Imagine a box with N knobs each calibrated like the combination wheel on a safe. When one turns a knob other knobs turn, unless perhaps you hold them still so they cannot. But then other knobs turn, but just when you turn some knob. Your job is to describe what combinations of knob settings are possible as constrained by invisibe mechanisms inside the box. There are of course many possible mechanisms in the box. One possible outcome of the task is to write down a long list of N-tuples of settings organized somehow to indicate which settings are near which others.
There are many things I want to say about the box to limit the possibilities:
The box can be characterized by its set of possible knob position combinations. Indeed this set is a sub-manifold of ℝN. The manifold is k dimensional where k≤N. (Actually we get the N torus since our knob setting values are circular. We will assume that there is an odometer on each knob counting full turns so as to stick to ℝN.) If N=3 we can provide a box that describes a torus. Each triple of knob settings <x, y, z> is possible just in case the point <x, y, z> lies on that torus. In that case the manifold is 2 dimensions. Fixing two knobs in such a 2D box tends to fix the third; there are 2 degrees of freedom—thus the 2D manifold.
An even simpler version is to imagine an ‘Etch A Sketch’ whose stylus is stuck in a circular groove on the screen. N=2 and k=1. The manifold is the 1D circle.
Some extreme boxes:
There is nothing new here except perhaps a new physical metaphor for a manifold.
If we add a flywheel (equal rotational inertia) to each knob but no inertia inside, then we can set the wheels turning and they will follow a trajectory which is a geodesic on the manifold; the metaphor is good for something! With this inertia perhaps we have a 2N dimensional symplectic manifold. Liouville’s theorem might live there.
This model may also serve to aid the intuition for dependent and independent variables. Any k of the knobs may serve as independent variables while the other N−k knobs are dependent.
These manifolds are all embedded in ℝN. Mathematicians have a somewhat more general notion of manifold. See the Atlas which describes a manifold which needs no space in which to be embedded.