# Covariant geometric characterization of slow invariant manifolds: New concepts and viewpoints

## TL;DR

This paper introduces a new geometric perspective on slow invariant manifolds in dynamical systems, using curvature and Hamiltonian mechanics to define and analyze SIMs in a coordinate-independent manner.

## Contribution

It presents a novel covariant geometric framework for characterizing SIMs, integrating curvature and Hamiltonian mechanics for a coordinate-independent analysis.

## Key findings

- New geometric criteria for SIM characterization
- Application to Davis-Skodje model demonstrating the approach
- Conjecture of a differential geometric definition of SIMs

## Abstract

We point out a new view on slow invariant manifolds (SIM) in dynamical systems which departs from a purely geometric covariant characterization implying coordinate independency. The fundamental idea is to treat the SIM as a well-defined geometric object in phase space and elucidate characterizing geometric properties that can be evaluated as point-wise analytic criteria. For that purpose, we exploit curvature concepts and formulate our recent variational approach in terms of coordinate-independent Hamiltonian mechanics.Finally, we combine both approaches and conjecture a differential geometric definition of slow invariant manifolds. For the Davis-Skodje model the latter can be completely expatiated.