Definition: Preliminaries

Consider a smooth manifold $M$ with a smooth vector field $\xi$. Then, the vector field defines a system of ODEs on M: $\frac{d}{dt}x = \xi(t)$

A smooth curve $\gamma(t), t \in (-\epsilon, \epsilon)$ is an **integral curve** of $\xi$ if $\frac{d}{dt}\gamma(t) = \xi(\gamma(t))$ (in other words, $\gamma(t)$ is a solution of $\frac{d}{dt}x = \xi(t)$)

Existence and uniqueness theorem: For any $x \in M$ there is a unique integral curve $\gamma_x(t)$ passing through it (s.t. $\gamma_x(0) = x$).

Flow $\phi_t$: To each vector field $\xi$, we can assign (at least locally) a diffeomorphism $\phi_t$ which shifts each point $x$ along $\xi$ by time $t$. In other words, $\phi_t(x) = \gamma_x(t)$.

"Locally" means that usually the flow $\phi_t$ is defined only for sufficiently small $t$ (depends on $x$).

**Completeness**: If each integral curve $\gamma_x(t)$ can be extended (in the sense of $t$) to $\mathbb{R} = (-\infty, \infty)$, then $\xi$ is called complete. Equivalently, completeness of $\xi$ means that the flow $\phi_t: M \to M$ is globally defined for all $t \in \mathbb{R}$.

Since $\phi_t\circ phi_s = \phi_{t + s}$, the flow can be thought of as a group under composition.

The **differential** of a smooth map $F: M \to N$ is the map $dF: TM \to TN$ defined by $dF(\frac{d}{dt}\gamma(t)) = \frac{d}{dt}F(\gamma(t))$. (sends tangent vectors to tangent vectors)

**Lie bracket of vector fields**: Given smooth vector fields $\xi$ and $\eta$ on $M$, we can introduce a new vector field $[\xi, \eta]$ s.t.:

- $[\xi, \eta]^k = \xi^i \frac{d}{d x^i} \eta^k - \eta^i \frac{d}{d x^i} \xi^k$ (in local coords)

- $[\xi, \eta](f) = \xi(\eta(f)) - \eta(\xi(f))$

.#-> This lie bracket represents taking the differential of $\xi$ with respect to $\eta$, or is it the other way around?

If $F: M \to N$ is a diffeomorphism, with $dF: TM \to TN$ its differential, then $dF([\xi, \eta]) = [dF(\xi), dF(\eta)]$ #-> This is obvious because $F$ is a transformation of local coords and the Lie bracket is independent on local coords?

Vector fields $\xi, \eta$ commute (i.e. $[\xi, \eta] = 0$) if and only if the corresponding flows commute (i.e. $\phi_{\eta}\circ \phi_{\xi} = \phi_{\xi}\circ \phi_{\eta}$)

Throughout, we'll use the notation $L_a: G \to G, x \mapsto ax$ (left translation by $a \in G$) and $R_a: G \to G, x \mapsto xa$ (right translation by $a \in G$)

Clearly $L_a$ and $R_a$ are diffeomorphisms of $G$ onto itself

We can use $dL_a: TG \to TG$ and $dR_a: TG \to TG$ for the differentials, and use the same notation for the differential at some fixed point $x \in G$ as well.

Left and right translations commute: $L_a\circ R_b = x \mapsto axb = R_b\circ L_a$

But in general, $L_a\circ L_b \neq L_b\circ L_a$ and $R_a\circ R_b \neq R_b\circ R_a$

also $L_b\circ L_a = L_{ba}$ and $R_b\circ R_a = R_{ba}$

The same applies to differentials

Definition: Left Invariance

A vector field $\xi$ is called **left invariant** if it is preserved under left translations.

In other words, for any $a \in G$, $dL_a(\xi(x)) = \xi(L_a(x))$.

Similarly, a vector field $\eta$ is called **right invariant** if $dR_a(\eta) = \eta$ for any $a \in G$.

In other words, if we consider the values $\xi(x)$ and $\xi(y)$ of our vector field $\xi$ at two distinct points $x \in G$ and $y = ax \in G$, then $dL_a(\xi(x)) = \xi(ax)$. Similar for right invariance.

Theorem: Left-invariant Construction

Take an arbitrary vector $\xi_0 = \xi(e) \in T_e G$ at the identity $e \in G$, and define a tangent vector $\xi(a) \in T_a G$ at any other point $a \in G$ by putting $\xi(a) = dL_a(\xi_0)$.

Then, we get a tangent vector $\xi(a)$ for any $a \in G$. $\xi$ is smooth because $L_a$ depends on $a \in G$ smoothly.

Such a vector field is left-invariant

Proof:

We only need to verify the condition that $dL_a(\xi(x)) = \xi(ax)$ for any $x, a \in G$. Notice that for $x = e$, this condition holds by construction.

For any other $x \in G$, we have $dL_a(\xi(x)) = dL_a(dL_x(\xi_0)) = dL_a\circ dL_x(\xi_0) = dL_{ax}(\xi_0) = \xi(ax)$

Corollary:

A left invariant vector field $\xi$ is uniquely defined by its "initial" value $\xi_0 = \xi(e)$ at the identity. Moreover, $\xi_0$ can be chosen arbitrarily

Corollary:

The set of left invariant vector fields is a vector space of dimension $\text{dim } G$, which is naturally isomorphic to the tangent space $T_e G$ to $G$ at the identity $e$.

This isomorphism is established by the construction $\xi(a) = dL_a(\xi_0)$

**Properties**:

If $\xi$ is a left invariant vector field on $G$, then

**Proposition 1**:

Let $\gamma_e(t)$ be the integral curve of $\xi$ passing through the identity $e$ (i.e. $\gamma_e(0) = e$). Then, the integral curve $\gamma_x(t)$ of $\xi$ passing through $x$ is $x \gamma_e(t) = L_x(\gamma_e(t))$.

**Proof**:

$\frac{d}{dt} L_x(\gamma_e(t)) = dL_x(\frac{d}{dt} \gamma_e(t))$ by def of differential, $dL_x(\frac{d}{dt} \gamma_e(t)) = dL_x(\xi(\gamma_e(t))$ by $\gamma_e(t)$ being an integral curve of $\xi$, and = $\xi(L_x(\gamma_e(t))$ by $\xi$ being left-invariant.

Thus, $L_x(\gamma_e(t)) = x \gamma_e(t)$ is an integral curve of $\xi$. It suffices that $x \gamma_e(0) = xe = x$ to show that $x \gamma_e(t)$ passes through $x$.

**Corollary**:

The left translation of any integral curve of $xi$ is again an integral curve

**Corollary**:

The flow $\phi_{\xi}^t: G \to G$ of $xi$ is defined by $\phi_{\xi}^t(x) = x \gamma_e(t)$, where $\gamma_e(t)$ is the integral curve of $\xi$ through the identity.

**Proposition 2**:

$\xi$ is complete, i.e. the flow $\phi_{\xi}^t: G \to G$ of $\xi$ is well defined for all $t \in \mathbb{R}$

**Proof**:

The definition of the flow for a left-invariant vector field $\xi$ is defined as $\phi_{\xi}^t(x) = x \gamma_e(t)$. If $\gamma_e(t)$ is defined on $(-\epsilon, \epsilon)$, then $\phi_{\xi}^t$ is defined on the whole group for $t \in (-\epsilon, \epsilon)$.

Then, $\phi^t$ can naturally be defined for all $t \in (-\infty, \infty)$ just by iterating: $\phi_{\xi}^t = \phi_{\xi}^{\frac{t}{k}}\circ\dots\circ\phi_{\xi}^{\frac{t}{k}}$ $k$ times where $\frac{t}{k} \in (-\epsilon, \epsilon)$.

Definition: One-parameter Subgroup

Definition: A smooth map $f: \mathbb{R} \to G$ is called a **one-parameter subgroup** of $G$ if $f(s + t) = f(s)f(t)$ for any $t, s \in \mathbb{R}$ #-> i.e. it can be parameterized by $\mathbb{R}$