## The Quotient Manifold Theorem

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 $G$ acts smoothly, freely and properly on $M$, the set of orbits form a foliation of $M$ whose leaves are embedded submanifolds diffeomorphic to $G$. Flat charts for the foliation can then be used to construct coordinates on the orbit space.

## The Unitary Group (Lie Groups)

For any complex matrix $A$, the adjoint of A is the matrix $A*$ formed by conjugating the entries of $A$ and taking the transpose: $A* = \bar{A}^{T}$. Observe that $(AB)* = (\bar{A}\bar{B})^{T} = \bar{B}^{T}\bar{A}^{T} = B*A*$. For any positive integer $n$, the unitary group of degree n is the subgroup $U(n) \subseteq GL(n, \mathbb{C})$ consisting of complex $n \times n$ matrices whose columns form an orthonormal basis for $\mathbb{C}^{n}$ with respect to the Hermetian dot product.

We will show that $U(n)$ is a properly embedded Lie subgroup of $GL(n, \mathbb{C})$ of dimension $n^{2}$.

## The Cotangent Bundle (Natural Transformations)

11.8 Suppose $C$ and $D$ are categories, and $\mathscr{F}, \mathscr{G}$ are (covariant or contravariant) functors from $C$ to $D$. A natural transformation $\gamma$ from $\mathscr{F}$ to $\mathscr{G}$ is a rule that assigns to each object $X \in Ob(C)$ a morphism $\lambda_{X} \in Hom_{D}(\mathscr{F}(X), \mathscr{G}(X))$ in such a way that for every pair of objects $X, Y \in Ob(C)$ and every morphism $f \in Hom_{C}(X,Y)$, the following diagram commutes…

## Level Sets

If $M$ and $N$ are smooth manifolds and $\Phi: M \rightarrow N$ is a smooth submersion, then each level set of $\Phi$ is a properly embedded submanifold whose codimension is equal to the dimension of $N$.