  Some model reduction techniques for multiple time-scale dynamical systems
make use of the identification of low dimensional slow invariant attracting
manifolds (SIAM) in order to reduce the dimensionality of the phase space by
restriction to the slow flow. The focus of this work is on a proposition and
discussion of a general viewpoint using differential geometric concepts for
submanifolds to deal with slow invariant manifolds in an extended phase space.
The motivation is a coordinate independent formulation of the manifold
properties and its characterization problem treating the manifold as intrinsic
geometric object. We formulate a computationally verifiable necessary condition
for the slow invariant manifold graph stated in terms of a differential
geometric view on the invariance property. Its application to example systems
is illustrated. In addition, we present some ideas and investigations
concerning the search for sufficient, differential geometric conditions
characterizing slow invariant manifolds based on our previously developed
variational principle.
