What are the possible applications in maths and physics of vector fields along smooth maps?

Question

I am currently working on a problem related to singularities of mappings between manifolds with metrics and the interplay of metric singularities with mapping singularities. Given a smooth map $F:M\longrightarrow N$, a vector field along $F$ is just a smooth section in the set $\Gamma(M,TN)$. I wanted references/books/papers (anything) related to applications of these objects in any context in maths and physics, where they are relevant, why they are studied, anything will be great. Thanks in advance!

Answer 1

I'm not sure the notation $\Gamma(M,TN)$ is exactly right here. The bundle $TN$ has base space $N$ so what I think you really want to write is $\Gamma(M,F^*TN)$ where $F^*TN$ is the pullback bundle on $M$ induced by $F$.

With this in mind, note that in physics, gauge fields are sections of principle bundles i.e. functions $s:M \rightarrow G$ for some Lie group $G$. These sections obey differential equations formulated in terms of connections. Connections are a kind of directional derivative that take a direction $v$ at some point $p \in M$ and return the directional derivative of $s$ as a member of $T_{s(p)}G$. This means that locally connections are of the form $t \otimes \omega$ where $t \in \Gamma(M, s^*TG)$ and $\omega \in \Lambda^1(M)$. Hence if you take $M$ as spacetime and $F$ as some gauge field on $M$, then we can think of a connection locally as being the tensor of a vector field along $F$ with a one-form.