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 faithfully embeddable into the category of locally ringed spaces (LRS) in a way that respects the underlying topological space.

Let's break down why this is the case, why some of your examples work, and why the proposed constructions don't quite get there.

The Crux of the Matter: Functions vs. Tensors

The "locally ringed space" approach (1) is fundamentally about defining a geometry by specifying a special class of scalar-valued functions. The ring structure is pointwise: (f+g)(p) = f(p)+g(p) and (fg)(p) = f(p)g(p). A morphism of LRS (φ, φ#): (X, O_X) -> (Y, O_Y) is a continuous map φ: X -> Y and a sheaf map φ#: O_Y -> φ_*O_X. For sheaves of functions, this φ# is just the pullback f ↦ f ∘ φ. The entire structure is determined by which functions you call "admissible" and how they behave under composition with maps.

In contrast, structures like Riemannian metrics and symplectic forms are not about selecting a special subring of functions. They are tensor fields. They live not in the sheaf of functions O_X, but in sheaves of sections of tensor bundles, e.g., Γ(X, S²(T*X)) for a metric g.

A morphism preserving such a structure (like an isometry or a symplectomorphism) φ: X -> Y is defined by a condition on its derivatives (the pushforward/pullback on tangent/cotangent bundles).

• Isometry: φ*g_Y = g_X
• Symplectomorphism: φ*ω_Y = ω_X

An LRS morphism f ↦ f ∘ φ only "sees" the values of functions, not their derivatives. The LRS framework is not naturally equipped to handle conditions on the derivatives of the underlying map φ.

Why Does It Work for Complex Manifolds?

This is the key example that makes the question so tantalizing. A complex structure is given by a tensor J: TX -> TX with J² = -id (and an integrability condition). A map φ is holomorphic if its differential commutes with J: dφ ∘ J_X = J_Y ∘ dφ. This is a condition on derivatives!

The "miracle" of complex analysis is that this condition on dφ is equivalent to a condition on a sheaf of functions.

Theorem (Bochner-Martin): A smooth map φ: X -> Y between complex manifolds is holomorphic (dφ ∘ J_X = J_Y ∘ dφ) if and only if for every open set V ⊂ Y and every holomorphic function f ∈ O_Y(V), the pullback f ∘ φ is a holomorphic function in O_X(φ⁻¹(V)).

This theorem establishes a perfect bridge: the category of complex manifolds with holomorphic maps is the category of complex analytic spaces (which are LRS) with their morphisms. The tensor condition on dφ is precisely captured by the LRS morphism condition on φ*.

Why Does It Work for Foliations?

This case also works. A function f is "admissible" for a foliation if it is constant along the leaves. This means X(f) = 0 for any vector field X tangent to the leaves. The set of such functions over an open set U forms a subring of C^∞(U). If f, g are constant on leaves, so are f+g and fg. This gives a sub-sheaf of rings O_F ⊂ O_X, and (X, O_F) is an LRS. A map φ preserves the foliation if and only if it pulls back leaf-constant functions to leaf-constant functions. This gives a fully faithful embedding.

Why It Fails for Riemannian and Symplectic Manifolds

No such miracle occurs for Riemannian or symplectic geometry. There is no special subring A ⊂ C^∞ such that a map φ is an isometry if and only if φ* maps A_Y to A_X.

• Riemannian: What would be the "special" functions? Harmonic functions (Δf=0)? The product of harmonic functions is not harmonic. Eigenfunctions of the Laplacian? Same problem. The metric g is used to define operators on functions (like the gradient and Laplacian), but it doesn't naturally single out a subring of functions.
• Symplectic: The symplectic form ω defines the Poisson bracket {,} on C^∞(M). A map φ is a symplectomorphism if and only if it preserves the Poisson bracket: φ*{f,h}_Y = {φ*f, φ*h}_X. This gives a beautiful algebraic description, but it's a category of Poisson manifolds, not LRS. The Poisson bracket is extra algebraic structure on the ring of functions; it's not encoded in the commutative ring structure itself.

Analyzing Your Proposed Construction

Your idea to form O_X[g] is clever. Let's analyze it for the Riemannian case. A section of O_X[g] over U is a polynomial in the tensor g with C^∞(U) coefficients: s = f_0 + f_1 g + f_2 g⊗g + .... Let's define an LRS morphism (φ, φ#) by having φ# act as pullback: φ#(f) = f∘φ and φ#(g_Y) = φ*(g_Y).

For φ# to be a map from O_Y[g_Y] to O_X[g_X], the element φ*(g_Y) must be in the ring O_X[g_X]. That is, φ*(g_Y) must be a polynomial in g_X with smooth coefficients.

1. Is every local isometry an LRS morphism? Yes. If φ is a local isometry, φ*(g_Y) = g_X. This is a very simple polynomial in g_X (degree 1, coefficient 1). So your construction gives a functor from (Riemannian mfds + local isometries) to LRS. The functor is also faithful.

2. Is it full? No. This is the crucial failure. Consider a map φ which is a homothety, e.g., φ*(g_Y) = c g_X for a constant c ≠ 1. This φ is not an isometry. However, φ*(g_Y) is a polynomial in g_X, so φ is a morphism in your LRS category. The category of LRS (X, O_X[g]) has more morphisms than the category of Riemannian manifolds with isometries. The embedding is not full.

The same problem arises for the symplectic case. A map φ with φ*ω_Y = c ω_X for c ≠ 1 is not a symplectomorphism, but it would induce a morphism in your LRS category (X, O_X[ω]).

Conclusion

The general principle seems to be:

A geometric structure can be described in the LRS framework if and only if its defining properties and the properties of its structure-preserving maps can be completely encoded by specifying a sub-sheaf of rings of the sheaf of smooth/continuous functions.

This works for structures whose definition is fundamentally about restricting the class of allowed functions (algebraic geometry, complex analytic geometry) or whose defining tensor's annihilator on the algebra of functions forms a sub-algebra (foliations).

It does not work for structures defined by non-degenerate tensors (g, ω) where the structure-preserving maps are defined by how they transform the tensor itself, a condition that does not, in general, translate into the preservation of a special subring of functions. The algebraic viewpoint for these geometries lies elsewhere: in Poisson algebras for symplectic geometry, and perhaps in sheaves of differential operators (with a distinguished Laplacian) for Riemannian geometry.