Is a 2D object on a plane that gets curved still considered 2D or is it now 3D?

Question

Imagine a 2D square drawn on a flat membrane. The square has length and width, but no depth (obviously; it's a square). Now take said membrane and place it with the middle of the square centered atop some round object. Then press down on the membrane to curve it around said round object.

Would the square on the membrane still be considered two dimensional since it hasn't gained depth relative to the membrane or would it now be considered a 3D object since to an outside observer, the corners of the square - due to the curvature of the membrane - are now lower than the center of the square, thus technically giving the square depth relative to the round object?

Answer 1

It depends on what kind of Dimension the letter D stands for. There are many.

1. The topological dimension of the deformed square is still $2$, since it's homeomorphic to a part of a plane.
2. The embedding dimension, defined as the smallest $n$ such that the space has an isometric embedding into $\mathbb{R}^n$, is $2$ for the flat square but $3$ for the deformed square.