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Some model reduction techniques for multiple time-scale dynamical systemsmake use of the identification of low dimensional slow invariant attractingmanifolds (SIAM) in order to reduce the dimensionality of the phase space byrestriction to the slow flow. The focus of this work is on a proposition anddiscussion of a general viewpoint using differential geometric concepts forsubmanifolds to deal with slow invariant manifolds in an extended phase space.The motivation is a coordinate independent formulation of the manifoldproperties and its characterization problem treating the manifold as intrinsicgeometric object. We formulate a computationally verifiable necessary conditionfor the slow invariant manifold graph stated in terms of a differentialgeometric view on the invariance property. Its application to example systemsis illustrated. In addition, we present some ideas and investigationsconcerning the search for sufficient, differential geometric conditionscharacterizing slow invariant manifolds based on our previously developedvariational principle.

Towards differential geometric characterization of slow invariant manifolds in extended state space: Sectional Curvature and Flow Invariance