# Differentiable Manifolds/The definition of differentiable manifolds

**Definition (differentiable manifold)**:

Let . Then a **differentiable manifold** of class is a topological space together with a family of functions , such that each is a homeomorphism defined on an open subset whose image is

- either an open subset of
- or an open subset of the half-space with respect to the subspace topology

satisfying the following conditions:

- For all , the function is times continuously differentiable on its domain of definition
- For all there exists an such that

**Definition (atlas)**:

Let be a differentiable manifold of class that is defined using a family of functions. An **atlas** of is a family , such that each is a homeomorphism defined on an open subset whose image is

- either an open subset of
- or an open subset of the half-space with respect to the subspace topology

so that the family is compatible with the family in the sense that for all and , the two functions and its inverse are times continuously differentiable on their respective domains of definition, and so that for each there exists a such that .

**Definition (chart)**:

Let be a differentiable manifold. Then a **chart** of is a function for some , where is any atlas of .

**Definition (boundary)**:

Let be a differentiable manifold equipped with an atlas . Further, let be the set of all such that maps to an open subset of the half-space equipped with its subspace topology w.r.t. . The **boundary** of , commonly denoted by , is defined as follows:

**Definition (differentiable manifold with boundary)**:

A **differentiable manifold with boundary** is a differentiable manifold equipped with an atlas such that the image of at least one chart is an open subset of (equipped with its subspace topology w.r.t. ) that intersects the boundary set .

**Proposition (the boundary of a differentiable manifold with boundary is a differentiable manifold)**:

Let be a differentiable manifold with boundary of class and let be an atlas of . Then is a differentiable manifold with boundary of class , and the family

constitutes an atlas of , where is defined as follows:

**Proof:** First, we prove that for each , the function is a homeomorphism.

To this end, it is prudent to observe that whenever and such that contains (where shall denote the domain of definition of ), then . This is because by the definition of , there exists a and an such that

- ;

yet the function is a homeomorphism, whence so is its inverse, so that upon assuming that , the closedness of the latter set permits the choice of an open neighbourhood of that does not intersect , and Brouwer's invariance of domain theorem then implies that

is an open neighbourhood of with respect to the Euclidean topology of , whereas the same set must be contained within the image of , which is in turn contained within , so that cannot intersect , for otherwise it would contain one of its boundary points and hence be not closed, contradicting the assumption that .

This proves that whenever , the function maps to . Hence, when restricted to the image of , the function is invertible and in fact a homeomorphism between a subset of endowed with its subspace topology and . In fact, restricted in this way, is a diffeomorphism of class .

Moreover, is a homeomorphism since the restriction of a homeomorphism is again a homeomorphism. Hence,

is a homeomorphism as the composition of homeomorphisms; indeed, is a homeomorphism between a subset of and a subset of .

Let now . Then

- ,

and the differentiability condition now follows from the fact that the composition of the three functions , and [[is times differentiable as the composition of times differentiable functions]].

Finally, by the very definition of the domains of definition of the functions in the family cover all of .