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!
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2$\begingroup$ isn't this at the heart of general relativity? $\endgroup$– Carlo BeenakkerCommented Feb 12 at 9:29
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2$\begingroup$ Have you read any differential geometry textbooks? Pretty much every one of them will explain how vector fields along maps of intervals and surfaces are used. The question itself is more appropriate for Math Stack Exchange. $\endgroup$– Moishe KohanCommented Feb 12 at 9:47
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$\begingroup$ @MoisheKohan, I wanted to know if there are any particular places where this thing was relevant/important like if for example we have a sigma field $F:\mathbb{R}^4_1\xrightarrow{}(N,h)$ and we want to understand the physical significance of vector fields along the parameter space which in this case is the minkowski space. Thanks for your response. $\endgroup$– Siddharth PanigrahiCommented Feb 13 at 12:53
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$\begingroup$ @CarloBeenakker, could you please elaborate on it a little more? Thanks for your response. $\endgroup$– Siddharth PanigrahiCommented Feb 13 at 12:53
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$\begingroup$ I do not know anything about sigma-models. But if this is what yoy are actually interested in, you should edit your question to make it clear that you are asking about sigma-models. $\endgroup$– Moishe KohanCommented Feb 21 at 18:35
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1 Answer
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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.
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1$\begingroup$ Thanks a lot for your answer. $\endgroup$ Commented Feb 22 at 6:51