User:Silly rabbit/Sandbox/connection

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A connection transporting lines from O to P

In the classical geometry of the Euclidean plane, a connection associates to each line through the origin the family of all lines parallel to it. The connection then allows us to move (or transport) lines from the origin to other points in the plane. It thus connects the geometry of the origin to that of other points of the plane.

The connection in the plane is rather trivial because the plane is flat. Other surfaces (or, more generally, manifolds) have more interesting connections. A common feature of many connections is that they determine a way to transport lines^[1] from one point to another along a path, thus connecting the geometry of one point to another. However, usually the direction of the transported line will depend on the path chosen. This path-dependence is one way to interpret the mathematical notion of curvature.

Parallel transport on a sphere depends on the path. Transporting from A → N → B yields a different vector than the initial vector.

[edit] Notes

1. ^ Such connections are, aptly named, linear connections or affine connections.