In many papers and textbooks, there is always a brief comment that differential-algebraic equations (DAE) can be considered as a system of ordinary differential equation (ODE) on manifold. But then nowhere is it explained what it actually means.
I desperately want to see what does it mean and visualize that manifold. What are the necessary steps to do? Is the algebraic part defining some sort of manifold and if so, how?
Some (possibly schematic only) concrete example is much appreciated.
Edit. To keep things simple, lets focus only on index-1 DAEs as suggested in the comment.
Edit2 Is it like the follwoing? For every (consistent) initial values, I have to find all solutions and the result is the manifold? Is the resulting quite different from what I would get when DAE is further reduced do ODE?