The Pantheon of Derivatives - Sections, Pullbacks And Pushforwards (III)

[fresh_42]

Some Topology​

Whereas the terminology of vector fields, trajectories, and flows almost by itself suggests its origins and physical relevance, the general treatment of vector fields, however, requires some abstractions. The following might appear to be purely mathematical constructions, and I will restrict myself to a minimum, but they actually occur in modern physics: from the daily need to solve differential equations on various (non-Euclidean) geometric objects like in general relativity or quantum field theory, to the front end research in cosmology.

Vector Bundles​

The tangents on a manifold $M$ define a vector field in a natural way. That is, at each point $x \in M$ there are the tangents to all possible curves through $x$ and they span the tangent (vector) space $TM|_x$ at this point. If $M$ is an m-dimensional manifold, then $\left. TM\right|_x$ is an m-dimensional vector space with the local coordinates $\frac{\partial}{\partial x^1},\ldots , \frac{\partial}{\partial x^m}$. Now we consider the collection of all these tangent spaces, i.e. for all points of $M$. This gives us a collection
\begin{equation}\label{TM}
TM= \bigcup_{x \in M}\left. TM\right|_x
\end{equation}
which we call tangent bundle of $M$. This can be generalized to an arbitrary vector field, in which case it is called a vector bundle. Note that these objects are actually tangent space bundles, resp. vector space bundles.


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[jedishrfu]

These are great articles!

How many more parts?

 

[Greg Bernhardt]

How many more parts?

two

 

[zwierz]

There is also a simple example of a nontrivial tangent bundle. Namely it is $TS^2$; here $S^2$ is the two dimensional sphere. If $TS^2=\mathbb{R}^2\times S^2$ then there exists a vector field $v(x)$ such that $v(x)\ne 0$ for all $x\in S^2$. But we know that this is not true

And another pretty thing is: why does physics need all these objects ? Simplest example: In classical mechanics the Lagrangian $L=L(q,\dot q)$ is defined on a tangent bundle of configuration manifold while the Hamiltonian $H=H(q,p)$ is a function of cotangent bundle of the configuration manifold: $L:TM\to\mathbb{R},\quad H:T^*M\to \mathbb{R}$