What is a geometric shape? I thought that the concept of "gemoetric shape" is clear enough - squares, ellipses, triangles, you know. But then I found several papers, such as this one, which define "shape" as "an un-parameterized immersed sub-manifold". I have read about immersion and submanifolds, but I still don't understand, how is this definition related to the intuitive meaning of a "geometric shape"? And why is it formally defined in such a way?
 A: First, note that in order to talk about a submanifold, we must already have in hand a background manifold in which the submanifolds live, i.e., in which they are subsets---your examples are all immersed $1$-manifolds in $\mathbb{R}^2$ (just the usual plane of high school plane geometry).
Now, the most important feature of manifolds is that they be "locally Euclidean", that is that around a point they look like Euclidean space, $\mathbb{R}^n$---the upshot of this condition is that we for many questions about Euclidean space we can ask the same or at least analogous questions about manifolds. One practical consequences of this definition is that we can always write down coordinates for a manifold---this is roughly idea of using latitude and longitude to specify places on a sphere, and what gives us a consistent way of producing maps of the world, despite that the world is round and maps are (usually) flat. This also means that in a sense, each point looks like every other point in a sense. (For $1$-dimensional manifolds. this more or less means we can trace out the shape by drawing a single curve, maybe closing up but definitely not crossing itself.)
One can write describe geometric objects that don't fit this description but they're always "irregular" in some way. For example, if we glue two circles at a point, then we've made a space that has a point (where the circles meet) that are different from all other points on the shape, a notion which we can make formal.
