Before this page I already had these pages of the same title:
one,
two and
three.
I must consolidate those but meanwhile:
There is a vital aspect of affine connection not yet captured in my other pages.
An affine connection on a manifold defines the meaning of parallel motion of a vector along a definite path in the manifold.
(Actually same direction and magnitude—same vector.)
- If there is a path from A to B to C and we move a vector from A to B, and then from B to C, that gives the same result as moving directly from A to C, always staying on the same path.
- The motion along some fixed path from A to B provides a linear transformation (or linear operator) of the tangent space at A to the tangent space at B.
The affine transformation derived from a metric has the additional property that the inner product, as defined by that metric, of two transported vectors remains constant.
An unnamed class of affine transformations have the property of maintaining the determinant of the transform between any two points.
When the path is a loop, the determinant of the resulting transformation is 1.
Perhaps these should be called unimodular affine connections.
The affine connection from a metric is unimodular.
Shild’s ladder provides an intuitive construction of parallel transport given the notion of geodesics, which in turn stems from a metric.
Connections by PlanetMath
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That page considered the ring of continuous real valued functions on a differentiable manifold.
It then considered the nature of a derivative operator and what the natural values of such an operator.