  The theory of slow invariant manifolds (SIMs) is the foundation of various
model-order reduction techniques for dissipative dynamical systems with
multiple time-scales, e.g. in chemical kinetic models. The construction of SIMs
and many approximation methods exploit the restrictive requirement of an
explicit time-scale separation parameter. Most of those methods are also not
formulated covariantly, i.e. in terms of tensorial constructions. We propose an
intrinsically coordinate-free differential geometric approximation criterion
approximating normally attracting invariant manifolds (NAIMs). We translate
some ideas behind existing approximation approaches, the stretching based
diagnostics (SBD) and the flow curvature method (FCM) to tensors of Riemannian
geometry, specifically to spacetime curvature in extended phase space. For that
purpose we derive from flow-generating smooth vector fields a metric tensor
such that the original dynamical system is a geodesic flow on a Riemannian
manifold. We apply the resulting method to test models.
