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There are, among others, three general ways of equipping a "space" (which for the purposes of this question could be a topological space or a differentiable manifold, according to the case) with further structure:

(1) "Specifying regular funcions", which leads to locally ringed spaces, e.g. real-analytic manifolds or holomorphic manifolds.

(2) "Choosing a section (perhaps with some nondegeneracy properties) of some (usually tensor) bundle", which leads for example to Riemannian manifolds, quasi-symplectic manifolds and quasi-holomorphic manifolds.

(2') A variant of the latter: "Choosing a nondegenerate section with some integrability properties of some (usually tensor) bundle", which leads e.g. to symplectic geometry and holomorphic geometry.

(3) "Choosing a sub-bundle (with some properties) of some bundle" , and this leads for example to foliated manifolds and contact structures.

There are certainly some overlaps between the above approaches. For example, one can think of a complex manifold as a topological space equipped with a sheaf of local rings (the sheaf of holomorphic functions of the complex manifold), which is an instance of (1), or as an even dimensional smooth manifold equipped with an "integrable" field of endomorphisms of its tangent bundle which square to $-\mathrm{id}$, and that's an instance of (2), or rather (2'). Another example is that of a foliated manifold: it's usually seen as an instance of (3), but you can construct the sheaf of locally-constant-on-the-leaves smooth functions, and that reduces, in some sense, to the point of view (1).

Furthermore, smooth (in the sense of differentiable) maps between complex manifolds that preserve the tensor $J$ giving the complex structure correspond exactly to locally ringed space maps between the corresponding complex analytic spaces. And I would imagine that some foliation-preserving map in the case of foliated manifolds translate fully faithfully into maps for the locally ringed structure mentioned above.

In either case, if I'm not mistaken, you get a fully faithful embedding of some "geometric category" into the category of locally ringed spaces (such that, say, the forgetful functor to $\mathrm{Top}$ is respected). So it seems that the approach (1) is quite general. It's then spontaneous to ask:

Is the category of symplectic manifolds (with symplectomorphisms as morphisms) fully faithfully embeddable (say, preserving the forgetful functor to $Top$) into the category of locally ringed spaces?

and

Is there a good notion of morphisms between Riemannian manifolds (e.g. local isometries? Riemannian submersions? compositions thereof?) such that the resulting category has a fully faithful embedding into locally ringed spaces as above?


The first (perhaps useless?) construction that comes to my mind is the following. Given a smooth manifold $X$ with sheaf of differentiable functions $\mathcal{O}_X$, let $\mathcal{T} en_X^{\bullet}:=\oplus_{i\geq 0} (T_X^{\; *})^{\otimes i}$ be the sheaf of covariant tensors on $X$. If $g$ is a Riemannian metric on $X$, one could consider the sheaf of (commutative) local rings generated by $\mathcal{O}_X$ and $g$:

$\mathcal{O}_X\[g\]\subseteq \mathrm{Sym}^{\bullet}(T_X^{\; *})\subseteq \mathcal{T} en_X^{\bullet}$

and then consider the locally ringed space $(X,\mathcal{O}_X\[g\])$.

One can do an analogous thing with a symplectic manifold $(X,\omega)$ viewing $\omega$ as an anti-symmetric tensor:

$\mathcal{O}_X\[\omega\]\subseteq \Omega_X^{\mathrm{even}} \subseteq \mathcal{T} en_X^{\bullet}$

to get a locally ringed space $(X,\mathcal{O}_X\[\omega\])$.

Can this construction be of any help to answer the above questions?

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Maybe it could be useful to generalize the notion of locally ringed space, and allow "structure sheaves" with values in categories other than rings (e.g. differential graded rings). –  André Henriques Feb 27 '11 at 16:24
    
@André: In the theory of "supermanifolds" people already use sheaves of $\mathbb{Z}/2\mathbb{Z}$-graded rings. And there are some recent theories of "derived manifolds" (of which I don't know anything). But my question was perhaps more down to earth. Anyway, why do you suggets using differential graded rings? Is it to keep track that the "generator" $g$ in $\mathcal{O}_X[g]$ is placed in (tensor) degree 2? –  Qfwfq Feb 27 '11 at 16:31
    
You cannot recover $g$ from $\mathcal{O}[g]$, only up to conformal equivalence. –  Johannes Ebert Feb 27 '11 at 17:40
    
Isn't $\mathcal{O}[\omega]$ just $\mathcal{O}[x]/x^{n+1}$? –  Zack Feb 27 '11 at 18:18
    
@Zack: ya.. it looks like so. Perhaps my definition is just useless. But I don't know how morphisms of symplectic/Riemannian manifolds induce homomorphisms on that sheaves. –  Qfwfq Feb 27 '11 at 18:26
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