Abstract
This chapter focuses on linear connections. Tangent spaces play a key role in differential geometry. The tangent space at a point, x, is the totality of all contravariant vectors, or differentials, associated with that point. By means of an affine connection, the tangent spaces at any two points on a curve are related by an affine transformation, which will, in general, depend on the curve. Linear connections of another kind are associated with each point of a given n-dimensional manifold a space of m dimensions. Schouten has proposed the use of linear connections in handling a scheme by which differential geometry is based on group theory, in the spirit of Klein's Erlanger Program. According to Klein, the associated spaces are to be the spaces of some group, and are related through linear displacement by transformations of this group.
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