Question: How can one generalize parallelograms to non-Euclidean spaces?
In particular, how can one generalize parallelograms to Finsler manifolds which are not necessarily affine spaces (i.e. to "spaces with norms which don't satisfy the parallelogram law")?
Or at least to Riemannian manifolds which are not affine spaces?
The dream of course would be a generalization to arbitrary geodesic spaces, but an appropriate notion of parallelism may not be available for them.
A pointer to a reference will suffice for an answer.
Problem: The standard definition of parallelogram is only valid for Euclidean geometry.
In Euclidean geometry, a parallelogram is a non-self-intersecting quadrilateral with two pairs of parallel sides.
The quadrilateral part should generalize simply enough to any space with geodesics, with arbitrary geodesics replacing line segments.
However, the choice of how to generalize "parallel" is not so straightforward. The Wikipedia article on parallelism mentions at least 3 distinct concepts which are equivalent in Euclidean geometry and in that case correspond to the notion of parallelism -- in non-Euclidean geometries they correspond to (i) equidistant curves, (ii) parallel geodesics, and (iii) geodesics sharing a common perpendicular. The article about the parallel postulate mention 4 distinct concepts which are equivalent in Euclidean geometry and in that case correspond to the notion of parallelism -- (i) constant separation, (ii) never meeting, (iii) same angles where crossed by some third line, (iv) same angles where crossed by any third line.