Why are affine connections called so? I have been going through some books regarding Smooth Manifolds and Riemannian Manifolds. But I haven't been able to get an answer to one question.
Could you explain what is the intuition behind naming the affine connection so?
Wikipedia says that the tangent space at a point of a surface is an affine subspace of the Euclidean space. However, I do not understand the explanation or its consequence. Any help would be very much appreciated.
Thanks.
 A: I actually find the Wikipedia page very instructive notably regarding the motivation of the concept as well as its name. The simplest answer is: assume your manifold $M$ is embedded in a real vector (or affine) space (e.g. by the Whitney embedding theorem). Then an affine connection on $M$ is a way to "connect" (via parallel transport) different tangent spaces (which are affine subspaces of the vector space) by affine transformations. The more advanced paragraph presenting affine connections as "affine" Cartan connections is relevant as well.
Note that today the terms "affine connection" and "linear connection" are used indistinctly (I believe) and most of the time just saying "connection" is enough. I think that oftentimes when "affine"/"linear" is specified, the point is to insist that it's not a principal connection, or a Cartan connection, or a projective connection etc; it's just a good old connection in a vector bundle (often the tangent bundle), most of the time seen as a linear differential operator.
