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For a $\mathbb{R}$-flow (X, $\Phi_{\mathbb{R}}$), the (local) cross-section is well defined (recall that a subset $S\subset X$ is a cross section of time $\xi>0$ if $S\cap \Phi_{[-\xi, \xi]}(x)=\{x\}$ for any $x\in S$). I would like to know whether there is some analogy for $\mathbb{R}^n$-flow (X, $\Phi_{\mathbb{R}^n}$). I don't know if someone already defined such cross-section for $\mathbb{R}^n$-flow or not. My thinking is that a subset $S\subset X$ is a cross section of time $\xi>0$ if $S\cap \Phi_{[-\xi, \xi]^n}(x)=\{x\}$ for any $x\in S$. Is this definition good? If the definition is as above, Whitney theorem (existence of a family of cross-sections for regular flow) holds for $\mathbb{R}^n$-flow? As well, is there a notion of expansive $\mathbb{R}^n$-flow which generalizes the $\mathbb{R}$-flow in the sense of Bowen-Walter?

Thanks.

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  • $\begingroup$ You might find some answers in the theory of foliations. $\endgroup$
    – Lee Mosher
    Commented Dec 2, 2021 at 19:02
  • $\begingroup$ Your question seems rather vague to me. You are asking if a definition is good, but good for what purposes? What do you want to accomplish? $\endgroup$
    – Ryan Budney
    Commented Dec 2, 2021 at 20:06
  • $\begingroup$ @RyanBudney I would like to ask whether there is some analogy for $\mathbb{R}^n$-flows. I want to know how the theory passes to high rank flows in general. $\endgroup$
    – user119197
    Commented Dec 3, 2021 at 8:07
  • $\begingroup$ @LeeMosher Thanks. I am not familiar with foliations. I would be appreciated if you could name some references. $\endgroup$
    – user119197
    Commented Dec 3, 2021 at 8:09
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    $\begingroup$ The theory of foliations is a gigantic field, and while there are some analogies with the theory of flows, understanding the analogies even in broad terms is going to be very difficult without any knowledge of the field. So I would suggest starting slowly, maybe with the Frobenius integrability theorem which you can read in many differential geometry/differential topology textbooks. I would suggest Spivak's "Differential Geometry", Volume 1, Chapter 6. $\endgroup$
    – Lee Mosher
    Commented Dec 3, 2021 at 13:52

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