Differential structure

In mathematics, an n-dimensional differential structure (or differentiable structure) on a set M makes M into an n-dimensional differential manifold, which is a topological manifold with some additional structure that allows for differential calculus on the manifold. If M is already a topological manifold, it is required that the new topology be identical to the existing one. For a natural number n and some k which may be a non-negative integer or infinity, an n-dimensional Ck differential structure is defined using a Ck-atlas, which is a set of bijections called charts between a collection of subsets of M (whose union is the whole of M), and a set of open subsets of : which are Ck-compatible (in the sense defined below): Each such map provides a way in which certain subsets of the manifold may be viewed as being like open subsets of but the usefulness of this notion depends on to what extent these notions agree when the domains of two such maps overlap. Consider two charts: The intersection of the domains of these two functions is and its map by the two chart maps to the two images: The transition map between the two charts is the map between the two images of this intersection under the two chart maps. Two charts are Ck-compatible if are open, and the transition maps have continuous partial derivatives of order k. If k = 0, we only require that the transition maps are continuous, consequently a C0-atlas is simply another way to define a topological manifold. If k = ∞, derivatives of all orders must be continuous. A family of Ck-compatible charts covering the whole manifold is a Ck-atlas defining a Ck differential manifold. Two atlases are Ck-equivalent if the union of their sets of charts forms a Ck-atlas. In particular, a Ck-atlas that is C0-compatible with a C0-atlas that defines a topological manifold is said to determine a Ck differential structure on the topological manifold. The Ck equivalence classes of such atlases are the distinct Ck differential structures of the manifold.