Can a Wild Metric on a 2D Disc Prevent Isometric Embedding in R^3?

[wofsy]

Can you show me a metric on the 2 dimensional disc so wild that no open subset can be embedded in R^3?

 

[zhentil]

Embedded isometrically? Otherwise, it's false.

 

[wofsy]

Embedded isometrically? Otherwise, it's false.

yes. No open subset can be embedded isometrically in R^3