Consider a 2D video game world (e.g., something like a traditional side-scroller). Suppose this world represents our space of interest - it's 2D Euclidean space. When rendering this 2D world onto a screen, I can apply a distortion effect to the image, as in the screenshot below (I found this image online):
So, the idea is that a flat 2D space is projected onto a flat screen ($p: \mathbb R^2 \rightarrow \mathbb R^2$) in a way that makes the image distorted. Suppose the mapping is a bijection, and smooth.
Now, this distortion does not affect the game world at all - from a perspective internal to the game world, everything functions in exactly the same way. If in-game characters were physicists, all measurements they'd be able to make would produce exactly the same results. There is no way to detect the distortion "from within"; the space looks flat.
Q1:
I know there is a mathematical notion of distortion. But, from a certain point of view, given that the structure internal to the game world is distortion-agnostic, can the distortion effect be seen as a form of extrinsic curvature (with respect to the flat in-game space, the domain of $p$), under some notion of curvature? In other words, does $p$ somehow induce intrinsic curvature on the target space, that can then serve as ambient curvature for the source space?
Consider now two perspectives: (1) one internal to the game world, and (2) another one that I'll call external.
By external perspective I mean the one where the distortion is apparent.
Q2:
From an external point of view, there's this strangely distorted space, but the objects embedded in it behave in the previously described way. Does this situation correspond to a (Riemannian) metric being defined on the "external" space (which is just a plane) in a way that makes the resulting geometric structure equivalent* to flat Euclidean space?
*Not sure about the precise notion of equivalence to be used here, but intuitively, it accounts for the internal perspective being distortion-agnostic. Diffeomorphism? Isometry?
Regarding the external perspective: initially, the idea that this was a form of extrinsic curvature seemed reasonable to me, but if the extrinsic curvature is related to the rate of change of the unit normal and tangent vectors across the (hyper)surface, then there's (apparently) no curvature because the surface is flat - all distortion happens in-plane. (Or am I now talking about an extra ambient space, on top of the two previously considered?)
Does it even make sense to use this idea as the original space is not embedded in a higher-dimensional space?
On the other hand, I feel like it should be possible to define objects that exhibit similar kinds of relationships (act as tangents and normals) without requiring such an embedding. I suppose one could use the projection map $p$ to see how a projected infinitesimal vector behaves during parallel transport (parallel within the "internal" space), and get some notion of extrinsic curvature from that?
Q3:
If this is a form of extrinsic curvature (or can be treated as such), what kind of curvature it is? How is it defined? Also, how is the tangent space defined? Or more specifically, how is the tangent plane at a point defined? (And if there are many ways to define these, is there a "natural" way, given the context?)