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KANGTOTO

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KANGTOTO

Thanks a lot! And I’m sorry I posted doubly the same thing.

Yes, it’s correct. An arc with Euclidean metric is topologically contractible but has non-trivial metric pi1. Also, a sphere with standard Riemannian metric is topologically simply connected but has non-trivial pi1.

I think finding a metric fibration in nature is a bit hard. It is because metric fibration is define to be locally a l^1 product, while wild objects usually have l^2 structure. For example, the standard projection of a plane to a line is not a metric fibration. Once I tried to change the monoidal structure of positive reals from l^1 to l^2 and develop the analogue of metric fibration, but it became too complicated to handle, at least at that time.

Yes, it’s correct. An arc is topologically contractible but has non-trivial metric pi1. Also, a sphere with standard Riemannian metric is topologically simply connected but has non-trivial pi1.

I think finding a metric fibration in nature is a bit hard. It is because metric fibration is define to be locally a l^1 product, while wild objects usually have l^2 structure. For example, the standard projection of a plane to a line is not a metric fibration. Once I tried to change the monoidal structure of positive reals from l^1 to l^2 and develop the analogue of metric fibration, but it became too complicated to handle, at least at that time.

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Based only on the examples Yasuhiko gave above, I’ll venture a guess at an example: an arc of a circle with the induced Euclidean metric.

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