Question about the Ehrenfest paradox

I posted this question on stackexchange and I am posting this here too because I had no answer there so far.

The Ehrenfest paradox exemplifies that a rigid body can't be defined in special relativity, but I can't convince myself that it implies non-Euclidean geometry as some sources say (http://ift.tt/1Lu3N0p).

In this post question(http://ift.tt/2zR9c0W), the answers vary from saying that it only implies that rigid bodies can't exist in SR or that it implies, in addition to the former, that the geometry is non-Euclidian.

In my point of view, it only tell us that the theory is not consistent with this thought experiment.

Could someone elaborate on what the Ehrenfest paradox really tell us?

I'am an undergraduate in physics and I will be talking about rigid bodies in one of my classes. I want to clarify what would be non-sense to say to my professor. I appreciate the attention.