Expansion of references section at differential topology.

]]>Have added more of the original (“historical”) References with brief comments and further pointers.

]]>created *hyperbolic manifold* in order to record a reference on the relation between volumes of hyperbolic 3-folds and their Chern-Simons/Dijkgraaf-Witten invariants by Zickert.

I am aware of the following: in the context of synthetic differential geometry (SDG) one obtains a Lie algebra by exponentiating a microlinear group by a standard infinitesimal object and taking the infinitesimal commutator, and that the functor expressed by this operation factors through formal group laws (FGLs) in the usual way. This reveals that Lie groups are FGLs with respect to first-order infinitesimals.

Now I would like to consider a lined topos equipped with higher-order infinitesimals, and develop in this context a modified notion of microlinearity. I have not yet developed the details of this. But does modifying microlinearity in this way, to yield R-modules by exponentiating FGLs with higher-order infinitesimals, sound reasonable? It is worth saying that in general we want certain polynomial identities to hold in the resulting R-modules, e.g. the Jacobian identity.

While FGLs have been thought of in this way (e.g. Didry in [1], an attempt to extend Lie theory to include Leibniz algebras), I have not found sources discussing modifications of microlinearity to subsume FGLs in the language of SDG. Some suggestive remarks can be found in Nishimura’s work, such as in the introduction of the paper [2], where the author discusses prolongations of spaces with respect to polynomials algebras as generalizations of Weil algebras. What do you think, nForum?

[1] Didry, M. Construction of Groups Associated to Lie- and to Leibniz- Algebras

[2] Nishimura, H. Axiomatic Differential Geometry II-2, Chapter 2: Differential Forms

]]>New entry distribution of subspaces and a disambiguating remark at distribution.

]]>I was in the differential cohesion group at the HOTT MRC last week and one thing we struggled with was that we only knew one model (sheaves over formally thickened Cartesian space) which is pretty complicated to construct. Also this specific model has extra properties so not all of the axiomatics can really be explained with just one model, for instance Felix told me that every object in this topos is formally smooth.

Plain cohesion, on the other hand, has some very elementary toy models like reflexive graphs and the sierpinski topos that are very nice for getting intuition.

Do we have any analogous toy models for differential cohesion? All the better if they are “thickenings” of sierpinski or reflexive graphs. Ideas are also welcome, I’d be happy to work through some details myself.

]]>Can we view the Bianchi identity as a local coherence law?

Let me explain. On a vector bundle with connection and holonomy group $G$, each path on the base manifold gives an isomorphism between the fibers at the endpoints. The fibers are all isomorphic, but if the connection is not exact, there is no canonical choice of isomorphism between two fibers.
If we now pick two homotopic paths $\gamma,\gamma'$ with the same endpoints, the “difference” (in a multiplicative sense) between the two holonomies is given by the parallel transport along the loop $\gamma'\circ\gamma^{-1}$. It’s an element in the holonomy group $G$. Let’s call it “difference of holonomies”.
Now, we said that to compare two fibers (at different points on the base space) we have to choose not just the two fibers, but also a path.
But to compare two *paths* (with the same endpoints), we don’t have to make any choice, right?
The “difference of holonomies”, as I called it above, only depends on the paths, not on a 2-cell connecting them. (I’m talking of 1-bundles here.)
Is this an example of coherence law?

Now, take two different 2-cells $\sigma,\sigma'$ connecting $\gamma,\gamma'$. We can “integrate” the curvature form on these 2-cells (possibly in a suitable non-Abelian sense), the result of the integral will be the same, and it will be equal to the “difference of holonomies” above.

Now there are two possibilities:

1) $\sigma,\sigma'$ are homotopic;

2) $\sigma,\sigma'$ are not homotopic.

If 1): then I can find a 3-cell connecting them. We can “integrate” the (exterior covariant) derivative of the curvature form on this 2-cell, that form is zero because of the Bianchi identity, and so the “integral” is the identity in $G$. So, the Bianchi identity is a local way to say that the difference of holonomies will not depend on the choice of the 2-cell, as long as it is between the right 1-cells.

If 2): the Bianchi identity is useless here, because there is no 3-cell along which we can do that “integral”. But the difference of holonomies still *has* to depend only on the two paths! So we can take the 2-cell $\sigma'\circ\sigma^{-1}$, and “integrate” the curvature form on it. The result should still be the identity in $G$.

Now as an example consider the tangent bundle to an orientable surface with Riemannian metric, with the Levi-Civita connection. This way the holonomy group is $SO(2)$, which is Abelian, and so all the “integrals” above are really (exponentials of) integrals of differential forms in the usual sense. In particular, curvature 2-form $R$ evaluated on the 2-cell $\sigma'\circ\sigma^{-1}$ is simply:

$\exp \Bigg( i\int_{\sigma'\circ\sigma^{-1}} R \Bigg)\;.$I have argued above that by “coherence”, this quantity should be the identity on $G=SO(2)$. This implies that:

$\int_{\sigma'\circ\sigma^{-1}} R$*must* be an integer multiple of $2\pi$.

Now, this is not the entire Gauss-Bonnet theorem, but part of it! Is this reasoning correct? Can it be generalized?

Sorry if what I ask is trivial (or wrong), I’m moving my first steps in the n-world. Please also tell me if anything is unclear.

]]>I have listed today’s arXiv preprint

- Marco Benini, Alexander Schenkel,
*Poisson algebras for non-linear field theories in the Cahiers topos*, arxiv/1602.00708

at variational calculus despite the title, as it seems that the construction of presymplectic current after Zuckerman’s idea on geometry of variational calculus is very central to the paper.

]]>I have just now two new master students who are going to look into certain geometric aspects of physics. Also a colleague just asked me for suggestions for a course on “geometry and physics”. I kept pointing to Frankel’s book. That’s great as far as it goes, but it misses on a lot of clarifications available meanwhile.

So I thought it’s about time that I start making notes for a modern introductory course on

I put that into the $n$Lab proper, instead of on my personal web. One reason is that otherwise hyperlinking becomes a pain. Another reason is that this should really not be hidden and reserved somewhere but be out there in the open for everyone to join in. Though I do have a certain strategy in mind, which I would like to ask to follow.

You’ll see what I mean when you look at the entry. It’s so far just a first sketch of a section outline with some keywords and notes to indicate what is eventually to go there. That’s how far I got tonight. (And I really need to sleep now to be ready for my homological algebra course tomorrow…) But I guess the idea and the intended structure is already visible. Will be expanded and edited in the course of the next weeks.

]]>stub for *moduli space of connections*, started to collect some references

I added redirets Atiyah sequence, Atiyah class, Atiyah algebroid to Atiyah Lie algebroid. Maybe we want to have Lie algebroid aspect (concentrating on bracket) and the cohomological/derived category aspect (cohomology class of the exact sequence of modules) separate in fuiture, but now the material is still too small. I added a number of interesting references and a sentence on the class.

]]>Some time ago I started a stub characteristic variety to record few references, mainly in D-module context. Regarding that the related notion of a characteristic ideal also appears in the treatment of Iwasawa polynomial and Alexander polynomial which Urs wants to understand from the point of view of connections and differential refinements of cohomology, maybe we should do some effort to make some pages which will connect various notions of characteristic ideals and their loci across various subjects. I just recorded

- Andrea Bandini, Francesc Bars, Ignazio Longhi,
*Characteristic ideals and Iwasawa theory*, arxiv/1310.0680;*Characteristic ideals and Selmer groups*, arxiv/1404.2788

at characteristic ideal for the version in the context of Iwasawa theory.

]]>New entry fundamental vector field which covers also the somewhat dual notion fundamental differentiable form. Redirecting also fundamental form. The entry is partly intended to support the content in the entry Ehresmann connection. Please check the content.

]]>M M Postnikov’s books on geometry and topology are among my personal favourites. Careful teahcing with love and elegance, precision in theory and with lots of examples elaborated in great detail. It is also very reliable. I have however problem with one statement which I found few times in his books and which I have problem with:

Let $U$ be an open set on a smooth Hausdorff paracompact manifold $M$ of dimension $m+n$. The following is equivalent for a distribution $H$ of subspaces in the tangent bundle $TM$: (i) There are $n$ smooth forms on $U$ such that $H_p$ is the common annihilator of them at every point $p\in U$; (ii) there exist $m$ smooth vector fields such that $H_p$ is the span of those at every point $p\in U$.

So, I have no problem in proving that this is true locally, or I think it holds more strongly, over a contractible open $U$. But I do not see that the trivializing $m$-tuple on the form side would imply global trivializing $n$-tuple on the vector field side for general open set $U\subset M$.

Any help ?

]]>New entry directional derivative, redirecting also Gâteaux derivative. Much of the material is adapted from Fréchet space (which also calls for this entry). Somebody should write more on the (possibly infinite-dimensional) manifold case.

]]>I am unhappy with Lie derivative. In the previous version it defined the Lie derivative as a secondary notion, using the differential and the Cartan homotopy formula (for which I finally created an entry). I have added a bit mentioning vector fields etc. and a formula using derivatives for forms but this is still not the right thing. Namely, in my understanding the Lie derivative is a **fundamental notion** and should not be *defined* using other differential operators, but by the “fisherman’s derivative” formula. Second it makes sense not only for differential forms but for any geometric quantities associated to the (co)frame bundle, and in particular to any kind of tensors, not necessarily contravariant or antisymmetrized. For this one has a prerequisite which will require some work in $n$Lab. Namely to a vector field, one associated the flow, not necessarily defined for all times, but for small times. Then for any $t$ one has a diffeomorphism, which is used in the fisherman’s formula. But fisherman’s formula requires the pullback and the pullback is usually defined for forms while for general tensor fields one may need combination of pullbacks and pushforwards. However, for diffeomorphisms, one can define pullback in both cases, and pullback for time $t$ flow corresponds to the pushforward for time $-t$. To define such general pullback it is convenient to work with associated bundles for frame or coframe bundle and define it in the formalism of associated bundles. In the coframe case, this is in Sternberg’s Lectures on differential geometry (what returns me back into great memories of the summer 1987/1988 when I studied that book). So there is much work to do, to add details on those. If somebody has comments or shortcuts to this let me know.

However, there is a scientific question here as well: what about when frame bundle is replaced by higher jet bundles, and one takes some higher differential operator for functions and wants to do a similar program – are there nontrivial extensions of Lie derivative business to higher derivatives which does not reduce to the composition of usual Lie derivatives ?

]]>Affinity in the context of D-modules, as defined by Alexander Beilinson is the subject of a new stub D-affinity. There is a categorical generalization in the MPI1996-53 preprint (pdf) of Lunts and Rosenberg in terms of differential monads. Many generalizations of Beilinson-Bernstein localization theorem have their intuitive explanation in a two-step reasoning. First the noncommutative algebra in question is understood as a noncommutative (or maybe categorical) resolution of singularities of a commutative object. Then the latter satisfies D-affinity and one can localize.

]]>I started an important entry differential monad. According to Lunts-Rosenberg MPI 1996-53 pdf differential calculus on schemes and noncommutative schemes can be derived from the yoga of coreflective topologizing subcategories in the abelian category of quasicoherent sheaves on the scheme, like the $\mathbb{T}$-filtration, and $\mathbb{T}$-part, in the case when the topologizing subcategory is the diagonal in the sense of the smallest subcategory of the category of additive endofunctors having right adjoint which contains the identity functor – in that case we say differential filtration and differential part. The regular differential operators are the elements of the differential part of the bimodule of endomorphisms. Similarly, one can define the conormal bundle etc.

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