Work In Progress!
A Bit About Manifolds
A manifold, in the broadest sense, is a structure which is locally homeomorphic to $\mathbb{R}^n$
at each of its open sets. In layman's terms, this means that while a manifold may be sufficiently "abstracted"
such that its elements look nothing like those of $\mathbb{R}^n$, we can regardless treat the manifold as if
it's $\mathbb{R}^n$, with a few potential caveats. But first, the formal definition.
Definition 1 A manifold is a second countable Hausdorff space $M$ equipped with a collection of
"charts" (called an atlas) such that for each open $U \subset M$ there exists a homeomorphism $\phi: U \to V \subset \mathbb{R}^n$
where $V$ is itself open. This $\phi$ is called a chart. We require that each $x \in M$ be in the domain of some
chart. We require that the collection of charts be maximal subject to the conditions above.
If we want to be able to perform calculus on our manifolds, however, a bit more is required.
Definition 2 A smooth ($C^\infty$) manifold is a manifold with the additional
restriction that the change of coordinates defined by the composition of one chart and another's inverse is itself
$C^\infty$. That is, we require for two charts $\phi$ and $\psi$ that given $\phi: U \to U' \subset \mathbb{R}^n, \psi: V \to V' \subset \mathbb{R}^n$, the following be continuously differentiable of all orders
$$\phi \psi^{-1} : \psi (U \cap V) \to \phi (U \cap V)$$
Both of these definitions look like quite the mouthful at first, but there's really little in the way of hidden complexity.
The defacto differentiability requirements on the change of coordinates mappings make inherent sense. If this differentiability
was not presupposed, there'd be nothing restricting the manifold from being put together as a patchwork of locally smooth
neighborhoods that don't inherently connect in a "nice" way, i.e. there may exist discontinuities at the points where they join.
The notion of second countability may throw some for a loop, so I'll take some time to review it. A topological space is called
second countable if it exhibits a countable base $\mathcal{B} = \bigcup_{i=1}^\infty B_i$. This is a property that's characteristic
of many conventionally "nice" mathematical spaces, and its presence in this definition is a natural consequence of the homeomorphicity
with $\mathbb{R}^n$. $\mathbb{R}^n$ is second countable, and second countability is a topological invariant. More concretely,
since there exists a countable collection of open sets $\mathcal{B}$ which form a base for $\mathbb{R}^n$, we can take this collection and form
its equivalent in $M$ by applying the appropriate inverse chart $U_i = \phi_i^{-1}(B_i)$ for $B_i \in \mathcal{B}$ to get a countable
base $\mathcal{U} = \bigcup_{i=1}^\infty U_i$ for $M$.
Similarly, when we look at the Hausdorff condition in the manifold definition, it follows naturally from the homeomorphic nature of
the neighborhoods of the manifold with those of $\mathbb{R}^n$. Recall that a Hausdorff space is one for which every pair of distinct
points there exists a corresponding pair of disjoint neighborhoods of those points. This is another standard regularity condition which
is also a topological invariant.
A slightly more general way of thinking about manifolds is to consider them as topological spaces endowed with some additional structure
which varies based on the specific type of manifold under consideration. We can denote this as a pair $(T, S)$ where $T$ is the space
and $S$ is the structure. For example, we can specify the smooth $n$-dimensional real manifold as the pair $(\mathbb{R}^n, C^\infty)$.
When viewed in this light, any map from one manifold to a counterpart which "satisfies" the underlying structure of the two spaces is
called a morphism. We can further refine this set by considering only those morphisms whose inverses are themselves morphisms.
A map satisfying this additional condition is called an isomorphism. As an example, for standard manifolds the canonical
isomorphisms are the homeomorphisms. For smooth manifolds, the isomorphisms are diffeomorphisms. To formalize this, we define the
following.
Definition Suppose $X$ is a topological structure. We say that the function $F_X$ is a functional structure on $X$
if and only if for all open $U \subset X$
i
1. $F_X(U)$ is a subalgebra of the algebra of all continuous real-valued functions on $U$.
2. $F_X(U)$ contains all constant functions.
3. $V \subset U, f \in F_X(U) \implies f|_V \in F_X(V)$
4. $U = \bigcup U_\alpha$ and $f|_{U_\alpha} \in F_X(U_\alpha)$ for all $\alpha \implies f \in F_X(U)$.
Then the notions of morphism and isomorphism can be formulated accordingly as...
Definition A morphism of functionally structured spaces
$$(X, F_X) \to (Y, F_Y)$$
is a map $\phi:X \to Y$ such that composition $f \to f \circ \phi$ carries $F_Y(U)$ into $F_X(\phi^{-1}(U))$. An isomorphism is a morphism $\phi$ such that $\phi^{-1}$ exists as a morphism.
Tangent Spaces, Differentials, and All Things Adjacent
The fundamental concept of differential calculus is the derivative, which allows us to compute the line tangent to a curve at any
given point. The analogue for manifolds is the differential, which allows us to define tangent vectors to arbitrary trajectories
along a surface. It will turn out that the set of these vectors in fact comprises a vector space
Definition Take a smooth manifold $M$ and a smooth, parametrized curve $\gamma: \mathbb{R} \to M$ satisfying
$\gamma(0) = p$. Let $f$ be a smooth function defined on an open neighborhood of $p$. Then we define the directional derivative
of $f$ along $\gamma$ at $p$ to be
$$D_\gamma(f) = \frac{d}{dt} f(\gamma(t))|_{t=0}$$
We call the operator $D_\gamma$ the tangent vector to $\gamma$ at $p$. For a point $p \in M$ we denote by $T_p(M)$ the vector
space of all tangent vectors to $M$ at $p$.
To understand the operator $D_\gamma$ in a more global context, it behooves us to introduce algebras. Algebras exist as a simple
extension of vector spaces. Specifically, they are vector spaces having an associated bilinear product. That is they take elements from
two vector spaces and the product mapping sends these to a third vector space. In order to understand this view of derivatives along
manifolds, we need to present the algebra on which the tangent operator is defined.
To do so, consider the following definition. Suppose you have a smooth, real-valued function $f:M \to Y$ defined at some point $p \in M$ where
$M$ is a smooth manifold. We call the equivalence class of $f$ defined under the equivalence relation
$f_1 \sim f_2 \iff f_1(x) = f_2(x)$ for all $x$ in a neighborhood $U$ of $p$ a germ. We can extend the notion
of a germ to sets $S$ and $T$ via the relation $S \sim_p T$ if there exists a neighborhood $U$ of $p$ such that
$$S \cap U = T \cap U \neq \emptyset$$
The way we're defining a germ here is more specific than is necessary. Germs are
a more general concept which capture notions of local equivalence via some property for objects acting on a topological space.
Most often these objects are functions or maps with some sort of additional structure, e.g. smoothness or continuity, although this is
not strictly necessary. To define a germ one needs only a notion of equality and a topological space, as these are the most abstract
structures which allow us to define notions of locality via neighborhoods. When discussing germ equivalence between two maps $f$ and
$g$ we can consider that these maps need not be defined on the same domain, provided some caveats are satisfied. We require that
if $f$ has domain $S$ and $g$ has domain $T$ then $S$ and $T$ are germ equivalent via the definition for sets given above. We also
require that $f|_{S \cap V} = g|_{T \cap V}$ for some smaller neighborhood $V$ of $p$ satisfying $p \in V \subseteq U$.
We denote the germ at $p$ of a function $f$ as $[f]_p$ and we denote the equivalence relation it defines as $f \sim_p g$.
A $K$-linear derivation $D:A_1 \to A_2$ where $A_1, A_2$ are algebras is a a $K$-linear map satisfying the product rule
$$D(ab) = aD(b) + D(a)b$$
Defining Tangent Vectors via Derivations
Suppose $M$ is a $C^\infty$ manifold. For any $x \in M$ we define a derivation by choosing a linear map $D:C^\infty(M) \to
\mathbb{R}^n$ satisfying
$$\forall f, g \in C^\infty(M) \quad D(fg) = f(x)D(g) + D(f)g(x)$$
We can define addition and scalar multiplication on the set of derivations in the same way we do for elements of a vector space.
We call the vector space obtained the tangent space to $x$ in $M$ and denote it as $T_x(M)$.
Let $\gamma: (-1, 1) \to M$ be a differentiable curve with $\gamma(0) = x$. Then the derivation $D_\gamma$ at $x$ is defined by
$D_\gamma(f) := (f \circ \gamma)'(0)$
Getting Infinitesimal with Differentials
When calculus is taught, the notion of a differential is frequently introduced and is usually vaguely described as the
infinitesimal change in some variable. Haphazard teachers may introduce the chain rule by making generalizations such as
that one can cancel differentials like $dx$ in the equation
$$\frac{df}{dz} = \frac{df}{dx} \frac{dx}{dz}$$
but this isn't entirely true, or at least it doesn't tell the whole story. We also see the differential pop up in the notation for
integration, i.e. we write $\int_{a}^b f(x) \ dx$ but we never really explain what the differential is and what it's doing there.
The general structure in which differential geometry operates makes things much clearer. What we'll see is that the differential
is actually an object which instructs us in how to map between the tangent spaces of different manifolds. The differential is
defined with respect to such a smooth map. In fact, the differential is an operator that takes in a map between manifolds and
spits out a map between their tangent spaces. For more specifics, let's get to the definition.
Definition Let $\phi : M \to N$ be a smooth map between manifolds. Then we say the the differential of $\phi$ is
the linear map $d\phi$ satisfying
$$d\phi(D)(g) = D(g \circ \phi)$$
and also satisfying
$$d\phi d\psi = d(\phi \circ \psi)$$
Taking this Tangent Thing a Step Further - Tangent Bundles
Before giving the formal definition, I'd like to give a brief intuitive explanation of what a tangent bundle constitutes. The
tangent bundle is unique in that it is itself a manifold $T(M)$, obtained by construction from the tangent spaces at each point
of some different manifold $M$. It can also be thought of as a cartesian product between points $p \in M$ and the associated tangent
vectors $v_p \in T_p(M)$. The cartesian product formulation naturally gives rise to a projection map $\pi:T(M) \to M$ from the
tangent bundle back to the manifold from whence it was constructed. Formally, we define the tangent bundle as the union of
tangent spaces
$$T(M) = \bigcup \{T_p(M) | p \in M\}$$
However, to make the manifold valid, we still need to specify an appropriate set of charts.
A manifold is said to be parallelizable if its tangent bundle is trivial.
An Instance of a More General Phenomenon
The tangent bundle is a specific instance of a concept called the vector bundle, which provides a method for constructing
a vector space parameterized by a different sort of space such as a topological space $X$ or a manifold $M$. One of the characteristics
of vector bundles is that they are locally trivial. They are also consist of a base space $V$ and a total space $M$. The base
space is formed as a family of vector spaces over points defined in $M$, and that family is required to be smoothly varying.
The formal definition involves both a base space $B$ and a total space $E$. A tangent
A General Instance of an Even More General Phenomenon
A vector bundle, as general as it is, is actually an instance of an even more general general phenomenon. The idea is we replace the
vector part of vector bundle with a fiber of some mapping. This mapping can be anything we want, subject to a few
conditions. This is used to construct spaces which look locally like product spaces but from a global view have a more general
topological structure.
Formal Definition A fiber bundle is a tuple $(B, E, \pi, F)$ consisting of a base space $B$, a total space $E$,
a continuous surjective projection map $\pi:E \to B$ and a fiber $F$, all satisfying the following
1. For every $x \in E$, there exists an open neighborhood $U \subset B$ of $\pi(x)$ such that there exists a homeomorphism
$\varphi: \pi^{-1}(U) \to U \times F$