Definition: Tangent Space

The **tangent space** $T_P M$ to $M$ at point $P$ is just the set of all tangent vectors at $P$.

$T_PM$ is obviously a vector space of the same dimension as $M$ #-> because even though $M$ may be embedded in a higher dimensional space, the tangent space has same dimension as $M$'s local euclidian space

Definition: Tangent Bundle

The tangent bundle $TM$ of a smooth manifold $M$ is the set of pairs $(P, \xi)$ where $P\in M$ and $\xi \in T_PM$.

The observation is that the tangent bundle is also a smooth manifold of dimension $2M$. #-> Because the points have dimension $M$ and the vectors have dimension $M$

The charts for $TM$ are defined by taking a chart $U$ for $M$, then considering the subset $TU$ that corresponds to restricting $TM$ to points in $P$.

Then, because we have a fixed local coordinate system on $U$ (because $M$ is a smooth manifold), each tangent vector for some $P\in U$ is uniquely determined by its components.

This means that there is a natural bijection between $TU$ and $\phi(U) \times \mathbb{R}^m$, which is a subset of $\mathbb{R}^{2m}: (P, \xi) \leftrightarrow (x_1, \dots, x_m, \xi_1, \dots, \xi_m)$

The transition functions between charts $TU_1$ and $TU_2$ are smooth, so $TM$ carries a natural structure of a smooth manifold of dimension $2m$.

If we assume we have a particular tangent vector $\xi(P)$ at each point $P \in M$, then we have a vector field on $M$. We can think of components of $\xi$ being functions of the local coordinates of $P$.

The vector field is said to be smooth if these functions are smooth.

We see that for every vector field $\xi = \xi(P)$, we can assign it a system of first-order ODEs, where $\frac{d}{dt}\mathbf{x}(t) = \xi(\mathbf{x})$. In other words, we think of the vector assigned to each point as a derivative of that point, with each component of that vector being an ODE wrt components of the original point.

In other words, vector field as a "flow"

The opposite holds true as well: any system of ODEs of this kind determines a vector field on $M$.

Smooth curves satisfying $\frac{d}{dt}\mathbf{x}(t) = \xi(\mathbf{x})$ are called integral curves.

According to the uniqueness and existence theorem for ODE, for each $P \in M$ there exists a unique curve $\gamma(t)$ s.t. $\gamma(0) = P$ and $t \in (-\epsilon, \epsilon)$.

If each curve $\gamma(t)$ can be defined for $t \in (-\infty, \infty)$, then the vector field is called "complete."

If $\xi$ is complete, then for any $t \in \mathbb{R}$ we can define a diffeomorphism $\phi^t = \phi^t_\xi: M \to M$ which is just the translation along integral curves by t.

To find the image of $P$ along $\phi^t$ we take the integral curve $\gamma(t)$ s.t. $\gamma(0) = P$ and by definition $\phi^t(P) = \gamma(t)$. A property of this is that $\phi^s\circ \phi^t = \phi^{t + s}$. So, we say that $\phi^t$ is a one-parameter group of diffeomorphisms.

On the other hand, a one-parameter group of diffeomorphisms $\phi^t$ on $M$ defines a vector field by $\xi(P) = \frac{d}{dt}|_{t=0} \phi^t(P)$

Conclusion: vector fields on $M$ = autonomous ODE on $M$ = one-parameter groups of diffeomorphisms

Example:

If we have a vector field $\xi = (x, y) \in \mathbb{R}^2$, then our integral curves are $\gamma(t) = (x_0 e^t, y_0 e^t)$

the diffeomorphisms $\phi^t$ are homotheties centered at the origin and with dilation factor $e^t$ #-> homotheties are translations of affine spaces, determined by a scaling point and a scaling factor

Example 2:

If on $S^3 = \{x^2 + y^2 + u^2 + v^2 = 1\}$ we consider a vector field $\xi(P) = (-y, x, -v, u)$, where $P \in S^3$, then the integral curve passing through $P = (x_0, y_0, u_0, v_0)$ can be described as follows:

$\begin{align*} (x_0 \cos{t} - y_0 \sin{t}, y_0 \cos{t} + x_0 \sin{t}, u_0 \cos{t} - v_0 \sin{t}, v_0 \cos{t} + u_0 \sin{t}) \end{align*}$

This is a circle of radius 1 centered at the origin and lying in the 2-dim plane spanned by the vectors $P = (x, y, u, v)$ and $\gamma(P) = (-y, x, -v, u)$.

The corresponding $\phi^t$ can be understood as the composition of two rotations by angle $t$ in the planes $O_{xy}$ and $O_{uv}$.

This is an example of a left invariant vector field on $SU(2)$.

Definition: Lie Bracket

The **Lie bracket** of vector fields is a bilinear differential operator which assigns to any vector fields $\xi$ and $\eta$ on a smooth manifold $M$, a new vector field denoted by $[\xi, \eta]$.

In local coordinates, the components of $[\xi, \eta]$ are defined by $[\xi, \eta]^k = \xi^i (\frac{d}{d x^i} \eta^k) - \eta^i (\frac{d}{d x^i} \xi^k)$

where we sum along the $i$ index

If we think of vector fields as derivations then $[\xi, \eta]$ is defined by $[\xi, \eta](f) = \xi(\eta(f)) - \eta(\xi(f))$ #-> What exactly is a derivation? Derivative as an abstract operation? Note to reader: At this point I didn't understand derivations

Properties:

- Bilinearity over $\mathbb{R}$: $[a\xi_1 + b\xi_2, \eta] = a[\xi_1, \eta] + b[\xi_2, \eta]$

- Skew-symmetry: $[\xi, \eta] = -[\eta, \xi]$

- Leibnitz rule: $[\xi, f(\eta)] = f[\xi, \eta] + \xi(f)\eta$ for any $f$ smooth over $M$ #-> Can't conceptualize what this means without fully understanding

- Jacobi identity: $[\zeta, [\xi, \eta]] + [\xi, [\eta, \zeta]] + [\eta, [\zeta, \xi]] = 0$

- if $\xi_1, \dots, \xi_k$ are pairwise commuting, linearly independent vector fields, there is locally a coordinate system $(x_1, \dots, x_m)$ s.t. $\xi_1 = \frac{\partial}{\partial x_1}, \dots, \xi_k = \frac{\partial}{\partial x_k}$

- If $N$ is a submanifold of $M$ and $\xi$ and $\eta$ are tangent to $N$, then so is $[\xi, \eta]$

- If $\xi$ and $\eta$ commute (i.e. $[\xi, \eta] = 0$), then the corresponding flows also commute

Definition: Differential

Let $F: M \to N$ be a smooth map, $P \in M$ and $Q = F(P)$.

Then, the **differential** $dF$ (at point $P$) is a linear map between tangent spaces $T_PM$ and $T_QN$. The following are each definitions of $dF$:

- Take $\xi$ in $T_P(M)$ and choose any curve $\gamma(t)$ through $P$ such that $\gamma'(0) = \xi$. Then, $\eta = dF(\xi)$ is the tangent vector to the image of $\gamma$ at the point $Q$:

$\eta = dF(\xi) = \frac{d}{dt}|_{t=0} F(\gamma(t)) \in T_Q(N)$ #-> In other words, we take some tangent vector, find its antiderivative, take its image, and then take its derivative

- Consider $\xi \in T_P(M)$ as a derivation. Then, $\eta = dF(\xi)$ is the derivation at $Q$ which acts on an arbitrary function $g: N \to \mathbb{R}$ s.t. $\eta(g) = \xi(g\circ F)$

- Choosing local coordinates $(x_1, \dots, x_m)$ and $(y_1, \dots, y_n)$ in neighborhoods of $P$ and $Q$, letting $\xi = (\xi_1, \dots, \xi_m)$ then the components of $\eta = dF(\xi)$ are as follows:

$\eta_j = \sum_{i=1}^m \frac{\partial}{\partial x_i} f_j \xi_i$ for $j = 1, \dots, n$ #-> in other words, the sum of partials of f wrt each coordinate times corresponding $\xi$

In matrix form, $\eta = dF(\xi) = \mathbf{J} \xi$, where $\mathbf{J}$ is the Jacobi matrix of $F$ at point $P$

Definition: Immersion

A smooth map $F: M \to N$ is called an **immersion** if its differential is a monomorphism (i.e. one-to-one) for any $P \in M$

Definition: Submersion

A smooth map $F: M \to N$ is called a submersion if its differential is an epimorphism (i.e. onto) for any $P \in M$

Definition: Embedding

A smooth map $F$ is called an **embedding** if $F$ is an immersion and $M$ is homeomorphic with $F(M)$

Definition: Covering

If we assume $dF$ to be an isomorphism for each point, then we have a local diffeomorphism between some neighborhoods of $P$ and $F(P)$. If we assume that for each point $Q$ there is a neighborhood $V(Q)$ such that $F^{-1}(V)$ is a disjoint union of some neighborhoods $U1, U2, \dots$ of $M$ s.t. $F|_{U_i}: U_i \to V$ is a diffeomorphism, then $F$ is called a **covering**. #-> how to interpret...

Examples:

- Smooth regular curve $\gamma: \mathbb{R} \to \mathbb{R}^2$ with self-intersections represents an immersion

- Letting $M = T^2$ a two-dimensional torus with standard angle coords $\phi_1, \phi_2$, then the map $\mathbb{R} \to T^2$ by $F(t) = (at, bt)$ where $(a, b) \neq (0, 0)$ is an immersion. If $\frac{a}{b} \in \mathbb{Q}$, then image of $\mathbb{R}$ is a closed curve on $T^2$; if $\frac{a}{b} \notin \mathbb{Q}$, we have the so-called irrational winding on $T^2$. #-> This is intuitive, because if $\frac{a}{b} \in \mathbb{Q}$ then $(a, b)$ is proportional to $(c, d) \in \mathbb{Z}^2$ so $F(4\pi) = F(0)$

- Consider $F: O(3) \to S^2$ which assigns to each $A \in O(3)$ its first row $(a_{11}, a_{12}, a_{13}) \in S^2$. Exercise: $F$ is a submersion. #-> the Jacobian of $F$ looks like just the first row?

- If $F: G_1 \to G_2$ is a smooth homomorphism of Lie groups then $F$ is either immersion or submersion. If $F: G_1 \to G_2$ is such that $dF: T_{e_1}G_1 \to T_{e_2}G_2$ is a linear isomorphism, then $F$ is a covering.

- The orthogonal projection $p: S^2 \to \mathbb{R}^2 = O_{xy}$ is neither an immersion nor submersion #-> Why? at the very least, the surjectivity seems like it would work. Maybe the intersection portion breaks?

- $F: \mathbb{R} \to S^1$ defined by $F(x) = e^{ix}$ is a covering (and, consequently, both submersion and immersion) #-> $dF = ie^{ix}$? I guess I don't get the way the differentials on tangent spaces work

- The map $f: (-2\pi, 2\pi) \to S^1$ defined by $f(x) = e^{ix}$ is not a covering (but still submersion and immersion) #-> I'm guessing that around $f(0)$, the neighborhood will include $(-2\pi, a)$ for some $a$ that will screw things up? Maybe not diffeomorphism?

Definition: Submanifold

A subset $N \subseteq M$ is called a **submanifold** if $N$ is a smooth manifold and the inclusion $p: N \hookrightarrow M$ is an embedding. #-> As long as the inclusion is homeomorphic, $N$ is a submanifold (as the inclusion is injective by definition)

Equivalent definition: $N \subseteq M$ is a submanifold if locally $N$ can be given by a system of equations $f_1(x_1, \dots, x_m) = 0, \dots, f_{m - k}(x_1, \dots, x_m) = 0$ satisfying the regularity condition (i.e. the rank of the Jacobi matrix = $m - k$)

If we let ourselves use any regular coordinate changes, we can always choose local coordinates in $M$ that $N$ will (locally) be given by $z_{k+1} = 0, z_{k+2} = 0, \dots, z_{m} = 0$.

Theorem: Embedding of Manifolds

Theorem: Any connected smooth $m$-dimensional manifold $M^{m}$ can be smoothly embedded into the Euclidian $2m$-space $\mathbb{R}^{2m}$.