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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.

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    $\begingroup$ What about defining a parallelogram as a quadrilateral whose diagonals bisect each other? $\endgroup$ – Aretino Apr 2 '17 at 12:15
  • $\begingroup$ @Aretino Yes I think this is a good idea, in fact probably better than the original definition. It works very well with more abstraction friendly versions of the parallel postulate, for instance en.wikipedia.org/wiki/… If you want to write this as an answer (maybe with one or two more sentences) I will accept it. It works very nicely in any geodesic space. But yes, I think this is exactly what I am looking for, I really appreciate you pointing this fact out to me, since I did not remember it from Euclidean geometry. $\endgroup$ – Chill2Macht Apr 2 '17 at 12:18
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While the standard euclidean definition refers to parallel lines, several other definitions are possible (and widely present in the literature). For instance, one can define a parallelogram as a quadrilateral having a center of symmetry, or as a quadrilateral whose diagonals intersect at their midpoint. The latter definition might be more suited to generalize the notion of parallelogram in non-euclidean spaces.

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