### An Intuitive Explanation of the Affine Connection

At each point in a manifold there is a vector space often called the tangent space at that point. When we think of a vector in a manifold we must generally be specific where in the manifold and then say that the vector is in the tangent space there. It is generally impossible to compare vectors in the tangent spaces at different points except for a topological construct called a fiber-bundle which provides a notion of a vector at one point being close to a particular vector at a nearby point of the manifold. On a manifold an affine connection is a mathematical structure on the manifold that allows parallel transport of vectors along any path in the manifold.

If the path is a loop and we carry vectors around the loop then they are generally changed and we demand that this be a linear transformation on the tangent space to itself. If vectors are changed then the affine connection defines a curved space. This concept of curvature is more general and less symmetric than curvature computed from a metric.

A metric for a manifold determines an affine connection and the curvature determined by that connection is the usual curvature determined by the metric.

Relative to some coordinate system for the manifold, the Christoffel Symbol of the Second Kind describes an affine connection. A coordinate system already suggests an affine connection—don’t change the vector components as you move the vector. A different coordinate system, however, suggests a different connection. If a metric is clearly in sight then the Christoffel Symbol usually denotes the affine connection stemming from the metric. This may lead to some confusion.

#### Conjecture

A ‘small’ ball, by definition a sphere, in a connected manifold, can be carried about with a symmetric affine connection to define a metric. If the affine connection is symmetric then this sphere may rotate as it traverses a loop, but it will not expand or shrink.

See this too. This is another more mathematical approach.