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Because the theory of sheaves is a functorial theory, it has been adopted in algebraic geometry (both using the functor of points approach and the locally ringed space approach) as the "main theory" used to describe geometric data. All sheaf data in the LRS approach can be described by bundles using the éspace étalé construction. It's interesting to notice that the sheafification of a presheaf is the sheaf of sections of the associated éspace étalé.

However, in differential geometry, bundles are for some reason preferred. Is there any reason why this is true? Are there some bundle constructions which don't have a realization as a sheaf? Are there advantages to the bundle approach?

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I must note that by "any bundle constructions that don't have a realization as a sheaf", I mean any useful bundle constructions that don't have a realization as a sheaf. I am aware of the adjunction between $Psh(X)$ and $Bundle(X)$, which restricts to an equivalence of categories on $Sh(X)$ and some subcategory of $Bundle(X)$ that I don't remember offhand (they are covering spaces in the continuous case.) –  Harry Gindi Mar 9 '10 at 0:10
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There is the obvious answer: a lot in mathematics depends on your point of view. Sometimes you care more about spaces themselves, and sometimes you care more about the functions/sections on the spaces. It seems to me to be counterproductive to force yourself to translate everything to the same language, regardless of what the problem you're dealing with is best suited to. That said, I'd be quite interested in what other people have to say about less subjective advantages. –  Ilya Grigoriev Mar 9 '10 at 2:22
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Sure, but my point was more about the overwhelming prominence of bundles in DG, while sheaves get relatively little exposure in that setting. Bundles are used much more often in AG than sheaves are used in DG, at least in my experience. –  Harry Gindi Mar 9 '10 at 2:46
    
More part of differential topology than of differential geometry, but Gromov made good use of (microflexible) sheaves with his h-principle about partial differential relations, giving you theorems about immersions, submersion etc. –  Lennart Meier Jun 5 at 8:36
    
In a different vein: While only few people would define a manifold as a locally ringed space such that ..., this viewpoint is quite common if one treats supermanifolds. –  Lennart Meier Jun 5 at 8:40

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up vote 65 down vote accepted

If $X$ is a manifold, and $E$ is a smooth vector bundle over $X$ (e.g. its tangent bundle), then $E$ is again a manifold. Thus working with bundles means that one doesn't have to leave the category of objects (manifolds) under study; one just considers manifold with certain extra structure (the bundle structure). This is a big advantage in the theory; it avoids introducing another class of objects (i.e. sheaves), and allows tools from the theory of manifolds to be applied directly to bundles too.

Here is a longer discussion, along somewhat different lines:

The historical impetus for using sheaves in algebraic geometry comes from the theory of several complex variables, and in that theory sheaves were introduced, along with cohomological techniques, because many important and non-trivial theorems can be stated as saying that certain sheaves are generated by their global sections, or have vanishing higher cohomology. (I am thinkin of Cartan's Theorem A and B, which have as consequences many earlier theorems in complex analysis.)

If you read Zariski's fantastic report on sheaves in algebraic geometry, from the 50s, you will see a discussion by a master geometer of how sheaves, and especially their cohomology, can be used as a tool to express, and generalize, earlier theorems in algebraic geometry. Again, the questions being addressed (e.g. the completeness of the linear systems of hyperplane sections) are about the existence of global sections, and/or vanishing of higher cohomology. (And these two usually go hand in hand; often one establishes existence results about global sections of one sheaf by showing that the higher cohomology of some related sheaf vanishes, and using a long exact cohomology sequence.)

These kinds of questions typically don't arise in differential geometry. All the sheaves that might be under consideration (i.e. sheaves of sections of smooth bundles) have global sections in abundance, due to the existence of partions of unity and related constructions.

There are difficult existence problems in differential geometry, to be sure: but these are very often problems in ODE or PDE, and cohomological methods are not what is required to solve them (or so it seems, based on current mathematical pratice). One place where a sheaf theoretic perspective can be useful is in the consideration of flat (i.e. curvature zero) Riemannian manifolds; the fact that the horizontal sections of a bundle with flat connection form a local system, which in turn determines the bundle with connection, is a useful one, which is well-expressed in sheaf theoretic language. But there are also plenty of ways to discuss this result without sheaf-theoretic language, and in any case, it is a fairly small part of differential geometry, since typically the curvature of a metric doesn't vanish, so that sheaf-theoretic methods don't seem to have much to say.

If you like, sheaf-theoretic methods are potentially useful for dealing with problems, especially linear ones, in which local existence is clear, but the objects are suffiently rigid that there can be global obstructions to patching local solutions.

In differential geomtery, it is often the local questions that are hard: they become difficult non-linear PDEs. The difficulties are not of the "patching local solutions" kind. There are difficult global questions too, e.g. the one solved by the Nash embedding theorem, but again, these are typically global problems of a very different type to those that are typically solved by sheaf-theoretic methods.

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That is a great answer! –  Sam Derbyshire Mar 9 '10 at 3:42
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Dear fpqc, In complex geometry sheaves and cohomology certainly play a role, although I'm not close enough to the field to know whether they are as dominant as they were in the heyday of Oka and Cartan. I would guess that the closer the investigations are to algebraic geometry, the more likely these methods are to play an important role. –  Emerton Mar 9 '10 at 3:42
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Regarding the statement that studying bundles allows one to stay in the realm of manifolds. I'm curious, do differential geometers not usually care that bundles do not form an Abelian category, i.e., that kernels and cokernels of maps between bundles are not necessarily bundles? When working with locally free sheaves in an algebro-geometric setting, I tend to find it necessary to view them in the larger category of coherent sheaves for this reason. –  Mike Skirvin Mar 9 '10 at 3:44
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Typically, one considers maps of bundles which are locally of the form $\mathbb R^m \hookrightarrow \mathbb R^n$, so that the quotient is again a bundle. In fact, even more geometrically inclined algebraic geometers tend to distinguish between a map of bundles (in the above sense), and a map of sheaves (such as the map $\mathcal O \hookrightarrow \mathcal O(1)$) given by choosing a hyperplane in projective space.) One point to consider is that the latter kinds of maps work well in algebraic geometry in part because the singularities of maps are so tame (zeroes or poles) ... –  Emerton Mar 9 '10 at 4:03
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@fpqc, Emerton, I would say that most definitely one uses sheaves when studying complex manifolds. Indeed, on an arbitrary complex manifold, cohomology of the "obvious" sheaves is just about all we have. Slowly, however, people are turning to more differential geometric ways to try and understand complex non-Kähler manifolds. In brief, one looks for a Hermitian metric whose 2-form satisfies some PDE analogous to being closed. Eg. work of Streets-Tian on pluriclosed metrics and work of Fu-Li-Yau on balanced metrics. –  Joel Fine Mar 9 '10 at 9:11

In differential geometry one often also has connections on the bundles, e.g. the Levi-Cevita connection on the tangent bundle. Many concepts of differential geometry use connections, such as holonomy or geodesics.

I'd love to learn the opposite to the following statement, but I have the feeling that there is no "nice" definition of a connection in the sheaf-theoretical approach. So I think this is another aspect why bundles are often preferred.

As a little side remark, I'd like to point out that in higher differential geometry the story continues. Curiously, the word "gerbe" - which comes originally from the sheaf-theoretical side - is often at the same time used for the higher analog of a bundle. This is one of the reasons why different people associate so diverse meanings with "gerbe".

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My experience with algebraic geometry is limited, but connections (and their cousins the crystals) seem to be used quite often in a sheaf-theoretic context. The connections tend to be flat, i.e., I rarely see people using nonzero curvature to prove things. –  S. Carnahan May 22 '10 at 5:10
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Connections not only can be defined as a concept for any sheaf of modules whatsoever on the manifold (or analytic space), but on a complex manifold one can show that a coherent sheaf admitting a connection is automatically a vector bundle. See Deligne's SLN book on diff'tl eqns. But the sheaf-theoretic viewpoint is extremely handy for studying these matters, among those who already care about sheaves for other reasons. –  BCnrd May 22 '10 at 7:18

In addition to Emerton's answer (which is great), it should be noted that in the complex algebraic and analytic categories, the study of singularities is both tractable and necessary. So it is natural to study "generalized" vector bundles that allow singular fibers, but this is possible only using local sections, i.e. sheaves. On the other hand, in the smooth category, singularities can be quite nasty and therefore are almost always avoided by differential geometers. If everything is smooth and nonsingular, there is little to be gained by using sheaves. I am not familiar with singular differential geometry, but it seems to me that using sheaves in that context might be just as essential as it is in algebraic geometry.

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Great answer as well! –  Harry Gindi Mar 9 '10 at 6:28

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