Curvature as the atlas failing to close
A frame carried around a closed chain of overlapping charts comes back visibly rotated against its ghost.
The relation on screen
Transport a frame around a loop and its failure to return is curvature, measured per unit enclosed area.
Symbols
- closed loop in the chart
- curvature, the deficit density
the surveyor's definition
Curvature does not need an embedding space. Carry a frame around a closed loop using nothing but the rules for translating between neighbouring charts, and if it comes back rotated the space is curved. On a sphere you can do this with three right-angled turns and end up facing ninety degrees off. Set the curvature slider to zero and the frame returns exactly as it left.
what to look for · The deficit scales with enclosed area, which is what distinguishes curvature from a bookkeeping error.
Consensus to physics
- 01carry a frame from chart to chart around a closed chain of overlaps
- 02each hop applies that pair's transition map
- 03the frame returns rotated relative to its start
- 04that holonomy is curvature, measured entirely from inside
In standard physics
Parallel transport and the Riemann tensor
Curvature defined the way a surveyor would define it. No observer steps outside; curvature is an internal consequence of how charts fail to agree globally.
boundary · The chart layout in the picture is illustrative. This is not a measurement of a spacetime Riemann tensor and no metric has been attached.