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 C^k differential structure is defined using a C^k-atlas, which is a set of homeomorphisms called charts between open subsets of M (whose union is the whole of M) and open subsets of ${\displaystyle \mathbb {R} ^{n}}$:

which are C^k-compatible (in the sense defined below):

Each chart allows an open subset of the manifold to be viewed as an open subset of ${\displaystyle \mathbb {R} ^{n}}$, but the usefulness of this depends on how much the charts agree when their domains overlap.

Consider two charts:

The intersection of their domains is

whose images under the two charts are

The transition map between the two charts translates between their images on their shared domain: