User talk:Silly rabbit

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[edit] Welcome

Welcome!

Hello Silly rabbit, and welcome to Wikipedia! Thank you for your contributions. I hope you like the place and decide to stay. Here are a few good links for newcomers:

I hope you enjoy editing here and being a Wikipedian! Please sign your name on talk pages using four tildes (~~~~); this will automatically produce your name and the date. If you have any questions, check out Wikipedia:Where to ask a question or ask me on my talk page. Again, welcome! 

I will keep in mind the lager offer. Oleg Alexandrov (talk) 03:31, 7 November 2005 (UTC)

I intend to honor that offer! Silly rabbit kicks are for trigs.

[edit] Torsion, specifically Torsion (differential geometry)

I don't know if you do requests, but I've always wanted to gain a better understanding of torsion in differential geometry, and I suspect you might be able to write something interesting about that. Books on diff geom typically state "Let the torsion be zero. Then..." and that is the last one hears of it. I personally have never seen a good writeup. Once, long long ago, I even had a professor express confusion over the issue (although it was in the context of supersymmetry): he stated something like "... and in this gauge, the curvature can be made of vanish, but then the torsion is not zero; and so it is not clear if this is a physical theory..." Ever since then, I've wondered if or when one can trade torsion for curvature (was the profs statement only true for supersymmetry?), and what that might mean. I'm hoping you can help.

Your former professor's confusion over whether the theory was physical most likely resulted from the fact that torsion can only be specified in the presence of some additional data besides a (pseudo-)Riemannian metric (or a discrete symmetry, as with a spin structure). I'm not much of a super-symmetrist myself. But many (ten?) years ago I attended a lecture in which the speaker axiomatically characterized the Einstein equation from GR, and then hedged by saying (roughly): "Of course, this only holds if the connection is torsion-free. And there may or may not be a good physical reason for this, but the issue deserves to be investigated." The reason for relativity theorists to reject the idea of connections with torsion is clear (see below): torsion usually depends on some preferred class of frames (or higher-order frames, jets). I think the speaker was referring to some structure above and beyond the usual relativity principle. In my opinion he was trying to say, in an obscure and carefully couched fashion, that the current theory was incomplete. I don't know if this helps to vindicate your former professor. Silly rabbit 18:50, 8 November 2005 (UTC)
We do have an article Parallelizable manifold but it never mentions torsion. The article on jet bundles mutters something about holonomic coordinates, which I think are supposed to be torsion-free coordinates on a G-bundle, but I'm not sure ... Are there smooth homotopies that take one from coordinate systems on manifolds with torsion but no curvature to manifolds with curvature but no torsion? Are there obstructions (homologies) to this? As I said, I'm quite confused on this. linas 14:22, 8 November 2005 (UTC)
Assuming I understand you correctly: There are obstructions, but they probably all arise in the existence of a connection with torsion but no curvature. As a first-guess approximation, in the Riemannian case I would say that the manifold must be parallelizable. Silly rabbit 15:42, 8 November 2005 (UTC)

And, as Oleg says: Welcome! Glad your here! linas 14:03, 8 November 2005 (UTC)

How deep should I go, then? Torsion is a tricky thing. Silly rabbit 15:23, 8 November 2005 (UTC)
Ok, let me try to explain at least some of the difficulties with torsion. Let's begin with the case of torsion on a (pseudo-) Riemannian manifold. We know that it is possible to eliminate all torsion using the Levi-Civita connection. But suppose instead that we wish to eliminate all curvature. Locally, this is possible. In a coordinate patch, pick a particular trivialization of the bundle of orthonormal frames, and simply declare the connection form to be the Maurer-Cartan form of O(n). There will be lots of torsion, but no curvature. Of course, this isn't of much use since the torsion has no nice properties (just like the "connection" we've chosen, it too depends on our preference of frame).
Well, sure, but I feel comfortable with the connection because I know how it transforms, and I know how to build various things from it. So the fact that the torsion is frame dependent is not a problem, per se. Perhaps I can build invariants from the torsion; I just don't know what these invariants are.
In (pseudo-)Riemannian geometry, the torsion is basically just frame-dependent garbage. Unless you have something else to pin it down with (for instance, a class of frames or sections of the frame bundle which is already preferred). That's why the torsion is usually eliminated from the connection.
For other geometrical structures, there are various integrability conditions which can lead to the presence of intrinsic torsion. For example, consider a complex analytic manifold with a hermitian metric. Here there is a natural tension between the data of the metric and the integrability condition of the manifold (Cauchy-Riemann equation). The U(n) connection can still be put in a normal form, but this form will have some associated torsion. In fact, a complex manifold admits a torsion-free U(n) connection if and only if it is Kähler. (I believe this is due in large part to Nijenhuis.)
Ah! Well, there!
In general, torsion is the stuff you try to get rid of in Cartan's equivalence method. Of course, as in the case of complex hermitian manifolds, it isn't always possible to eliminate all of the torsion. So, just as curvature is an invariant of Riemannian manifolds, there are geometric structures with torsion invariants as well. Silly rabbit 16:07, 8 November 2005 (UTC)
So none of this really answers the question: What is torsion? Even in this more general context, answers like "stuff you can't get rid of", or even "just another invariant of the structure, like curvature" are unsatisfactory. It is an unfortunate deficiency of mathematicians that we tend to overlook the forest for the trees. I need some time to think about this. Silly rabbit 16:15, 8 November 2005 (UTC)
Yes, thank you, that was a good reply. And there's no hurry or urgency nor is a personal reply even needed; its just that I noticed you were working on nearby articles, so I thought I'd ask. I'm not sure I know what my question is. Perhaps it is this: "when is torsion important, and when isn't it?", and you've partly answered it: its important for complex manifolds, and not important for Riemann surfaces. Seems like its important when Lie groups enter the picture (e.g. in fiber bundles) because these are in a sense naturally "parallelizable"; but I have no particular intuition. I have a few questions which I am too embarrassed to ask since I suspect they're shallow and can be answered with only a minor bit of effort.
As to supersymmetry, I don't exactly understand it myself. It seems to be the result of a tension between the adjoint rep (for example, that thing that acts naturally on a tangent bundle) and the fact that adjoint reps are decomposable into sums involving the fundamental rep (which then tie to spinors in a strange related way). And so there's this soup of ingredients that promises a geometrical outcome, but I haven't quite made sense of it. linas 02:57, 10 November 2005 (UTC)
For a truly complete answer of the question of torsion, one needs to incorporate Spencer cohomology. This is the cohomological data for a Lie group G that can yield a unique prolongation of G-structures on a manifold. Spencer cohomology captures the ambiguity in selecting the torsion for prolongation equivalence problems. Anyway, it's very hard stuff for intrinsic geometry, although it can always be described in local coordinates using jets. The precise higher-order behavior is controlled by the curvature of the connection and its torsion.
Consider, for example, the first order differential equation for geodesics on a (pseudo-)Riemannian manifold. This is well-understood. But what if we only have a conformal structure to work with? In this case, then, there will be torsion in the first order because of the fact that the first-order connection involves some derivatives of an arbitrary function. But this torsion is not intrinsic, since the connection itself changes if we change our representative of the conformal metric. Nevertheless, it is possible to absorb all of this torsion in higher-order. This yields a connection on a higher-order G-structure: a Cartan connection. Furthermore, the associated geodesic equations require a second-order initial condition, as opposed to a first-order condition (as is the case in a Riemannian manifold.)
The only conclusion I can make is that torsion is such a tricky thing that it can't possibly be explained to someone who isn't an expert. I mean no offence, but ultimately it has no interpretation apart from that of D.C. Spencer. Cheers, and sorry for dumping all of this on my talk page, Silly rabbit 20:15, 15 November 2005 (UTC)
No offense taken, and no apologies needed; by contrast, thanks. These questions have been a stone in my shoe for far too long; I'm slowly getting around to actually doing something about it. Please don't delete the above, I'll have to read it a few more times before I am through. linas 22:55, 15 November 2005 (UTC)

[edit] The trick to editing wikipedia sites is...

Indtrouce as mnany splelling erors as posible. Thenn you ar shoor too get attencion!

But seriously, I'm sorry about the simplectic versus symplectic. I was wrong, it was late at night (for me anyway), etc. I hope some good content comes out of my error by bringing more people to the page cotangent bundle.

--Silly rabbit, kicks are for trigs!

[edit] Char poly to trace

Not exactly the right place for such a question (although it does pertain to how the total Chern class and the Chern character can be expressed in terms of one another). Is there an article which might be contain a sort of determinant to trace formula? The formula is simple enough:

\log \det(I-tA)=-\sum_{i=1}^\infty t^i\frac{\hbox{tr}(A^i)}{i}

(where A is a matrix with entries in a commutative rational algebra and blah, blah, blah...) I've had occassion to use this formula in the past (with Pontrjagin classes).

'Proof.' This follows from the identity for an n×n matrix A

\det(A-tI)=\sum_m (-t)^{n-m}\sum_\pi (-1)^{\epsilon(\pi)}\frac{tr(A)^{\pi_1}tr(A^2)^{\pi_2}\dots tr(A^m)^{\pi_m}}{\pi_1!\pi_2!\dots\pi_m!1^{\pi_1}\,2^{\pi_2}\dots m^{\pi_m}}

The inner sum extends over all partitions π=(π1,...,πm) of m of the form π1+2π2+...+mπm=m with each πi a non-negative integer. The sign is determined by \epsilon(\pi)=\sum_{i=0}^{[m/2]}\pi_{2i}. This can be proven combinatorially.  ;-)

Alternatively, we can prove it using Cauchy's theorem. Let z and t be complex variables, and let f(z)=det (A-zI). Let w1,...,wn be the zeros of f(z). These zeros are just the eigenvalues of the matrix A. Let γ be a contour enclosing all of the wi. We have

\frac{1}{2\pi i}\int_\gamma t^k z^k\frac{f'(z)}{f(z)}dz=t^k\sum_iw_i^k=t^k\hbox{tr}(A^k)

Summing over all k and employing the geometric series

\frac{1}{2\pi i}\int_\gamma \frac{1}{1-tz}\frac{f'(z)}{f(z)}dz=\sum_kt^k\hbox{tr}(A^k).

By Cauchy's theorem, the LHS is equal to

-\left.\frac{1}{t}\frac{d}{dz}\right|_{z=1/t}\log f(z)

from which the result follows after some manipulation.

Techniques like this are also used to study functional determinants. I should learn more about such things.

Silly rabbit 21:13, 13 November 2005 (UTC)

I haven't seen an article on WP that resembles this. Seems that th article on determinant doesn't link to such a thing. WP does have the marginally-related Jacobi's formula. The intersting but troublsome case is where A is infinite-dimensional, possibly trace-class or a nuclear operator when its not hermitian. WP also does not currently have an article for the resolvent formalism, which deals with 1/det (A-\lambda I) and related constructions (the resolvent formalism is often used in quantum mechanics and dynamical systems, and is often accompanied by physics-type hand-waving, because its often not clear if the proofs are valid for the operators in question.)

One WP search technique is to click on "what links here", which shows all the other articles that reference this one. It can be an interesting read sometimes. This works even if the target article hasn't yet been created! (i.e. it will reveal the articles with the "red links").

Rather than posing questions on your talk page, I can recommend posting questions on Wikipedia talk:WikiProject Mathematics, which is followed by many, and will elicit a broader reply. linas 23:18, 15 November 2005 (UTC)

[edit] Differentiation in Fréchet spaces

So what actually is the difference between Differentiation in Fréchet spaces and the Gâteaux derivative? According to the article you've written, they look identical. Are you sure that we don't have two names for the same thing (in which case, would it not be better to have only one article?). -lethe talk + 01:25, 9 June 2006 (UTC)

There is no difference between the the definitions, except that the Gâteaux article defines them on locally convex topological vector spaces so it won't have very many nice properties. Over Fr\`{e}chet spaces, the chain rule holds. Over a certain subclass, you get an inverse function theorem. More on this later. Silly rabbit 03:39, 9 June 2006 (UTC)
Ah, I see. I'm still not convinced that it shouldn't all be in the same article, but I'll wait and see what you have up your sleeves. -lethe talk + 04:03, 9 June 2006 (UTC)
Perhaps I should rewrite a bit of the intro to say something like: "The Gâteaux derivative has many happy properties on the category of Fréchet spaces..." I gather that most folks around here know about the Gâteaux derivative on comparatively nice spaces such as Hilbert spaces (via quantum field theory no doubt). But the uglier spaces are important for variational calculus, embedding problems, and (in my own case) diffeomorphism groups. So I'll give the Fréchet stuff my top priority, and I hope not to disappoint. Thanks for the feedback, Silly rabbit 04:26, 9 June 2006 (UTC)

[edit] Hello?

Is this a "silly rabbit" with any connection to Boston Herald internet forums in the last 5 years?. Many thanks, Nesher 21:45, 14 June 2006 (UTC)

Sorry, no. It's good to know that I'm not the only sylvilagus sillius around, though. Must be a close cousin. Silly rabbit 03:09, 15 June 2006 (UTC)

[edit] invitation

I'm greatly surprised you aren't yet listed on Wikipedia:WikiProject Mathematics/Participants. Dmharvey 03:21, 18 June 2006 (UTC)

[edit] Cartan-Karlhede algorithm

You should write a review paper on this topic. Right now WP lacks needed background articles which might make it difficult to incorporate some of your suggestions, but a readable review paper would make an excellent source for the next revision. ---CH 20:06, 21 June 2006 (UTC)

Thanks. I've already added it to my todo list. Unfortunately, this list seems to grow rather than shrink with time ;-). Silly rabbit

[edit] Infinite-dimensional holomorphy

Hi Silly rabit. Thank you for your edits to Infinite-dimensional holomorphy. By the way, do you have the time to add a section about trully infinite dimensional holomorphic function, that is, not only with values in a Banach space, but also defined on a Banach space (instead of the complex numbers)? In that case by the way weak holomorphy is not equivalent to strong holomrphy (Gateaux ≠ Frechet). I know too little about this topic to attempt to write myself. Oleg Alexandrov (talk) 22:21, 21 June 2006 (UTC)

Actually I added there the little I knew. You are welcome to take a look and expand on my blurb. Oleg Alexandrov (talk) 04:29, 22 June 2006 (UTC)
Thanks. I'll see what I can do with this article. There are many different notions of holomorphy out there. For one thing, a substantial part of the theory carries over to Frechet spaces (with the Gateaux derivative). Although in the f:X->Y case, it may be best to stick with Banach spaces because the two theories do have substantial differences, and most people only seem to be interested in the Banach case anyway. Silly rabbit 11:50, 22 June 2006 (UTC)

[edit] The state of connections

Hi all. I'm a bit concerned about the state of affairs in connection theory on Wikipedia. I've collected some of the pages here:

  • connection (mathematics) -- this is a good starting point, but perhaps could be turned into Category:Connection (mathematics)
  • connection form -- strangely enough, the article actually fails to define the connection form (in either the sense of Ehresmann or Cartan). Perhaps that's the source of some of the confusion in this discussion.
  • covariant derivative -- This article ought to be about the covariant derivative in any old vector bundle. Instead it focuses exclusively on affine connections.
  • gauge covariant derivative -- This article seems to be about what covariant derivative ought to be about (but isn't), perhaps with a few physical idiosyncracies which may justify its existence as a separate article.
  • Cartan connection -- I abstain from comment, since I have been trying to bring this up to par lately. See the talk page.
  • Levi-Civita connection -- A fine stub, but this definitely needs some expansion. At least a brief discussion of parallel transport and geodesics are warranted.
  • affine connection -- This stub fails to distinguish between the infinitesimal notion of a connection, and the local notion of parallel transport. I like the fact that it at least makes clear what is affine about the connection, but it fails to show how the affine-ness also admits an infinitesimal interpretation which distinguishes the connection from a garden-variety linear connection.
  • parallel transport -- Why does this only discuss the parallel transport of tangent vectors? A notion of parallel transport can be defined for any connection in any vector bundle along any C1 curve. (More generally, it could talk about development in principal bundles too and thus tie in with holonomy.)
  • Riemannian connection -- The same thing as a Levi-Civita connection, but it's allowed to have torsion. Ok.
  • Christoffel symbols -- This is probably the best article in the entire group. (No mention of generalized Christoffel symbols or jet transformations, but you can't have everything.)
  • Weitzenbock connection

And here's a wish list:

Silly rabbit 00:06, 27 June 2006 (UTC)

Anything I can do to help? Maybe you should make a "to-do" list. -lethe talk + 03:27, 27 June 2006 (UTC)
Hi Lethe. I've started [Category:connection (mathematics)]. I've collected everything I could find which is of direct relevance to connections there. I'm also trying to write an intro which is (1) agnostic to whatever the connection du jour happens to be, (2) reasonably accessible to someone who is seeing connections for the first time, and (3) gives a broad overview of the history of connections and the various relationships between the different kinds of connections out there. I know that there is already a connection (mathematics) article. I hope to merge some of that into the category page, and assimilate the rest of it elsewhere. Silly rabbit 05:11, 27 June 2006 (UTC)

[edit] Todo List

Here's a (non-exhaustive) to-do list

  1. Categorize everything which is closely related to connection theory in [Category:Connection (mathematics)]. (I think I have mostly finished with this task.)
  2. Write a stub for Koszul connection. This is just the "usual" idea of a connection in a vector bundle (i.e., \nabla_X.)
  3. Make the other de facto references to a Koszul connection point back to Koszul when appropriate.
  4. Fix connection form. Define the connection form as a vertical vector valued form at the outset. Do it on a fibre bundle first, and then discuss the G-equivariance for a principal bundle separately. (Otherwise there is simply too much going on in the definition to digest at once.) The Vector bundles section should be more clearly delineated as different from the usual definition of the connection form. A couple of things need to be fleshed out here. First, the Koszul connection is indeed "invariant". So I don't like all the hedging at the beginning of the section. Second, the differential form which ultimately crops up is the "connection form" under consideration. I might call this a gauge connection (which is, I believe, what some physicists call them). Indicate that it comes from a connection form (in the sense defined earlier in that article) upon choosing a gauge.
  5. Fix parallel transport. This article needs a complete, and highly detailed rewrite. A topic as important as parallel transport shouldn't be marginalized like this. Parallel transport is, afterall, one of (if not the) centerpiece of connection theory. Specifically, (1) talk about the parallel transport along a Koszul connection, (2) parallel transport for a linear or affine connection and how this leads to geodesics, (3) parallel transport as a lifting of curves to a principal bundle, (4) some (albeit brief) mention of holonomy -- and maybe the relationship with curvature, (5) the more general notion of development along curves or families of curves (at this point, we enter various topics which are probably best treated elsewhere: monodromy, Cartan-Darboux theorem, and who-knows-what-else).
  6. Someone figure out what the heck a Grothendieck connection is, please ;-)
  7. Cartan connection already has an extensive to-do list.
  8. Projective connection needs to be started. This is a good application of Cartan connection.

More to come later. Silly rabbit 05:11, 27 June 2006 (UTC)

[edit] Some discussions with Lethe

I am unfamiliar with the name Koszul, but your description makes it sound like the definition we have at covariant derivative is the Koszul connection. Perhaps a redirect instead of a stub? Clean-up and point changing still needed, I suppose. Similarly, Ehresmann connection should probably redirect to connection form. -lethe talk + 05:28, 27 June 2006 (UTC)
A covariant derivative (defined as it is in covariant derivative) is a linear connection. A Koszul connection[1], named after Jean-Louis Koszul, is a connection on any vector bundle (not just one associated to the bundle of linear frames for the manifold). So a redirect certainly won't do the trick. Furthermore, even the Koszul version of the covariant derivative (with respect to a linear connection) is different from the classical one. Classically, one defines the connection in terms of some generalized Christoffel symbols Γ (which may be defined for a general linear connection by postulating their transformation law). This is unwieldy, as anyone who has ever tried to work with Christoffel symbols knows only too well. Koszul provided the required algebraic framework for describing the ∇ algebraically. Most modern approaches to covariant differentiation define the covariant derivative as a type of Koszul connection. Silly rabbit 06:17, 27 June 2006 (UTC)
Then again, I may be just arguing for its own sake here. The two ideas are compatible enough that, with a little effort, they can easily be merged. I do worry a little that the more general Koszul connection is a lot to hit first-time readers with. I could say something like: "More generally, covariant derivatives also make sense on general vector bundles" in the intro. But I really hate articles that do that sort of thing. Silly rabbit 07:23, 27 June 2006 (UTC)
I agree with the Ehresmann connection redirect to connection form. There is the possibility of confusion with a Cartan connection, but that (remote) possibility can probably be handled easily. Silly rabbit 06:30, 27 June 2006 (UTC)
Gauss-Manin connection, I've added it to Category:Connection (mathematics). -lethe talk + 15:24, 27 June 2006 (UTC)
Thanks. I think I saw that yesterday, but managed to take away the mistaken impression that it wasn't really a connection. (It was late, I was tired. I can't be bothered to read the first sentence of the article ;-D.) Anyway, yes, thanks. Silly rabbit 15:28, 27 June 2006 (UTC)

[edit] Torsion redux

I started reading the various articles on connections. I note you added Koszul connection yesterday, including the statement:

There exists a local trivialization of the bundle E with a basis of parallel sections if, and only if, the curvature vanishes identically.

Although you linked "parallel" to parallel transport, I mentally linked it to parallelizable manifold, which is perhaps closer to the intended meaning?

I may be able to explain what I found confusing about torsion. It's well known that S3 = SU(2) is parallelizable. That means that there exists a connection on S3 for which the curvature vanishes, right?. However, this connection has a non-zero torsion everywhere, right? But, of course, S3 also has a connection whose curvature is that of a sphere; this curvature is non-zero everywhere, but the torsion vanishes identically everywhere (its the Levi-Civita connection).

So ... are these two connections, one flat, and one not, "isolated points", or do there exist a continuum of connections interpolating between the two, having a little bit of curvature and a little bit of torsion? If there's a continuum, is there a canonical path through it (i.e. the path where each connection on the path has curvature distributed uniformly, and torsion distributed uniformly)? If there's a continuum of connections between these two, is this space of connections simply connected? Or is the structure more complex? What is the "shape" of the space of all possible connections? If there are homotopically inequivalent paths in the space of connections that takes one from S3 to parallelized S3, then one might argue that, in this sense, the parallelization isn't unique.

Since all bundles are locally parallelizable (as far as I know; its part of the definition of charts and atlases for a bundle, right?) does this mean that the geometry of the space of connections is trivial if and only if the bundle is trivial? In particular, one should be able to read the homotopy groups of one straight off from the other, right? Or, perhaps, this is more complex: did I just make the Poincaré conjecture generalized to parallelizable manifolds?

I am an amateur at geometry, and infrequently, at that. I suspect I could answer my own questions if I retreated and ground out some calculations, read a few books. However, being both lazy and social, I thought I'd pose them here. linas 15:02, 28 June 2006 (UTC)

Well, I re-read our earlier conversation on this topic. Perhaps I should shut up and just do the calculations. I probably just can't get farther without investing some hard work. linas 15:14, 28 June 2006 (UTC)


The confusion has to do with the two uses of the term parallel (both of which are, rather unfortunately, used in differential geometry). I meant parallel in the sense of parallel transport: that is, sections which are utterly annihilated by the connection. To be sure, a vector bundle will always be (locally) parallelizable in the topological sense by just giving a basis of local sections. But requiring these sections to be parallel with respect to the connection is a further constraint which will not be satisfied in general. Stated another way, a local parallelization is just a local section of the frame bundle. If we have a connection (on the frame bundle), then a basis of parallel sections of the tangent bundle defines a horizontal section of the frame bundle. So parallel for connections might roughly be equated with horizontal. (In fact, I might just do that in the article Koszul connection...)
To address your question about the shape of the space of all connections, it may get a little hairy depending on the particular "brand du jour" of the connection, the gauge group under considerations, and other sundry details.
For principal connections, i.e. Ehresmann connections, the space of connections admits parititions of unity over the base manifold and the space of connections over a fibre is convex. See the Chern-Weil homomorphism for at least one important topological consequence of these facts. Furthermore, a principal bundle admits a flat connection if and only if it is equivalent to a principal bundle whose transition functions are constant. This certainly implies some topological constraints on the bundle, but it is unclear (to me) what they are.
For Cartan connections, things get a little clearer. Basically, a connected manifold admits a complete flat Cartan connection if and only if it is (diffeomorphic to) the quotient of a homogeneous space by a discrete group.
Regards, Silly rabbit 15:59, 28 June 2006 (UTC)
Oh, I guess I didn't really answer your questions. I need to think about it. Sorry, Silly rabbit 16:05, 28 June 2006 (UTC)
Well at least you can glean one thing from my response: the space of connections is convex. So indeed, SU(2) will admit a continuous, even a linear, family of connections interpolating between the two. (Just take t∇1 + (1-t)∇2.) Silly rabbit 16:11, 28 June 2006 (UTC)
Oh, another thing. The Maurer-Cartan connection on SU(2) is flat and torsion-free. Silly rabbit 16:44, 28 June 2006 (UTC)
That didn't make a whole lot of sense, did it? Ok, I clearly need a break. Silly rabbit 19:02, 28 June 2006 (UTC)
I added something equally untrue to an article on my first days at WP, and boy, did I ever get walloped for it!. Thanks, though, this was helpful; I think I finally understand my question, whih is a big step. linas 23:34, 29 June 2006 (UTC)

[edit] More on holomorphy

Hi Silly Rabit. Thank you for working on infinite-dimensional holomorphy. I have a question. I think sections 3 and 4

3 Holomorphic functions defined on a Banach space
4 Weak holomorphy

need to be deleted or integrated I think as that stuff was already discussed in the sections above. Wonder what you think. Thanks. Oleg Alexandrov (talk) 19:35, 30 June 2006 (UTC)

They definitely need to go at some point. I'm not quite done fleshing some of the stuff out, but I've been engrossed with a major overhaul over at Category:connection (mathematics). I'll get back to it now and then. In the mean time, I'll see what I can do about trying to merge in the older material. Best regards, Silly rabbit 19:46, 30 June 2006 (UTC)

[edit] better off

We'd probably be better of with you here. At any rate, I intend to sit down and figure out a few things about Fréchet and Gâteaux derivatives, and I need you around for that, so you can't leave! :-) But I can certainly understand and appreciate how edit-warring and arguing can get you frustrated. -lethe talk + 01:57, 4 July 2006 (UTC)

[edit] a good edit

I'm usually loathe to remove content from articles. It's someone else's good faith work that we're supposed to respect. But sometimes, it just has to go. This was a good edit. The Lie derivative is not a generalization of the exterior derivative (nor the other way around), and there's pretty much no way I can see to save that text to make it right. So good work, sir, and I salute you. -lethe talk + 20:19, 5 July 2006 (UTC)

[edit] Infinite-dimensional holomorphy

Hi. It is me again. I wonder if you could work more on that article. There is good stuff in there now, but it does not yet feel coherent. Would be nice to integrate the bottom two sections too. Thanks. Oleg Alexandrov (talk) 17:16, 16 July 2006 (UTC)

Anybody home? :) Oleg Alexandrov (talk) 04:14, 9 August 2006 (UTC)

[edit] Today's featured article

Just wanted to let you know a featured article you worked on, 0.999..., was featured today on the Main Page. Tobacman 00:37, 25 October 2006 (UTC)

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