## Fundamental Theorem on Flows

Let V be a smooth vector field on a smooth manifold M. There is a unique smooth maximal flow whose infinitesimal generator is V. The flow has the following properties:

(a) For each point on the manifold, there is curve theta that is the unique maximal integral curve of V starting at said point.

(b) If there is another point in the flow domain… (need to reinterpret what this means, actually, as I can’t parse the proof as I’m writing this).

(c) The integral curves are diffeomorphisms, whose inverses can be computed by reversing time.

…we’ll get back to this and elaborate on what it means with respect to existence and uniqueness of solutions to ODEs. I just wanted you to see where my wordplay came from. Btw, integral curves are smooth immersions.
🙂

## 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…