Topology

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 ${\mathbb{R}}^n$, 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.

Manifolds are cool. Especially differentiable manifolds which we can do calculus on.

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:

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:

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:

An inner product $x\cdot y$ induces a norm with:

In a vector space, a metric may be induced from a norm by using subtraction:

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:

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

So simple!! You can either:

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.

sphere with two Möbius strips stuck into it as per the classification of closed surfaces.

Important 4D spaces:

Simulate it. Just simulate it.

This section is about the definition of the dot product over ${\mathbb{C}}^n$, which extends the definition of the dot product over ${\mathbb{R}}^n$.

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:

E.g. in ${\mathbb{C}}^1$:

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

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

Given: the norm ends up being:

E.g. in ${\mathbb{C}}^2$:

${\mathbb{R}}^n$ 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.