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October 9, 2012

Tangency

Posted by Urs Schreiber

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This is a guest post by Andrew Stacey (NTNU, currently on sabbatical at Oxford .

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Quiz Time

Here’s a question for you all. What is the tangent space at 00 of the unit interval, T 0[0,1]T_{0} [0,1]? To make it easier, I’ll make it multiple choice:

  1. {0}\{ 0\} ,

  2. [0,)[0,\infty ),

  3. \mathbb{R} ,

  4. None of the above.

What about \mathbb{R} \vee \mathbb{R} , the space formed by gluing two copies of \mathbb{R} together at their origins (you can think of this as the union of the xx and yy axes if it helps). Here are your options for this space.

  1. {0}\{ 0\} ,

  2. \mathbb{R} \vee \mathbb{R} ,

  3. 2\mathbb{R} ^{2},

  4. None of the above.

There’s no particular “right answer” to these (though your answers to the two questions should match up). I can justify all three of the concrete answers. I shan’t, yet, because I want to know what others think and why without tainting the survey.

Background

Those who know me will know that I like loop spaces. I’m pretty happy to meet them in any guise, but if I had to express a preference then it would be as a differential topologist (me, that is, not the loop spaces). This means that I’m considering them as some sort of infinite dimensional manifold.

It’s not a long, nor a difficult, path (ha ha) from loop spaces to more general smooth spaces. Spaces that are almost, but not completely, unlike manifolds. We’ve had many discussions here about what a generalised smooth space should look like. Right now, I don’t want you to think too deeply about that. I just want you to be aware of the fact that there are smooth spaces beyond manifolds. They don’t have charts, but they have a strong family resemblance to manifolds so a lot of intuition and ideas can be extended from manifolds to these more general spaces.

This is what I’m trying to do with tangent spaces: extend them from manifolds to generalised smooth spaces.

Tangent Spaces

The problem is that there is not a unique definition of “tangent space” in differential topology. There are several equivalent definitions, but they do not remain equivalent when one generalises them. That’s okay because actually I’m not after a unique definition. I’m after a characterisation. Thus the question I really want to ask is the following:

Suppose I gave you two smooth spaces, XX and YY, and told you that YY was a tangent space for XX (I’d probably better give you the projection π:YX\pi \colon Y \to X as well). What would you expect that to tell you about YY?

Note that I’m using “tangent space” here to mean all the pointwise tangent spaces put together into a new smooth space. I can’t say tangent bundle because they may not form a bundle. Note also that, following from what I said about the different definitions, I’m using the indefinite article: a tangent space.

Conclusion

My desired conclusion from this is to be able to give a characterisation of a tangent structure on a category of generalised smooth spaces. It will consist of an endofunctor, and one or two natural transformations, where the functor assigns to a smooth space a tangent space. But before I can characterise such functors, I need to know what characterises a tangent space. Hence the question.

The longer term goal is that I want to use tangent spaces as a tool to study smooth spaces. A finite dimensional manifold is actually modelled on its pointwise tangent spaces and this turns out to be a very important property in studying mapping spaces. More general smooth spaces will not have as close a relationship, but nonetheless there might still be enough of a relationship to be able to exploit it.

Posted at October 9, 2012 5:59 PM UTC

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Re: Tangency

An algebraic geometer will want to distinguish “tangent space” (answers 3 and 3) and “tangent cone” (answers 2? and definitely 2).

Posted by: Allen K. on October 9, 2012 7:01 PM | Permalink | Reply to this

Re: Tangency

That’s the word I was looking for!

Posted by: Mike Shulman on October 10, 2012 1:55 AM | Permalink | Reply to this

Re: Tangency

A-ha. Very interesting that there’s two notions. Can you give me a 2 second definition for a differential topologist?

I presume that there’s a natural morphism from the cone to the space. Is it true that the tangent space is the linearisation of the tangent cone?

Posted by: Andrew Stacey on October 10, 2012 10:26 AM | Permalink | Reply to this

Re: Tangency

I think of tangent vectors via physical intuition: direction and speed for a particle moving in the space in question. So I would answer 3 in both cases. I suppose the kinematic tangent space is the formalization of this for the category of Smooth Spaces?

This type of tangent space will not be a vector space at a singular point. But it is very important that the total differential of a smooth map between manifolds is linear (when restricted the fibers). So I wonder, what is the algebraic structure on more general tangent spaces, such that the differential of a map will respect this structure? At the very least, there will be an action of the positive reals, and a partially defined addition operation. And upon the extension to tangent multivectors, the exterior product of two tangent vectors will be defined precisely when addition is defined between the same. It seems to me that these structures are to vector spaces what groupoids are to groups, and that they formalize the algebraic structure of (kinematic) tangent spaces at singular points.

Posted by: Tyler Jones on October 10, 2012 2:53 AM | Permalink | Reply to this

Re: Tangency

Did you mean to say answer 2 in both cases?

Posted by: Todd Trimble on October 10, 2012 3:45 AM | Permalink | Reply to this

Re: Tangency

Ah, yes I did.

Posted by: Tyler Jones on October 10, 2012 4:15 AM | Permalink | Reply to this

Re: Tangency

Taking tangent vectors to be defined in terms of the velocities (velocity == speed &\& direction) of particles moving in the space seems reasonable. However, depending on what precisely one means by this, one will get different answers.

If XX is our space, then let’s define a “particle trajectory” to be a differentiable function f:RXf:\mathbf{\R}\to X. I think it is clear what ‘differentiable’ means in our two examples. We get the velocity at time t=0t=0 by considering the differential f(0)f'(0).

With this definition, we get T 0[0,1]={0}T_0[0,1]=\{0\}. The reason is that for any non-zero velocity, the trajectory would have to be outside of [0,1][0,1] either shortly before t=0t=0 or shortly after.

However, one could make many other definitions. For example, if we consider the domain of a trajectory to be [0,)[0,\infty) instead of requiring it to be defined for all times R\mathbf{R}, then we obtain T 0[0,1]=[0,)T_0[0,1]=[0,\infty). If we take the domain to be (,0](-\infty,0], then we get T 0[0,1]=(,0]T_0[0,1]=(-\infty,0]!

Posted by: Tobias Fritz on October 10, 2012 5:19 PM | Permalink | Reply to this

Re: Tangency

If I think of [0,1][0,1] as a manifold with corners, then T 0[0,1]T_0 [0,1] is T 0T_0 \mathbb{R}, which you may identify with \mathbb{R}. If I think of [0,1][0,1] as S 1/(/2)S^1/(\mathbb{Z}/2) then T 0[0,1]T_0 [0,1] is isomorphic to /(/2)\mathbb{R}/(\mathbb{Z}/2). Whether one should think of /(/2)\mathbb{R}/(\mathbb{Z}/2) as [0,)[0,\infty) is another story…

\mathbb{R}\vee \mathbb{R} maybe thought of as a subset of 2\mathbb{R}^2, i.e., as a subCartesian space. Or it may be thought of as a stratified space. And there choices of the stratification. So there is more than one “right” answer to your question.

My feeling is that answeres depend on what you are trying to accomplish.

Posted by: Eugene on October 10, 2012 3:13 PM | Permalink | Reply to this

Re: Tangency

My feeling is that answeres depend on what you are trying to accomplish.

That’s true.

What I’m trying to accomplish is a characterisation of what “YY is ‘the’ tangent space of XX” means. So I’m less interested in which answer you pick as to why. And in the “why” then I’m looking for reasons that are a bit space-independent. So a different way of looking at [0,1][0,1] to what I expected is mildly interesting, but not as interesting as knowing why for that way of looking at [0,1][0,1] you pick /(/2)\mathbb{R}/(\mathbb{Z}/2).

For the view of [0,1][0,1] as a manifold with corners, why do you allow the negative direction?

For \mathbb{R} \vee \mathbb{R} let’s pick a stratification - I’ll let you pick one, in fact - and tell me what answer you get (and why!) for that choice.

Posted by: Andrew Stacey on October 10, 2012 5:21 PM | Permalink | Reply to this

Re: Tangency

What I’m trying to accomplish is a characterisation of what YY is `the’ tangent space of XX means.

But wouldn’t the answer depend on the category that XX is in? More precisly, I would expect the tangent bundle TXTX of XX to be an object in the same category as XX and the tangent space at a point (if such a thing makes sense) to be a subobject of TXTX.

So if [0,1]=S 1/(/2)[0,1] = S^1/(\mathbb{Z} /2), it’s an orbifold, so its tangent bundle should be an orbifold too.

If XX is a stack, then TXTX is a stack. I am not sure if you could say that a fiber of TXXTX\to X is a vector space; probably not.

If [0,1][0,1] is a manifold with corners, its tangent bundle should be a manifold with corners. I like to think that manifolds with corners have differential forms so the tangent bundle of a manifold with corners should better be a vector bundle.

If XX is a stratified space in some sense of the word, I would expect its tangent bundle to be a stratified space too.

I know that there is a nice version of cotangent bundles of orbit spaces for proper group actions (see Examples of singular reduction). Such a cotangent bundle is not a vector bundles but it is a stratified space and it’s even symplectic.

(I guess I am also saying that you are better off thinking of tangent bundles rather than tangent spaces)

Posted by: Eugene on October 10, 2012 7:04 PM | Permalink | Reply to this

Re: Tangency

But wouldn’t the answer depend on the category that XX is in?

The precise form of the answer would, but I’m hoping for some thoughts on how to get to that answer.

Think of it this way: if I declare that YY is the tangent space of XX, what would make you reject my choice?

I’m happy for you to restrict to a particular category that you know about and have intuition for but I’m looking for something that can be applied more generally.

For example, with Euclidean spaces we have T n n× nT \mathbb{R}^n \cong \mathbb{R}^n \times \mathbb{R}^n. But you wouldn’t like me to say that for any smooth space, we’ll declare that TX=X×XT X = X \times X.

(This is a silly example because the Euclidean space version is only an isomorphism on the object-level, it doesn’t extend to a natural transformation T×T - \to - \times -. But hopefully it helps clarify what I’m after.)

(I guess I am also saying that you are better off thinking of tangent bundles rather than tangent spaces)

I’m trying to - but the language is getting in the way. I’m avoiding using the word “bundle” as that implies local structure that may not exist. So I’m using “tangent space” really to mean “total tangent space” (I did say this in the original). If there’s a better phrase, I’ll use that instead.

Posted by: Andrew Stacey on October 11, 2012 1:38 PM | Permalink | Reply to this

Re: Tangency

Since the unit interval is the naive quotient of \mathbb{R} by the natural action of the infinite dihedral group D D_\infty, as having once been a physicist and always fond of mirrors, I cannot think of a better tangent space T 0[0,1]T_0 [0,1] than \mathbb{R}, with exponential map the one that folds up \mathbb{R} — modify this as seems sensible for other points of [0,1][0,1]. You can tell that I like my interval to be metrized in a sensible way.

Posted by: Jesse C. McKeown on October 10, 2012 3:56 PM | Permalink | Reply to this

Re: Tangency

insert scare-quotes as needed, e.g. “the” and “natural”.

I see Eugene made a similar observation to me, a little earlier, but with different conclusions. Nifty!

Posted by: Jesse C. McKeown on October 10, 2012 4:08 PM | Permalink | Reply to this

Re: Tangency

Is there any way to naturally embed smooth spaces inside of smooth manifolds? My answer to both questions is 2, though that’s purely based on intuition. And my intuition in both cases (I realize after some consideration) is based on having spaces sit inside of 3\mathbb{R}^3 and sort of imagining particles traveling along and then “taking off” at some point, and what possible velocities and directions can they have as they take off.
Posted by: Jon Beardsley on October 10, 2012 10:05 PM | Permalink | Reply to this

Re: Tangency

To sing a song that others can surely sing better…

In a synthetic diff geom setting, we’ll have a “tiny” space TT — think of it as a real interval of infinitesimal length — with the property that for any space XX, the tangent bundle is X TX^T. Actually, TT is a pointed space, and the map 1T1 \to T picking out the point induces the projection map TX=X TXT X = X^T \to X. As you’d expect, its fibre over xXx \in X is the tangent space at xx.

So in this setting, it all comes down to (i) choosing your category, and (ii) choosing your tiny object TT. But if I’m correctly catching your drift, Andrew, you’re considering the possibility of multiple different TTs. Am I understanding?

Posted by: Tom Leinster on October 10, 2012 11:38 PM | Permalink | Reply to this

Re: Tangency

So in this setting, it all comes down to (i) choosing your category, and (ii) choosing your tiny object TT. But if I’m correctly catching your drift, Andrew, you’re considering the possibility of multiple different TTs.

I think Andrew is not even getting to (ii), if the conversation at the nforum is anything to go by.

Posted by: David Roberts on October 10, 2012 11:55 PM | Permalink | Reply to this

Re: Tangency

I agree with that. The feeling I get from Andrew is that he mostly has been working with concrete structures (i.e., forming a category CC where hom(1,):CSet\hom(1, -): C \to Set is faithful, where 11 is the one-point space), so that maps are determined by what they do on points. That would rule out the tiny object TT (the “walking tangent vector”).

Posted by: Todd Trimble on October 11, 2012 12:25 AM | Permalink | Reply to this

Re: Tangency

Todd, Tom,

Does the “walking tangent vector” definition work for differential stacks, which do not form a concrete category? In particular, how does it compare with: “represent your stack by a Lie groupoid and apply the tangent functor to the groupoid”?

Posted by: Eugene on October 11, 2012 12:39 PM | Permalink | Reply to this

Re: Tangency

Certainly my personal preference is for concrete categories, but maybe that’s just because I’m a hide-bound stone-age mathematician who hasn’t yet renaissed.

But I’m trying to put preferences aside and get at some intuition on how tangent spaces should behave no matter how they are defined and what category they are defined in. So if TT is available, fine - use it. But give me a reason for using it rather than just “It’s there, what else are you going to do with it?”.

In fact, let’s follow this idea. The claim is that TT is the “walking tangent vector” in that it exists purely to be a tangent vector. What is it about TT that makes it such?

We can think of the Lawvere theory for groups as the “walking group” in that it exists purely to support the notion of a group (please don’t pick apart my limited understanding!). In it, we have an object that is a group, and is nothing but a group. It’s easy to explain this since all we have of it are the group operations. So its quintessential groupiness is evident.

What is it about TT that makes it quintessentially tangential?

Posted by: Andrew Stacey on October 11, 2012 1:45 PM | Permalink | Reply to this

Re: Tangency

“Walking tangent vector” is a whimsical and probably really lousy term. (For one thing, ‘vector’ generally has little meaning to me outside of the context of an abstract vector space structure – it’s not as if a child can point and say, “Look, Mommy! A vector! Awww…”)

But the idea is that in synthetic differential geometry, TT is supposed to be an object that represents tangent vectors on ordinary finite-dimensional (paracompact, Hausdorff) manifolds without boundary MM, in the sense that there is a natural bijection between tangent vectors vTMv \in T M and smooth maps TMT \to M. (I choose such manifolds because there it is uncontroversial what we mean by tangent space. Right?) Under some assumptions, this ought to be enough to uniquely characterize TT, but I’d have to think a while longer on that to be sure what assumptions I might mean.

The idea anyway is that TT is forced to be the spectrum (in an algebraic geometry sense, or an analogous smooth geometry sense) of the algebra [x]/(x 2)\mathbb{R}[x]/(x^2). In other words, precisely the kind of algebraic-geometric or smooth-geometric gadget you’d use to represent first-order jets.

Incidentally, in the context of synthetic differential geometry, the “tangent bundle” X TX^T is not guaranteed to have vector-space fibers for every “smooth object” XX. (Put in different language: this TT doesn’t carry “co-vectorspace structure”.) So just because I answered a certain way in the nForum straw poll doesn’t mean I think that a tangent space has to be a vector space, always.

Another thing which I found interesting in the nForum discussion is that while TT may be uniquely characterized, there can be different ways of assigning smooth (SDG) structure to somewhat innocent-looking spaces like [0,1][0, 1], which leads to different answers to the question about tangent spaces. My own inclination was to treat [0,1][0, 1] more or less as the spectrum of C ()C^\infty(\mathbb{R}) modulo the ideal of smooth functions which vanish on [0,1][0, 1]. But as made clear to me by Urs in the nForum discussion, there is at least one other reasonable option, leading to a different tangent space.

Posted by: Todd Trimble on October 11, 2012 8:32 PM | Permalink | Reply to this

Re: Tangency

Incidentally, in the context of synthetic differential geometry, the “tangent bundle” X TX^T is not guaranteed to have vector-space fibers for every “smooth object” XX

…but there is a very general class of smooth objects, called microlinear ones, for which it does. Microlinear spaces are much more general than traditional smooth manifolds; for instance, they are closed under all limits and exponentials. Thus, for example, we can define a version of the “cross” as the zero-set of λx.λy.xy:R 2R\lambda x.\lambda y. x y : R^2 \to R, which is microlinear (being an equalizer of a pair of maps between microlinear spaces), and hence has a linear tangent space in this sense. Of course, this is not too surprising, since the SDG “line object” RR contains nilsquare infinitesimals δ\delta, for which points like (δ,δ)(\delta,\delta) will belong to that “cross”.

Posted by: Mike Shulman on October 12, 2012 12:37 AM | Permalink | Reply to this

Re: Tangency

True; I should have mentioned that. Thanks for doing so!

Posted by: Todd Trimble on October 12, 2012 1:27 AM | Permalink | Reply to this

Re: Tangency

Just a quick line to suggest reading this blog post by Jeffrey Morton about Cartan geometry and spacetime. He points out (and this is not new, but nicely explained), that Cartan geometries are like manifolds where the ‘tangent space’ is a homogeneous space more general than a vector space - in fact modelled on some coset space G/HG/H for a Lie group GG.

Posted by: David Roberts on October 11, 2012 2:34 AM | Permalink | Reply to this

Re: Tangency

Robin Cockett and I just wrote a paper on this exact subject! We were looking to give an axiomatization for abstract tangent functors, based on functors that occured in Cartesian differential categories. The axiomiatization we arrived at was similar to what you mention above: an endofunctor with various natural transformations and coherence equations.

Soon after we submitted the paper, however, Anders Kock kindly pointed out to us that Rosicky had come up with almost the exact same axiomatization almost 30 years ago! He was trying to understand the similarities between the tangent functor on the category of smooth manifolds, the tangent functors in SDG, and the tangent functors in algebraic geometry. His paper can be found here.

After we recovered from our shock, we realized that we still had something to say on the subject. Namely, (1) the “tangent spaces” of any abstract tangent functor give a Cartesian differential category, and (2) we could improve Rosicky’s result about representable tangent functors. Specifically, the end result is that if you start with an abstract tangent functor which you know is representable, you recover the setting of SDG: a commutative rig R whose infinitesimals D satisfy the Kock-Lawvere axiom: R^D = R x R.

So, on the one hand, restricted to particularly simple spaces, abstract tangent functors are the same as Cartesian differential categories, while on the other hand, if the tangent functor is representable, you recover SDG.

In between these two extremes are all sorts of other examples: the canonical example of finite-dimensional smooth manifolds, the convenient manifolds of Kriegl and Michor (where the tangent bundle is the kinematic one), and various categories of algebraic geometry. Hopefully other examples of generalised smooth spaces, such as diffeology or Fermat spaces, should also satisfy the axioms.

Anyways, our updated paper can be found here.

Posted by: Geoff Cruttwell on October 11, 2012 5:10 PM | Permalink | Reply to this

Re: Tangency

Thanks for joining the discussion!

Your article was pointed out to me when I first asked about this on the nForum. It looked very interesting (I haven’t digested it all as yet). The big bit that I wasn’t so sure I liked was the requirement that the fibres be vector spaces.

Can you explain why you put that in there?

Posted by: Andrew Stacey on October 11, 2012 5:57 PM | Permalink | Reply to this

Re: Tangency

Well, we only ask that the fibres be additive, not necessarily vector spaces; in fact, there is no explicit mention of a base field/ring in the axiomatization. Rosicky also asked that the fibres have negation, but we removed that requirement as many of the combinatorics/computer science type examples (polynomial functors, combinatorial species) need not have negation.

As to the question of why we wanted additivity, one of our original goals for tangent structure was to show that Cartesian differential categories were a special case of tangent structure; since these have additive hom-sets, we needed addition in the fibres. In addition, as all the other examples had addition, we saw no harm in including it.

Thus, the inclusion of additivity was more pragmatic than philosophical, so I can certainly see a case for removing it. I note in the nForum post that you object to additivity on the basis of quotient examples; this may well be a reasonable objection. You also mention that the canonical flip natural transformation may be useful. I would add that of all the structure in the axiomatization, it is the vertical lift natural transformation and its universal property that is the most important: it is needed to define the Lie bracket, to show that the tangent spaces form a Cartesian differential category, and to show that representable tangent structure gives SDG. So, I would recommend looking at the structure of the vertical lift: I could well imagine removing the additive requirement while keeping the canonical flip and vertical lift.

Posted by: Geoff Cruttwell on October 11, 2012 7:32 PM | Permalink | Reply to this

Re: Tangency

As a followup to my comment, looking at the axiomatization again, I note that the statement of the universality of the vertical lift uses the existence of addition. In Rosicky’s original formulation, the universality of the vertical lift was a bit different, but at the very least uses the existence of 0’s in the fibres (see proposition 2.15 in our paper). However, even there, it is not Rosicky’s formulation that is used in the proofs, but the form we give in the axioms, and to get from his formulation to ours requires negation in the fibres.

There may be some clever way to express the universality of the vertical lift without addition/negation/zeroes, but I’m not sure what it would look like.

Posted by: Geoff Cruttwell on October 11, 2012 7:46 PM | Permalink | Reply to this

Re: Tangency

Andrew, interesting that you posted this now, as I’m in the middle of writing a paper with Enxin Wu which includes a section about tangent spaces to diffeological spaces. It’s not ready for distribution yet, but I will try to remember to post a link here when it is.

In any case, I think that there is no one right answer, and that what you define your tangent vectors to be depends on what you’d like to do with them. If they should relate to flows, then you probably want to take something like equivalence classes of curves (and then you won’t end up with a vector space). Or you may take sums of such things to get a vector space, which is probably the same as taking the colimit of the tangent spaces of the plots. Or you might take derivations on an algebra associated to your space, e.g. an algebra of germs of functions, if your goal is to, well, differentiate functions.

There are maps between the various tangent spaces, and under certain assumptions the maps are isomorphisms.

Posted by: Dan Christensen on October 11, 2012 5:17 PM | Permalink | Reply to this

Re: Tangency

I’d like to read that - don’t forget to bring it to my attention when it’s done.

I certainly am not looking for One True Answer. You’ve mentioned a few constructions there and what I’d like is some characterisation that can encompass them all, but which also somehow excludes things that are definitely not tangent spaces (I gave the silly example above that TX=X×XT X = X \times X would be wrong even though it “works” for Euclidean spaces).

Posted by: Andrew Stacey on October 11, 2012 6:01 PM | Permalink | Reply to this

Re: Tangency

Andrew, do you know about the metric tangential calculus of Elisabeth Burroni and Jacques Penon?

The central definition — and it’s amazing if this is new — is as follows. Let WW and XX be metric spaces, and let f,g:WXf, g: W \to X be two maps. By “map” I just mean map of sets for the moment. Let wWw \in W. Then ff and gg are tangent at ww if f(w)=g(w)f(w) = g(w) and

d(f(u),g(u))d(u,w)0asuw(uw). \frac{d(f(u), g(u))}{d(u, w)} \to 0 \quad as u \to w \quad (u \neq w).

This is an equivalence relation on maps WXW \to X. You could define the “(W,w)(W, w)-tangent bundle” of XX to be the set of all equivalence classes of maps WXW \to X. (I guess we want to restrict to just nice enough maps: continuous or Lipschitz or something. Also, maybe we don’t want actual maps WXW \to X, but germs at ww of maps WXW \to X.)

If you’re also given a point xXx \in X, you could define the “(W,w)(W, w)-tangent space” of XX at xx to be the set of equivalence classes [f][f] such that f(w)=xf(w) = x. I haven’t read Burroni and Penon’s paper, but from the abstract it looks as if they make some such definitions (using “locally lipschitzian” maps).

Particularly interesting cases would be where W=W = \mathbb{R} (and w=0w = 0, say), or where W=[0,)W = [0, \infty) and w=0w = 0.

Posted by: Tom Leinster on October 11, 2012 9:59 PM | Permalink | Reply to this

Re: Tangency

Whoa! That sounds incredibly interesting!

And yes, it does seem amazing that it could be new. On the other hand, perhaps it takes a lot of bravado or optimism to pursue such an idea – many people would probably think, “if this really works, surely someone would have already thought of it…” :-)

Posted by: Todd Trimble on October 11, 2012 10:48 PM | Permalink | Reply to this

Re: Tangency

Hi,
I usually read you guys but this is my first post here.
In the past, I’ve came a cross infinite dimensional bundles in which it would be nice to do some Cartan calculus: given a connection, define the curvature, Lie derivative, contraction, etc… and obtain the well known identities (Cartan’s formula, Bianchi, etc…). Can you do this with any notion of tangent space on your infinite dimensional manifold?

Posted by: Manuel on November 20, 2012 2:20 PM | Permalink | Reply to this

Re: Tangency

I haven’t checked all the comments above, but I think the following idea hasn’t been mentioned.

Following the post Re: “How many sides does a circle have?” that’s been doing the rounds lately, one has the example of the surface of an ordinary cone. The author defines the tangent ‘space’ as a cone, and gives the following more abstract definition:

More seriously: the surface of a convex body is a classical example of an Alexandrov space (metric space of curvature bounded below in the triangle comparison sense). Perelman proved that any Alexandrov space can be stratified into topological manifolds. Lacking an ambient vector space, one obtains tangent cones by taking the Gromov-Hausdorff limit of blown-up neighborhoods of p{p}. The tangent cone has no linear structure either — it is also a metric space — but it may be isometric to the product of k{\mathbb{R}^k} with another metric space. The maximal k{k} for which the tangent cone splits off k\mathbb{R}^k becomes the rank of p{p}.

and then links to this paper which shows how this can break down under weaker assumptions.

I wonder if the result of Perelman can be used for generalised smooth spaces, perhaps in a restricted setting?

Posted by: David Roberts on August 19, 2013 3:29 AM | Permalink | Reply to this

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