Topology is the plumbing of calculus.

The key concept of topology is a neighbourhood.

Just by havin the notion of neighbourhood, concepts such as limit and continuity can be defined without the need to specify a precise numerical value to the distance between two points with a metric.

As an example. consider the orthogonal group, which is also naturally a topological space. That group does not usually have a notion of distance defined for it by default. However, we can still talk about certain properties of it, e.g. that the orthogonal group is compact, and that the orthogonal group has two connected components.

Basically it is a larger space such that there exists a surjection from the large space onto the smaller space, while still being compatible with the topology of the small space.

We can characterize the cover by how injective the function is. E.g. if two elements of the large space map to each element of the small space, then we have a double cover and so on.

The key concept of topology.

We map each point and a small enough neighbourhood of it to $R_{n}$, so we can talk about the manifold points in terms of coordinates.

Does not require any further structure besides a consistent topological map. Notably, does not require metric nor an addition operation to make a vector space.

A notable example of a Non-Euclidean geometry manifold is the space of generalized coordinates of a Lagrangian. For example, in a problem such as the double pendulum, some of those generalized coordinates could be angles, which wrap around and thus are not euclidean.

Collection of coordinate charts.

The key element in the definition of a manifold.

A generalized definition of derivative that works on manifolds.

TODO: how does it maintain a single value even across different coordinate charts?

TODO find a concrete numerical example of doing calculus on a differentiable manifold and visualizing it. Likely start with a boring circle. That would be sweet...

TODO what's the point of it.

Bibliography:

- www.youtube.com/watch?v=j1PAxNKB_Zc Manifolds #6 - Tangent Space (Detail) by WHYB maths (2020). This is worth looking into.
- www.youtube.com/watch?v=oxB4aH8h5j4 actually gives a more concrete example. Basically, the vectors are defined by saying "we are doing the Directional derivative of any function along this direction".One thing to remember is that of course, the most convenient way to define a function $f$ and to specify a direction, is by using one of the coordinate charts.We can then just switch between charts by change of basis.

- jakobschwichtenberg.com/lie-algebra-able-describe-group/ by Jakob Schwichtenberg
- math.stackexchange.com/questions/1388144/what-exactly-is-a-tangent-vector/2714944 What exactly is a tangent vector? on Stack Exchange

A member of a tangent space.

www.youtube.com/watch?v=tq7sb3toTww&list=PLxBAVPVHJPcrNrcEBKbqC_ykiVqfxZgNl&index=19 mentions that it is a bit like a dot product but for a tangent vector to a manifold: it measures how much that vector derives along a given direction.

A metric is a function that give the distance, i.e. a real number, between any two elements of a space.

A metric may be induced from a norm as shown at: Section "Metric induced by a norm".

Because a norm can be induced by an inner product, and the inner product given by the matrix representation of a positive definite symmetric bilinear form, in simple cases metrics can also be represented by a matrix.

Canonical example: Euclidean space.

TODO examples:

- metric space that is not a normed vector space
- norm vs metric: a norm gives size of one element. A metric is the distance between two elements. Given a norm in a space with subtraction, we can obtain a distance function: the metric induced by a norm.

In plain English: the space has no visible holes. If you start walking less and less on each step, you always converge to something that also falls in the space.

One notable example where completeness matters: Lebesgue integral of $L_{p}$ is complete but Riemann isn't.

Subcase of a normed vector space, therefore also necessarily a vector space.

Appears to be analogous to the dot product, but also defined for infinite dimensions.

Vs metric:

- a norm is the size of one element. A metric is the distance between two elements.
- a norm is only defined on a vector space. A metric could be defined on something that is not a vector space. Most basic examples however are also vector spaces.

Metric space but where the distance between two distinct points can be zero.

Notable example: Minkowski space.

When a disconnected space is made up of several smaller connected spaces, then each smaller component is called a "connected component" of the larger space.

See for example the

There are two cases:

- (topological) manifolds
- differential manifolds

Questions: are all compact manifolds / differential manifolds homotopic / diffeomorphic to the sphere in that dimension?

- for topological manifolds: this is a generalization of the Poincaré conjecture.Original problem posed, $n=3$ for topological manifolds.Last to be proven, only the 4-differential manifold case missing as of 2013.Even the truth for all $n>4$ was proven in the 60's!Why is low dimension harder than high dimension?? Surprise!AKA: classification of compact 3-manifolds. The result turned out to be even simpler than compact 2-manifolds: there is only one, and it is equal to the 3-sphere.For dimension two, we know there are infinitely many: classification of closed surfaces
- for differential manifolds:Not true in general. First counter example is $n=7$. Surprise: what is special about the number 7!?Counter examples are called exotic spheres.Totally unpredictable count table:$n=4$ is an open problem, there could even be infinitely many. Again, why are things more complicated in lower dimensions??
Dimension | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | Smooth types | 1 | 1 | 1 | ? | 1 | 1 | 28 | 2 | 8 | 6 | 992 | 1 | 3 | 2 | 16256 | 2 | 16 | 16 | 523264 | 24 |

So simple!! You can either:

- cut two holes and glue a handle. This is easy to visualize as it can be embedded in $R_{3}$: you just get a Torus, then a double torus, and so on
- cut a single hole and glue aMöbius strip in it. Keep in mind that this is possible because the Möbius strip has a single boundary just like the hole you just cut. This leads to another infinite family that starts with:

A handle cancels out a Möbius strip, so adding one of each does not lead to a new object.

You can glue a Mobius strip into a single hole in dimension larger than 3! And it gives you a Klein bottle!

Intuitively speaking, they can be sees as the smooth surfaces in N-dimensional space (called an embedding), such that deforming them is allowed. 4-dimensions is enough to embed cover all the cases: 3 is not enough because of the Klein bottle and family.

Important 4D spaces:

Simulate it. Just simulate it.

math.stackexchange.com/questions/466707/what-are-some-examples-of-infinite-dimensional-vector-spaces

This section is about the definition of the dot product over $C_{n}$, which extends the definition of the dot product over $R_{n}$.

Some motivation is discussed at: math.stackexchange.com/questions/2459814/what-is-the-dot-product-of-complex-vectors/4300169#4300169

The complex dot product is defined as:

$∑a_{i}b_{i} $

E.g. in $C_{1}$:

$(a+bi)⋅(c+di)=(a+bi)(c+di )=(a+bi)(c−di)=(ac+bd)+(bc−ad)i$

We can see therefore that this is a form, and a positive definite because:

$(a+bi)⋅(a+bi)=(aa+bb)+(ba−ab)i=a_{2}+b_{2}$

Just like the usual dot product, this will be a positive definite symmetric bilinear form by definition.

Given:
the norm ends up being:

$x=∑_{k=1}a_{k}+b_{k}i∈C_{n},a_{k},b_{k}∈R$

$∣x∣=∑_{k=1}a_{k}+b_{k} $

$R_{n}$ with extra structure added to make it into a metric space.

The identity matrix.

Each elliptic space can be modelled with a real projective space. The best thing is to just start thinking about the real projective plane.