Differentiable Manifolds and Matrix Lie Groups

Abstract

If X is a topological manifold and (U ₁, φ ₁) and (U ₂, φ ₂) are two charts on X with ${U}_{1} \cap{U}_{2}\neq \emptyset$, then the overlap functions ${\varphi }_{1} \circ{\varphi }_{2}^{-1} : {\varphi }_{2}({U}_{1} \cap{U}_{2}) \rightarrow{\varphi }_{1}({U}_{1} \cap{U}_{2})$ and ${\varphi }_{2} \circ{\varphi }_{1}^{-1} : {\varphi }_{1}({U}_{1} \cap{U}_{2}) \rightarrow{\varphi }_{2}({U}_{1} \cap{U}_{2})$ are necessarily homeomorphisms between open sets in some Euclidean space. In the examples that we have encountered thus far (most notably, spheres and projective spaces) these maps actually satisfy the much stronger condition of being C ^∞, that is, their coordinate functions have continuous partial derivatives of all orders and types (see Exercise 1.1.8 and (1.2.4)).