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The following construction is standard, and it deserves a name.

Suppose we need to construct a diffeomorphism from a manifold $M$ to itself with some additional properties. Observe that the flow $\phi^t$ for any reasonable vector field $v(t)$ on $M$ defines a diffeomorphism for any $t$. So it remains to find a vector field $v(t)$ such that the flow $\phi^1$ satisfies your properties.

Do you know a name for it?

If not, what would be a good name?

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    $\begingroup$ Physicists call this type of thing "infinitesimal generator". $\endgroup$
    – Tobias Fritz
    Commented Apr 10, 2020 at 19:50
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    $\begingroup$ @WillieWong, usually not, but sometimes it is useful. $\endgroup$
    – Anton Petrunin
    Commented Apr 10, 2020 at 21:35
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    $\begingroup$ I only ask because after I upvoted @TobiasFritz's comment, I begin to doubt whether the usual use of the "infinitesimal generator" allows for $v$ to be time dependent. $\endgroup$
    – Willie Wong
    Commented Apr 10, 2020 at 21:57
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    $\begingroup$ You see people give these flows various names. I use this construction in the time-dependent context. My vector fields are usually defined by an isotopy of a submanifold, you then extend the velocity vector field of the submanifold to the ambient manifold. So I just call it isotopy extension. But I see other people call them "point pushing" or "circle pushing" maps depending on what the submanifold might be. Your context is a little more general than my own, though. $\endgroup$
    – Ryan Budney
    Commented Apr 11, 2020 at 0:19
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    $\begingroup$ I would say "Moser trick", since it reminds me of the trick explained below for volume forms. There is still not too much named after Moser, so I don't think it would confuse anyone. "Euler trick" or "Arnold trick" or "Newton trick" might confuse people. $\endgroup$
    – Ben McKay
    Commented Apr 11, 2020 at 14:14

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This is often called "path method". In particular one has classically the Moser path method.

Let us prove for example (after Moser) that given a closed oriented $n$-manifold $M$ and two volume forms $\omega_0$, $\omega_1$ on $M$ with the same integral, there exists a diffeomorphism $f$ of $M$ such that $$f^*(\omega_1)=\omega_0$$ To this end, the path method considers $$\omega_t=(1-t)\omega_0+t\omega_1$$ and looks for a time-dependant vector field $X_t$ whose flow $\phi_t$ satisfies the condition $$\phi_t^*(\omega_t)=\omega_0$$ (Then, for $t=1$, $f=\phi^1$ will work)

The condition holds trivially for $t=0$. Deriving this condition with respect to $t$, one finds $$\phi_t^*(L_{X_t}\omega_t+\omega_1-\omega_0)=0$$ where $L$ is the Lie derivative. This amounts to $$L_{X_t}\omega_t=\omega_0-\omega_1$$ i.e. by Cartan's formula $$d\iota_{X_t}\omega_t=\omega_0-\omega_1$$ Since $\omega_0$ and $\omega_1$ have the same integral, they are cohomologous: one has a $(n-1)$-form $\alpha$ on $M$ such that $$\omega_0-\omega_1=d\alpha$$ Hence it is enough that $$\iota_{X_t}\omega_t=\alpha$$ The end of the argument is purely (multi)linear algebra: this last equation admits for every time $t$ and at every point $x$ a unique solution $X_t(x)$ since $\omega_t(x)$ is a nonzero $n$-form on $T_xM$. An analogous method applies to symplectic forms, nonsingular closed $1$-forms and contact forms. The path method also proves the $MJ^2$ Lemma (Lemma 3.2 in F. Laudenbach, A proof of Reidemeister-Singer's theorem by Cerf's methods. Ann. Fac. Sci. Toulouse Math. (6) 23 (2014), 1, 197–221, Arxiv 1202.1130); and in particular the Morse Lemma.

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    $\begingroup$ What is the $MJ^2$ Lemma? $\endgroup$
    – abx
    Commented Apr 11, 2020 at 9:19
  • $\begingroup$ Thank you, but maybe "Moser trick" is better than "path method" (?) $\endgroup$
    – Anton Petrunin
    Commented Apr 11, 2020 at 16:34
  • $\begingroup$ As you like it, but to me, this deserves the name of "method", being more interesting, more general, and involving more the nature of things, than a mere "trick". $\endgroup$
    – Gael Meigniez
    Commented Apr 12, 2020 at 16:32
  • $\begingroup$ why "path" --- say "flow method" would be better. $\endgroup$
    – Anton Petrunin
    Commented Apr 13, 2020 at 5:36
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This is a hugely studied field in the area of medical image processing. Goes by name of (large) diffeomorphic registration or diffeomorphic mapping

https://en.m.wikipedia.org/wiki/Large_deformation_diffeomorphic_metric_mapping

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