From: baez@galaxy.ucr.edu (John Baez) Subject: Re: exterior and geometric calculus Date: Tue, 4 Dec 2001 02:19:31 +0000 (UTC) Newsgroups: sci.physics.research Summary: Natural bundles on manifolds, connections with GL_n. In article <9uf7ou$3ec$1@glue.ucr.edu>, Toby Bartels wrote: >What I think is a more fundamental distinction >than that between and >is that between *topology* and *geometry*, >in the sense defined on another thread. Right! >Any spinor or tensor with various values >can be dealt with in differential topology, A small criticism of this point will be made below. >but there isn't much that you can *do* with them: >contraction of indices of opposite variance, >differentiation and integration of forms, >Lie differentiation, and bracketing of multivectors >(I've doubtless forgotten something). It's sort of hard to write down a complete list of things you can do with tensors without knowing exactly what "things you can do" means. However, if we make this notion precise in a certain way, we can develop a *complete classification* of what we can do with tensors in the context of differential topology. To do this, first we need the concept of a "god-given vector bundle". Crudely speaking, a god-given vector bundle is a vector bundle that we can cook up on a manifold using no extra structure on our manifold. I'll let Toby figure out how to make this into a precise definition! With the right precise definition, the following stuff is true: It turns out that any representation of GL(n) gives a recipe for constructing a god-given vector bundle over any n-dimensional manifold. For example, the trivial representation gives the trivial bundle. The fundamental representation gives the tangent bundle. The dual of the fundamental representation gives the cotangent bundle. And so on. Perhaps Toby can sketch this "and so on"! The basic idea is that all the ways of getting new representations from old ones yield ways of getting new vector bundles from old ones. It also turns out that *all* god-given vector bundles come from representations of GL(n) by this trick. Rather obviously, then, all god-given vector bundles are direct sums of vector bundles corresponding to *irreducible* representations of GL(n). These were classified by Hermann Weyl, and the Wizard sketched the classification a while back in this thread. This stuff is the secret reason the Wizard was so eager to force Eric Forgy to learn about Young diagrams - a trick for classifying irreducible representations of GL(n) and other related groups. Once Eric understands this classification, he will understand all the god-given vector bundles! The differential forms he's so fond of are on the list, but there are many more. As the Wiz hinted, they all correspond to densitized tensors and "pseudo" densitized tensors. With a bit more work, we can also classify all the god-given maps *between* god-given vector bundles. Toby can probably hazard a guess as to what's going on here, since he lives and breathes category theory - and category theory says maps between things are just as important as things! If we take these maps as our definition of "all the things we can do with tensors" in the context of differential topology, we are then in the happy position of having a THEOREM which lists all the things we can do! Of course, integration is not one of the things on this list. The list only includes "local" things that we can do. We could try to prove a more general theorem, though. If anyones wants to learn more about this, here is a book about it that one can download for free! Natural operations in differential geometry by Ivan Kolar, Jan Slovak and Peter W. Michor http://www.emis.de/monographs/KSM/ Here's a bit of the introduction: The aim of this book is threefold: First it should be a monographical work on natural bundles and natural operators in differential geometry. This is a field which every differential geometer has met several times, but which is not treated in detail in one place. Let us explain a little, what we mean by naturality. Exterior derivative commutes with the pullback of differential forms. In the background of this statement are the following general concepts. The vector bundle $\La^kT^*M$ is in fact the value of a functor, which associates a bundle over $M$ to each manifold $M$ and a vector bundle homomorphism over $f$ to each local diffeomorphism $f$ between manifolds of the same dimension. This is a simple example of the concept of a natural bundle. The fact that the exterior derivative $d$ transforms sections of $\La^kT^*M$ into sections of $\La^{k+1}T^*M$ for every manifold $M$ can be expressed by saying that $d$ is an operator from $\La^kT^*M$ into $\La^{k+1}T^*M$. That the exterior derivative $d$ commutes with local diffeomorphisms now means, that $d$ is a natural operator from the functor $\La^kT^*$ into functor $\La^{k+1}T^*$. If $k>0$, one can show that $d$ is the unique natural operator between these two natural bundles up to a constant. So even linearity is a consequence of naturality. This result is archetypical for the field we are discussing here. A systematic treatment of naturality in differential geometry requires to describe all natural bundles, and this is also one of the undertakings of this book. [Here $\La^kT^*M$ is the bundle whose sections are k-forms, as will be obvious to anyone who spends all day writing in TeX.] In fact, this introduction gives some really big hints concerning the questions I wanted Toby to answer! But I hope I managed to get him to think a bit before seeing these hints. :-) >Especially in the quest for background free theories, >I find it interesting to see what can be done >using only differential topology. Me too! >Then when differential geometry is needed, >I like to know what geometric structures are required. >Is a metric needed? a symplectic form? a volume form? >an orientation? a complex structure? nondegeneracy? Right! Now for my promised "small criticism": Spinor bundles are *not* on the list of god-given vector bundles. This is related to the fact that spinors form a representation of Spin(n), not GL(n). So, we need some extra structure - in fact some extra "stuff" - on our manifold before we can define spinors on it. Toby knows the technical category-theoretic definition of "structure" and "stuff", which we discussed with James Dolan a long time back here on s.p.r.. So, he'll know what subtle point I'm making above. Those who don't should read the thread entitled "Just Categories Now", starting here: http://groups.google.com/groups?hl=en&selm=72pusp%243o5%241%40pravda.ucr.edu >Then I know what background is required for a theory to work >and how the theory might couple to spacetime geometry >in a broader theory that has less background structure. Or background *stuff*! >You might notice that 2 of the geometric structures given as examples, >a symplectic form and a volume form, are differential forms. >So I don't see forms as any better or worse than, say, metrics. >It's the imposition of a *particular* structure on the manifold >that makes me sit up and notice . Right - but we *can* talk in a rigorous way about the concept of "more" structure, and try to do with as little as possible. >The geometric calculus of Hestenes and his crowd >uses the fixed background structure of a metric. >That's fine if that's what a theory has. ... but awful otherwise! It's really the inflexibility of their approach that makes it so limiting. >If we all learn anything from these discussions, >then I hope that it would be Ecclesiastes 3:1. Hear, hear. Actually what I've mainly learned so far is which verse "Ecclesiastes 3:1" must be! I know what you must be referring to, even though I hadn't known the verse number.... ============================================================================== From: baez@galaxy.ucr.edu (John Baez) Subject: Re: exterior and geometric calculus Date: Tue, 4 Dec 2001 08:15:19 +0000 (UTC) Newsgroups: sci.physics.research In article <9uhbrj$neu$1@glue.ucr.edu>, John Baez wrote: >It turns out that any representation of GL(n) gives >a recipe for constructing a god-given vector bundle >over any n-dimensional manifold. This is true. >It also turns out that *all* god-given vector bundles >come from representations of GL(n) by this trick. This is false: as James Dolan reminded me, there are also lots of other god-given vector bundles, like jet bundles. (The fiber at p of a jet bundle over M transforms in a way which depends, not just on the first derivative df(p) of the diffeomorphism f: M -> M, but also on the higher derivatives. To understand god-given vector bundles like this, we at least need to understand the representations of certain groups which have GL(n) as a quotient.) >If anyone wants to learn more about this, here is a book >about it that one can download for free! > >Natural operations in differential geometry >by Ivan Kolar, Jan Slovak and Peter W. Michor >http://www.emis.de/monographs/KSM/ Obviously I need to read this book! I know they talk a lot about jet bundles, but now I realize I need to better understand the complete classification of god-given vector bundles.