  Finally, Cartan’s generalization of Riemannian space (lower  horizontal red arrow) (figure 13) turns on the recognition that the  “Tangent Space” in Cartan’s sense is not the same, or need not be the same, as the ordinary tangent  space that arises naturally from the underlying differential structure of a Riemannian manifold. Cartan’s  “Tangent Space” \(\Sigma(M_{p})\) at \(p \in M\) denotes what is known as a fiber in modern fiber bundle language, where the manifold \(M\) is called the base space of the fiber bundle. 