Here's the question: Is $X_3 = X_1 \star X_2$ always a geodesic manifold w.r.t. $X_1$ and $X_2$.
I think it's an extremely insightful question. For whatever reason it was downvoted with no explanation by the downvoter. Also, the same person probably downvoted the other 3.
This strikes me as odd.
I'm having trouble figuring out how to improve the question. I spent close to 10 hours writing and re-drafting this question.