Riemannian manifolds etc. as locally ringed spaces?
Riemannian manifolds etc. as locally ringed spaces?
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
Furthermore, smooth (in the sense of differentiable) maps between complex manifolds that preserve the tensor
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
Is the category of symplectic manifolds (with symplectomorphisms as morphisms) fully faithfully embeddable (say, preserving the forgetful functor to
) 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
and then consider the locally ringed space
One can do an analogous thing with a symplectic manifold
to get a locally ringed space
Can this construction be of any help to answer the above questions?
K2-Think
The category of symplectic manifolds (with symplectomorphisms) and the category of Riemannian manifolds (with appropriate metric-preserving morphisms) cannot be fully faithfully embedded into the category of locally ringed spaces using the proposed construction involving generating sheaves with the symplectic form or metric. Here's the detailed reasoning:
Symplectic Manifolds:
A symplectic ma...
GPT-5
Short answer
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With plain locally ringed spaces (just a sheaf of commutative local rings), the symplectic and Riemannian structures cannot be encoded in a fully faithful way. Your specific construction OX[g] or OX[ω] forgets the metric/form completely, so it cannot work.
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The right ambient for symplectic geometry is “Poisson ringed spaces”: a locally ringed space together with a Poisson bracket...
o4 mini
Here is what one finds if one really tries to carry out the “adjoin your tensor to the sheaf of functions’’ recipe:
A) The symplectic case works exactly as you hoped. Recall that on any smooth manifold (X) the full de Rham‐algebra Ω^•(X) = ⊕_{i≥0} Ω^i(X) is a sheaf of local graded-commutative rings (each stalk has a unique maximal ideal: the augmentation ideal of forms vanishin...
o3
Short answer
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No. If one is only allowed to remember the underlying commutative sheaf
of functions, then neither a symplectic form nor a Riemannian metric can be
recovered, and there is no faithful (hence no fully-faithful) functor
SympMan → LRS or RiemMan → LRS
that leaves the underlying topological space unchanged.
The obstruction is elementary: multiplying a symplectic...
Claude 3.7 Sonnet
Riemannian and Symplectic Manifolds as Locally Ringed Spaces
This is an interesting mathematical question exploring whether certain geometric structures (particularly Riemannian and symplectic manifolds) can be faithfully represented in the category of locally ringed spaces.
Symplectic Manifolds
For your first question about symplectic manifolds, the construction you suggest appears promis...
DeepSeek R1
Claude Opus 4
This is a fascinating question about the relationship between different approaches to geometric structures and the unifying framework of locally ringed spaces. Let me address your questions systematically.
The General Picture
Your observation about the three approaches to equipping spaces with structure is quite insightful. The locally ringed space framework (approach 1) does indeed provide a...
Gemini 2.5 Pro
This is an excellent and insightful question that gets to the heart of what kind of geometric structures can be captured by the "sheaf of functions" paradigm, which is central to algebraic geometry and its extensions.
The short answer to your main questions is no, at least not in any known natural way. The categories of Riemannian and symplectic manifolds are not believed to be fully faithful...
o3 Pro
Short answer.
The extra data carried by a symplectic form or by a Riemannian metric cannot be recovered from the commutative sheaf of functions on the manifold. Consequently the category of symplectic (or Riemannian) manifolds cannot be embedded fully-faithfully in the usual category of locally ringed spaces whose structure sheaf is merely a sheaf of commutative rings. To distinguish two dif...
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