When recently reading about the remarkable (but somewhat intuitive) Fary's Theorem, I wondered about natural extensions of it to other spaces, none of which were listed on the Wikipedia page.
It's clear how we define concepts such as planarity in other well-known surfaces, such as tori or Klein Bottles etc., however it is less clear to me how to characterise whether a line is "straight" in a natural sense on tori, etc. So:
- Is there a natural way to characterise straight-ness in other spaces? (I have heard previously of the concept of a "geodesic", but am not at all familiar with it)
- Can we use this to find generalisations of Fary's Theorem in these other spaces?