Fields

• A vector or a tensor field is a mapping that assigns   a vector or a tensor   to every point of a   manifold.
• For this to make any sense, there must be a whole vector space available at each point of the manifold from which to choose a vector of the vector field. That space is called a tangent space.
• For this, in turn, to make any sense, we must have some means of comparing tangent vector spaces at nearby points. This is usually described in terms of a   parallel transport of a vector between those points. Remember that the tangent spaces at two different points are two completely different spaces.
• A structure that comprises a manifold (also called a base space, from every point of which grows a Lie group   (a tangent vector space attached to a point of the manifold is a special example of a Lie group) is   called a fibre bundle. The groups that grow from the points of the manifold are called fibres.
• All those fibres taken together form a manifold too which is called   a total space.