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Oct 30, 2020 at 17:26 comment added Malkoun Nice answer Prof. Bryant, as usual. And thank you for the pathological example too. I had never heard of "path geometry" before. It seems I ask a lot of classical questions. Note that your pathological example can be natural extended to $x^2+y^2 < 2$, and then $\Sigma$ becomes a boundary of $M$, in a suitable sense.
Oct 30, 2020 at 17:18 vote accept Malkoun
Oct 30, 2020 at 12:11 history answered Robert Bryant CC BY-SA 4.0