In this section we prove that smooth, free, and proper group actions always yield smooth manifolds as orbit spaces. The basic idea of the proof is that if acts smoothly, freely and properly on , the set of orbits form a foliation of whose leaves are embedded submanifolds diffeomorphic to . Flat charts for the foliation can then be used to construct coordinates on the orbit space.

# Category Manifolds

# The Unitary Group (Lie Groups)

For any complex matrix , the **adjoint of A** is the matrix formed by conjugating the entries of and taking the transpose:

. Observe that . For any positive integer , the **unitary group of degree n** is the subgroup consisting of complex matrices whose columns form an orthonormal basis for with respect to the Hermetian dot product.

We will show that is a properly embedded Lie subgroup of of dimension .

# The Cotangent Bundle (Natural Transformations)

**11.8 **Suppose and are categories, and are (covariant or contravariant) functors from to . A natural transformation from to is a rule that assigns to each object a morphism in such a way that for every pair of objects and every morphism , the following diagram commutes…

# Level Sets

If and are smooth manifolds and is a smooth submersion, then each level set of is a properly embedded submanifold whose codimension is equal to the dimension of .