadruped robots, which are characterized
by a non-constant metric tensor. This paper discusses different generalizations
of linear oscillation modes to nonlinear systems and proposes a definition of
strict nonlinear normal modes, which matches most of the relevant properties of
the linear modes. The main contributions are a theorem providing necessary and
sufficient conditions for the existence of strict oscillation modes on systems
endowed with a Riemannian metric and a potential field as well as a
constructive example of designing such modes in the case of an elastic double
pendulum.
