
\part{Fundamentals}

\chapter{Manifolds}




Informally, they capture the idea that some topological spaces 'look' Euclidean locally, but are possibly not Euclidean. For example, Earth's surface 'looks' like $\mathbb{R}^{2}$ from a human's point of view (ignoring bumps and mountains on the ground), but we know that the Earth is topologically a $2$-sphere; if you walk in a straight line for long enough, you eventually reach your starting point again. Alternatively, if one shines a light parallel with the ground, it will traject above a far enough observer due to the curvature of the earth; this is how we know the Earth isn't flat.

Manifolds are especially important in differential topology, which devotes itself to the studies of 'differentiable manifolds'.




\chapter{Curves and surfaces}
Differential topology is essentially the study of differentiable manifolds.


\section{Topological curves}


Consider the following example of a jump.

A dog jumps with ltitude according to $f(t)=\sin(\pi t)$ with $t \in [0,1]$ indexing the start and end of the jump.



\begin{definition}
A \emph{topological curve}
\end{definition}

\begin{definition}
A \emph{closed curve}
\end{definition}


\begin{definition}
A \emph{simple closed curve}
\end{definition}

This is also known as a Jordan curve.




We are familiar with topological spaces and for this we require a \emph{differentiable manifold}; This is a special topological space  that is Hausdorff, and is equipped with a smooth coordinate system in terms of $\mathbb{R}^n$ for neighborhoods of any point. The idea is that since we have a notion of differentiability in $\mathbb{R}^n$ due to vector analysis, we want to use a smooth coorcinate system to port this to more abstract spaces.


\begin{definition}[Manifold]
a \emph{manifold} is a Hausdorff space such that there exists some natural number $n$ where for every point $p \in X$ there is some neighborhood $V$ of $p$ homeomorphic to the Euclidean space $\mathbf{R}^n$.
\end{definition}

Informally, we say that manifolds are locally Euclidean Hausdorff spaces. One can study manifolds without any extra structure . That said, the coolest thing about manifolds is that with a bit more structure it allows a notion of differentiation.



Chart
\begin{definition}[Chart]
Given a topological space $M$, a \emph{chart on $M$} is an ordered pair $(U,\phi)$, where $U$ is open in $M$ and $\phi : U \to \mathbb{R}^n$ is a homeomorphism to some open set of $\mathbb{R}^n$ 
\end{definition}



\begin{definition}[Atlas]
Given a topological space $M$, a \emph{atlas on $M$} is  a set of charts on $M$ whose union equals $M$
\end{definition}


\begin{definition}[Differentiable atlas]
Differentiable atlas of $M$ is an atlas such that for any 2 charts $(U,\phi),(W,\psi)$, the function $\phi \circ \psi^{-1}$ is smooth.
\end{definition}


\begin{definition}[Compatible atlases]
2 atlases for $M$ are \emph{compatible atlases} if their union is a differentiable atlas for $M$
\end{definition}




\begin{definition}[Differentiable manifold]
Manifold with a maximum differentiable atlas (consider a base atltas and generate the 'maximal atlas' by taking the union with all other compatibel atlases to base atlas).
\end{definition}


\section{Differentiation on differentiable manifolds}

Differentiable manifolds are among the more gabstract spaces on which a notion of differentiation is possible; it does this by using charts to translate the function into a representation in terms of $\mathbb{R}^n$

\begin{definition}[Differentiable function onto $\mathbb{R}$]
Let $M$ be a manifold, $f : M \to \mathbb{R}$ is \emph{differentiable at $p$} iff for any differentiable chart $(U,\phi)$ of $p$ such that $f \circ \phi^{-1}$ is differentiable at $\phi(p)$
\end{definition}

\begin{definition}[Differentiable function between manifolds]
Let $M,N$ be differentiable manifolds, $f : M \to N$ is \emph{differentiable at $p$} iff for any differentiable chart $(U,\phi)$ of $p$ and $(W,\eta)$ of $f(p)$ such that $\eta \circ f \circ \phi^{-1}$ is differentiable at $\phi(p)$
\end{definition}

One can prove that if this holds for at least one chart, it holds of any chart! So we could have relaxed our definition to require any chart that makes such compositions differentiable to exist, and it would be logically equivalent.

\subsection{Diffeomorphism}

Let's say 

\begin{definition}[Diffeomorphism]
A \emph{diffeomorphism} between two differential manifolds $N$ and $M$ is a bijective function $f : N \to M$ such that both $f$ and $f^{-1}$ are continuously differentiable.
\end{definition}



\section{Tangent spaces}

When working in $\mathbb{R}^n$, a differentiable curve at some point could evolve in any direction; for any vector you can imagine, there is a differentiable curve that has that vector as its tangent at some point. When working generally on differentiable manifolds, we may be subject to a more limited amount of tangents; for instance when considering differentiable curves on a sphere at a certain point, we can't have tangents can have a component that would make the particle 'pop off' the sphere. The tangents are limited to the tangent plane of the sphere at that point, rather than any vector.


So given a point in some differentiable manifold, we need to consider that not all tangents are possible and we should enumerate a set that contains the possible tangents at that point.

How will we generate all the tangents of a point on the manifold? By considering all possible differentiable curves on that manifold passing through that point, and putting all the derivatives into a set. 

We'll define the tangent space is is a set of all the possible tangents of differentiable curves on the manifold pasing through $p$. 


\begin{definition}[Tangent vector]
Given a differentiable manifold $M$, point $p\in M$ and chart around $p$ $\varphi : U \subseteq M \to \mathbb{R}^n$, then a \emph{tangent vector of $M$ at $p$} is the equivalence class of some curve $\gamma : [-1,1] \to M$ with $\gamma(0)=p$, defined in the following way.  
\[ \gamma'(0) = \{ \eta : [-1,1] \to M : \gamma(0)=\eta(0)=p \phi \circ \eta = \phi \circ gamma\}\]
\end{definition}

We can indeed check that this set is an equivalence relation (i.e the sets generated by 2 curves are either identical or disjoint).

As an alternative to tangent vectors one may define many concepts of differential topology by means of derivations; a concept in differential algebra (this is found in the advanced part). A derivation is basically a linear map satisfying 'Leibniz's law' (i.e product rule) on an algebra; it models differentiation as a purely algebraic object.


We now develop tangent spaces by means of tangent vectors.

\begin{definition}[Tangent space]
Given a differentiable manifold $M$, point $p\in M$ and chart around $p$ $\varphi : U \subseteq M \to \mathbb{R}^n$, then the  \emph{tangent space of $M$ at $p$} is the set $T_{p}M$ of all tangent vectors of $M$ at $p$.
\end{definition}


The idea is that if we consider all possible curves through a point on a space and take their derivatives (i.e tangents), we end up with the space of all tangent vectors at a point.

If we were to use derivations, the idea would be that each derivation 'defines' a type of differentiation as the derivative along a specific curve (or curve equivalence classes). This means that all the notions of the derivative defined by derivations essentially correspond with a tangent vector.


\begin{definition}[1-form]
	Function between 
\end{definition}

Tangent vectors once again help us define pushforwards.
\begin{definition}[Differential (pushforward)]
	For a differentiable function between differentiable manifolds $\mathbf{f} : M \to N$, The \emph{differential of $\mathbf{f}$ at $\mathbf{x}_0$} is a function $\mathrm{d}\mathbf{f}_{\mathbf{x}_0} : T_p M \to T_{\mathbf{f}(p)}N $ defined as such.

	\[d \mathbf{f}_{p} (\gamma'(0)) = (\mathbf{f} \circ \gamma)'(0)\]
\end{definition}

\begin{theorem}
The differential is a linear map of the following form.
	\[ d \mathbf{f}_{p} (\gamma'(0)) = \mathbf{J}_{\phi \circ \mathbf{f} \circ \psi^{-1}}(\psi (p)) \gamma'(0) \]
\end{theorem}


Differentials act as a way to communicate a linear approximation of differential change from one manifold to the other.
It is a generalization of the total differential, which is what it reduces to when the manifolds are Euclidean spaces.


\begin{definition}[Critical point (differential geometry)]
	For a function $f : U \subseteq \mathbb{R}^n \to \mathbb{R}^m$, $p \in U$ is critical point iff its pushforward $df_p$ is not surjective
\end{definition}




\begin{definition}[Tangent bundle]
	\[TM = \bigsqcup_{p\in M} T_p M\]
\end{definition}
The tangent bundle, well, 'bundles up' all the tangent spaces (double counting elements in multiple tangent spaces, hence why disjoint product is used).


One priminent example are differentiable curves; diffeomorphic classes represent all the reparametrizations of the differentiable curves.

- autodiffeomorphism
- inverse function theorem (manifolds)

- compatible pair of charts
- compatible chart to atlas
- stereographic projection
- manifold

submanifold
Orientable atlas

- n-sphere
- real projective space
- complex projective space
- Grassmanian


- tangent space

- Fiber bundle


Diffeomorphism



De Rham cohomology

\part{Advanced}

Derivation space


\chapter{Lie theory}
\begin{definition}[Lie group]
Lie group is a group $(M,\cdot)$ such that $M$ is also a differentiable manifold and multiplication and the inverse opertation are differentiable functions.
\end{definition}
Lie algebra
