Concept

Transport

Transport is the carrying of a vector, frame or other structure continuously along a path, one small step at a time, without any sudden change. Comparing what returns at the end of a closed loop with what set out reveals twisting — a flipped orientation, a turned frame — that no single point can see.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Also named here as vector bundle — the same set of essays touches all of them, so they are one junction rather than several.

A frame carried round a Möbius band. A flat rectangle whose ends are about to be joined, with a pair of arrows carried along it — one along the band and one across it — and the sign of the frame at each station.

Orientation is a sign

Carry a pair of arrows once round a loop and compare them with the pair they started as. The comparison is a determinant, its sign is the whole of the answer, and no room, no side and no normal vector appears anywhere in the statement.

topology · Orientability

Every section of the Möbius band crosses the zero section. The Möbius band drawn flat, with 3 continuous sections; zeros per section: 1, 3, 1.

Every section of the band must vanish

Read the Möbius band as a line standing over each point of a circle, and choose a point on each line continuously: a section. On a cylinder a section can stay away from zero all the way round. On the band it cannot — every section crosses zero an odd number of times — and that single fact is what orientability means for a bundle. There are exactly two line bundles over a circle, told apart by one sign, and two Möbius bands added together make the plain cylinder's twin.

topology · Orientability

Named alongside it

The objects these essays reach for when they reach for this one.

Determinant Frame Orientability Orientation Vector bundle Handedness Intermediate value theorem Intrinsic property Möbius band Normal vector Volume form

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