Differential Geometry from a Modern Standpoint

Abstract

One is led to the concept of a differentiable manifold, if one tries to define spaces M more general than R ⁿ on which it is still possible to do calculus in some sense. For this purpose we demand first that any point of M — which is so far only a set — is contained in a subset U, which can be mapped onto an open set in R ⁿ by a 1–1 map h.

R ⁿ = (x ¹, …, x ⁿ);x ⁱ∈R together with its usual topology and vector space structure.