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Roger Penrose in his book "The Road to Reality" (Section 14.8 - Symplectic Manifolds) loosely defines a "floppy" structure to be one which if we apply two variants of it on two copies of the same manifold, the two manifolds are locally isomorphic (and thus locally indistinguishable from one another). The example he brings is that of two symplectic manifolds with the same dimension and signature. I think what he is referring to (although he does not make it explicit) is Darboux's Theorem which again asserts that two symplectic manifolds of the same dimension are locally isomorphic (more accurately: symplectomorphic).

In his exact words and I quote:

The local structure of a symplectic manifold is an example of what might be called a ‘Floppy’ structure. There is, for example, no notion of curvature for a symplectic manifold, which might serve to distinguish one symplectic manifold from another, locally. If we have two real symplectic manifolds of the same dimension (and the same ‘signature’, cf. §13.10), then they are locally completely identical (in the sense that for any point p in one manifold and any point q in the other, there are open sets of p and q that are identical). This is in stark contrast with the case of (pseudo-) Riemannian manifolds, or manifolds in which merely a connection is specified. In those cases, the curvature tensor (and, for example, its various covariant derivatives) defines some distinguishing local structure which is likely to be different for different such manifolds.

This makes sense to me. Then he goes on to describe two more examples of manifolds one of which is floppy while the other isn't. In particular he makes the following two claims:

  1. Let $M_1$ be a real manifold with a nowhere vanishing vector field $F$ on it. Then $M_1$ is a floppy manifold.
  2. In contrast, let $M_2$ be a real manifold with two general vector fields. Then $M_2$ is NOT a floppy manifold

Question 1: For the 1st part, I think the family of tori would be a good place to start since assigning a (smooth) nowhere vanishing vector field is always possible. I can also see how we can use $F$ to define a global frame on the entire $M_1$ starting by setting $\partial_1 := F \neq 0$ etc. However, in what sense are all tori (with $F$) locally indistinguishable? For example, shouldn't we able to (locally) distinguish a torus with circular cross sections versus one with elliptical just by looking at the difference in curvatures between a circle and an ellipse?

Question 2: Now to his second claim. First of all, what do you think he means by "general" vector fields? Am I right to assume that they are linearly independent and by extension neither one of these can be anywhere vanishing (as this would violate independence)? And how does the existence of the second field makes them distinguishable?

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    $\begingroup$ Your opening phrase regarding "floppy structure on 2 different manifold" seems off. Floppiness is a property of the structure itself, not of the manifold which supports the structure. Symplectic structures are floppy: a symplectic structure on any $n$ manifold is locally isomorphic at every point to the standard symplectic structure on $\mathbb R^n$ (yes, that is Darboux's theorem). $\endgroup$
    – Lee Mosher
    Jul 30, 2021 at 15:15
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    $\begingroup$ Nonvanishing vector fields are floppy: a nonvanishing vector field on any $n$-manifold is locally isomorphic at every point to a coordinate vector field on $\mathbb R^n$. So your question 1 regarding the torus is irrelevant. Yes, the torus does have nonvanishing vector fields, and each of them is locally isomorphic to a coordinate vector field on $\mathbb R^n$. $\endgroup$
    – Lee Mosher
    Jul 30, 2021 at 15:18
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    $\begingroup$ Whether or not a given $n$-manifold $M$ has a nonvanishing vector field is an interesting question, but not with regard to floppiness: any nonvanishing vector field that does happen to exist on $M$ is locally isomorphic to a coordinate vector field on $\mathbb R^n$. $\endgroup$
    – Lee Mosher
    Jul 30, 2021 at 15:18
  • $\begingroup$ @LeeMosher Thank you for your comment. This exactly why I posted this question, to clarify the definition. For example, is it just the structure or the manifold PLUS the structure? For example, what am I to make of the following statement (taken verbatim from Penrose): "A real manifold with two general vector fields on it would be floppy". Straightforward grammar suggests that the floppiness refers to the manifold. When paired with contexts it seems to refer to the structure PLUS the manifold. $\endgroup$
    – Pellenthor
    Jul 30, 2021 at 15:53
  • $\begingroup$ Then there is his other (verbatim) statement: "One such example [of a floppy structure] would be a real manifold with a nowhere vanishing vector field on it". Again, does he include the manifold to the structure? If not, why can't he simply skip the word manifold altogether and just say "a vanishing vector field"? $\endgroup$
    – Pellenthor
    Jul 30, 2021 at 15:55

1 Answer 1

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Question 1: I think Penrose is thinking of manifolds (which are floppy) as opposed to Riemannian manifolds (which are not). Specifically, the idea of a torus having a cross sectional circle vs a cross sectional ellipse is implicitly viewing tori as Riemannian manifolds, not as plain manifolds.

The sense in which a manifold with vector field is floppy is the following:

Suppose $M$ is a smooth manifold and $V$ is a smooth vector field on $M$ with $V(p)\neq 0$ for some $p\in M$. Then there is a chart $U$ containing $p$ for which $V$ is a coordinate vector field.

(See, e.g., Theorem 2.1 of these notes)

Question 2: I'm not exactly sure what Penrose means by "general", but his comment certainly applies to linearly independent vector fields. Specifically, given linearly independent vector fields $V_1$ and $V_2$ on (a neighborhood in) a manifold, one can ask whether or not there is a single chart where $V_1$ and $V_2$ are both coordinate vector fields.

In general, the Lie bracket $[V_1,V_2]$ measures the obstruction to this. That is, if $[V_1,V_2] = 0$, then the answer is yes, while if the bracket is non-zero, the answer is no.

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    $\begingroup$ As someone who is not a symplectic geometer, the floppiness has always seemed artificial to me. Namely the condition $d\omega = 0$ seems analogous to me of saying "flat" in Riemannian geometry. And flat Riemannian manifolds are floppy. I am totally unaware of any study of "fake-sympletic" forms where the closed assumption is dropped. $\endgroup$ Jul 30, 2021 at 13:24
  • $\begingroup$ This is helpful thanks. Since I am not a differential geometer, can you perhaps say a bit more about what is the notion of "sameness" for the tori then? In what way are two tori with different cross sections locally indistinguishable? Does he simply mean topologically (ie. homeomorphic to one another)? Do they have the same chart? $\endgroup$
    – Pellenthor
    Jul 30, 2021 at 14:40
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    $\begingroup$ @Pellenthor: I believe he means as a manifold (i.e. diffeomorphic to one another). $\endgroup$ Jul 30, 2021 at 15:53
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    $\begingroup$ No, I think that makes his second comment true. Specifically, the point is that a pair of vector fields is equivalent to coordinate vector fields iff their Lie bracket is zero. Thus, there are local obstructions to a pair of vector fields on one manifold being "the same" as a pair on another manifold, so these structures are not floppy. $\endgroup$ Jul 30, 2021 at 16:09
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    $\begingroup$ Yes, that sounds right. $\endgroup$ Jul 30, 2021 at 16:21

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