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have expanded the single sentence at differential geometry to something like a paragraph, indicating how differential geometry is the “higher geometry modeled on the pre-geometry $\mathcal{G} = CartSp$”
HELP:
at differential geometry I want under Related concepts a square table as in the introduction of Sharpe’s book.
I managed to produce parts of it (have a look) but I have not been able to insert the obvious vertical arrows without destroying the outcome. I don’t really know that table syntax. Can anyone help?
I’m afraid I don’t know how to fix the table, but I’d like to ask a question about its appearance there. Are we to think of $CartSp$ as the pregeometry corresponding to Euclidean and then Riemannian geometry, and for there to be other pregeometries for the Kleinian homogeneous spaces and associated Cartan geometries? Or do the latter arise from imposing further conditions on the geometry emerging from $CartSp$?
Scary entry so far. Do you know a differential geometer who can read it ?
Technically
the geometry modeled on the pre-geometry
is puzzling, as Lurie explains that there are many geometries corresponding to the same pre-geometry.
I’d like to ask a question about its appearance there. Are we to think of $CartSp$ as the pregeometry corresponding to Euclidean and then Riemannian geometry, and for there to be other pregeometries for the Kleinian homogeneous spaces and associated Cartan geometries?
I have to admit that I don’t fully buy into the idea that “Cartan geometry generalizes Riemannian geometry”. I rather think that the useful point of view is: “Cartan geometry is a special case of principal bundles over Riemannian spaces”.
Accordingly, I’d tend to answer your question with: “No.”
But I am open for discussion. Maybe I am missing something here.
I rather think that the useful point of view is: “Cartan geometry is a special case of principal bundles over Riemannian spaces”.
This is possible/likely, though I am not fully sure, if the unrolling the frame in Cartan’s moving frame business corresponds exactly to a special kind of charting of a principal bundle, or there are adiditonal global phenomena implied there. Who is an expert ?
What is the difference intended between the entry differential geometry and the entry generalized smooth space ? Both are now about a type of a space, but differential geometry is a subject and not a type of a space. In particular, differentiable manifolds can be studied in differential geometry, differential topology and analysis on manifolds. As my teacher in symplectic geometry Y-G. Oh used to say, the analysis on manifolds is mainly concerned with the maps out of a manifold (functions on a manifold), while differential geometry mainly with geometric objects in a manifold, that is maps into a manifold (e.g. curves).
This is possible/likely, though I am not fully sure, if the unrolling the frame in Cartan’s moving frame business corresponds exactly to a special kind of charting of a principal bundle, or there are adiditonal global phenomena implied there. Who is an expert ?
Not sure what you want an expert to say here? The definition of Cartan connection is simple enough. It’s just a principal connection with a simple extra constraint on it.
The question seems to be more philosophical. Of a principal bundle we usually think of as being extra structure over the base manifold. Should we change that point of view only because we impose an extra constraint?
What is the difference intended between the entry differential geometry and the entry generalized smooth space ?
These two entries weren’t created in parallel, I think. But to my mind, the entry “differential geometry” should say what differential geometry is about. In a useful generalized sense, this involves also generalized smooth spaces, and so there should be an entry focused on these. No?
Concerning #8:
In the very interesting foreword to Sharpe’s book (listed at Cartan geometry) Chern recalls that Cartan geometry was mainly forgotten by the community after Ehresmann produced the definition of principal connections. Chern lists two reasons:
Ehresmann was easier to read “at that time”, he says;
Ehresmann’s definition was more general anyway.
Chern then does come back to Cartan’s original point of view and emphasizes how it is useful. But I think there is a small bit of truth here to the saying: “A concept which has been forgotten – and rightly so.” :-)
Not to get me wrong, I think that the notion of Cartan connections (and of solder forms and vielbeins and so on) is important. But I think it is to be regarded a special case of Ehresmann theory / principal connections. Not as a stand-alone theory.
I wonder if there’s any abstract nonsense way of characterising Cartan theory as a special case of Ehresmann theory.
Are there cases where you sympathise with the point of view that something may be lost when a theory is subsumed within another. Probability theorists complained of their theory being seen as a mere chapter of measure theory.
I wonder if there’s any abstract nonsense way of characterising Cartan theory as a special case of Ehresmann theory.
I have thought about this a bit for the case of the Poincaré group. Because there it amounts to a famous open problem in quantum gravity: in the first order formulation of gravity (mandatory if spinors are to be included) the field content is an $Iso(d,1)$-connection. The metric itself is encoded in the $\mathbb{R}^{d+1}$-summand of the connection: the vielbein. Or rather, in general that vielbein encodes a symmetric tensor, not, however, necessarily a non-degenerate one: a Riemannian metric.
Now, the problem is this: classically we would tend to demand that indeed we have a non-degenerate 2-tensor, hence a metric. This demand is equivalent to requiring that the given $Iso(d,1)$-connection is in fact an $(O(d,1) \hookrightarrow Iso(d,1))$-Cartan connection.
But is this a good constraint to impose when looking at the quantum theory? Should not the path integral maybe be over all vielbein fields? Who says that fundamentally there must be a non-degenerate metric tensor?
Who actually says that the correct model of gravity by $Iso(d,1)$-geometry needs to have a non-degenerate vielbein? It’s not clear to me that it has to. The theory works much more smoothly without an extra constraints. Extra constraints are generally a source of pain in quantum theory.
I have decided to slightly change that little table at differential geometry. It now looks roughly like this:
local model | global geometry |
---|---|
Euclidean geometry | Riemannian geometry |
Klein geometry | Cartan geometry |
I’ll copy this over the the other three entries, too.
Not sure what you want an expert to say here? The definition of Cartan connection is simple enough.
The Cartan connection is just a basic input datum in a more complex method of studying geometry, the Cartan’s moving frame method in which one kind of scrolls out the charts to get geometry. I think this method is more than just charting a principal bundle.
I thought the definition of Cartan connection includes the idea of “moving frames”. Is this not just the special case of a $O(n) \hookrightarrow Iso(\mathbb{R}^n)$-Cartan connection? The moving frame is nothing but the vielbein that is part of this. No?
In a useful generalized sense, this involves also generalized smooth spaces, and so there should be an entry focused on these. No?
I think not. The entries for differential manifolds, orbifolds, polyfolds, differential stacks, smooth spaces, generalized smooth spaces etc. are object (rather than subject) pages which should exist on their own. Differential geometry should be different from those in the same sense as the entry topology is different from topological space or topological structure. So I would plan the chapters like classical differential geometry of curves, surfaces, the difference between internal and external geometry, then the geometry of manifolds, and manifolds with structure, and of smooth bundles with structure, then links to generalizations like generalized smooth space, differentiable stacks etc. All subfields like symplectic geometry, Riemannian geometry, Finsler geometry, contact geometry, generalized complex geometry could be inside.
Next as I said, the differential geometry is not decided by being ABOUT manifolds but rather about specific geometric aspect fo manifolds, differing from analysius on manifolds and differing from differential topology. Wikipedia is more along this line, though too short: differential geometry.
The Cartan’s METHOD Of moving frames is NOT the same as moving frame. (As car race is not the same as a car.) The essence is in certain kind of charting by rolling out the frame from a bundle to the manifold, and I am not sure what nontrivial global aspects come out of that rolling.
David,
one more comment re #12:
here is an important case to keep in mind:
quantum gravity is well understood in dimension 3. There the Einstein-Hilbert action functional is equivalent to the $Iso(2,1)$-Chern-Simons action functional (for a certain choice invariant polynomial). So one can quantize this. But not as a theory of Cartan connections, but as a theory of principal $Iso(2,1)$-connections. The metric can become degenerate. The Cartan-geometry condition can be violated.
I’ll try to write more about this at Chern-Simons gravity later.
re #18 That’s a telling piece of evidence.
Yes. On page 7 of Witten’s old article “2 + 1 DIMENSIONAL GRAVITY AS AN EXACTLY SOLUBLE SYSTEM” (sorry for the caps) you can find the argument sketched why 3d quantum gravity cannot make sense if one imposes invertibility of the vielbein.
This means: 3d quantum gravity does not make sense if we regard the gravitational field as an $Iso(2,1)$- Cartan geometry/Cartan connection. We have to take it to be more generally to be a principal connection.
Since the 4d case will of course not be simpler than the 3d case, it seems unlikely that the sitation will change there.
Urs, I asked above why do you say “the” geometry based on pregeometry of Cart if Lujrie says that more than one geometry (say n-cat truncations) can correspond to the same pregeometry.
I asked above
Sorry, I had missed that.
why do you say “the” geometry based on pregeometry of Cart if Lujrie says that more than one geometry (say n-cat truncations) can correspond to the same pregeometry.
Because of prop 3.4.10 in Structured Spaces:
the $n$-truncated geometric envelopes $\mathcal{G}_n$ of a pre-geometry $\mathcal{T}$ are such that the $\mathcal{G}_n$-structured $\infty$-toposes are precisely those $\mathcal{G}_\infty$-structured $\infty$-toposes (where $\mathcal{G}_\infty := \mathcal{G}$ is the full geometric envelope) whose structure sheaf happens to be $n$-truncated.
So this gives a way to characterize certain full sub-$\infty$-categories of all $\mathcal{G}$-structured $\infty$-toposes. No new information, no new “geometry” in the non-technical sense of the word. Just a sub-case of the full geometry induced by $\mathcal{T}$.
Good, thanks.
I have revised differential geometry inserting a title which more traditionally describes its scope and then putting the $n$POV into a separate section (which could eventually maybe be partly moved into generalized smooth space with a link and shorter description).
added pointer to:
added pointer (here and elsewhere) to:
added these pointers (also at Lie group):
Jean Gallier, Jocelyn Quaintance, Differential Geometry and Lie Groups – A computational perspective, Geometry and Computing 12, Springer (2020) [doi:10.1007/978-3-030-46040-2, webpage]
Jean Gallier, Jocelyn Quaintance, Differential Geometry and Lie Groups – A second course, Geometry and Computing 13, Springer (2020) [doi:10.1007/978-3-030-46047-1, webpage]
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