  Using nonlinear projections and preserving structure in model order reduction
(MOR) are currently active research fields. In this paper, we provide a novel
differential geometric framework for model reduction on smooth manifolds, which
emphasizes the geometric nature of the objects involved. The crucial ingredient
is the construction of an embedding for the low-dimensional submanifold and a
compatible reduction map, for which we discuss several options. Our general
framework allows capturing and generalizing several existing MOR techniques,
such as structure preservation for Lagrangian- or Hamiltonian dynamics, and
using nonlinear projections that are, for instance, relevant in
transport-dominated problems. The joint abstraction can be used to derive
shared theoretical properties for different methods, such as an exact
reproduction result. To connect our framework to existing work in the field, we
demonstrate that various techniques for data-driven construction of nonlinear
projections can be included in our framework.
