embedding of smooth manifolds into formal duals of R-algebras


Differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry



smooth space


The magic algebraic facts




tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)




(Milnor’s exercise)

The functor

C ():SmoothMfdAlg op C^\infty(-) \colon SmoothMfd \longrightarrow Alg_{\mathbb{R}}^{op}

which sends a smooth manifold (finite dimensional, paracompact, second countable) to (the formal dual of) its \mathbb{R}-algebra of smooth functions is a full and faithful functor.

In other words, for two smooth manifolds X,YX,Y there is a natural bijection between the smooth functions XYX \to Y and the \mathbb{R}-algebra homomorphisms C (X)C (Y)C^\infty(X)\leftarrow C^\infty(Y).

Proof is for instance in (Kolar-Slovak-Michor 93, lemma 35.8, corollaries 35.9, 35.10).


The statement of theorem 1 serves as the stepping-stone for generalizations of differential geometry such as to supergeometry. On the other hand, for transporting various applications familiar from algebraic geometry to differential geometry (such as Kähler differentials, see there) the above embedding is insufficient, and instead of just remembering the associative algebra structure, one needs to remember the smooth algebra-structure on algebras of smooth functions. See also at synthetic differential geometry.


If one drops standard regularity assumptions on manifolds then theorem 1 may break. For instance allowing uncountably many connected components, then there are counterexamples (MO discussion).

The analogous statement in topology is:


A proof of the statement appears in

Discussion that takes the dual algebraic formulation as the very definition of smooth functions is in

  • Jet Nestruev, section 6 of Smooth manifolds and Observables, Graduate texts in mathematics 218, 2003

The analog of the statement for real algebras refined to smooth algebras is theorem 2.8 in

Last revised on June 4, 2018 at 06:27:58. See the history of this page for a list of all contributions to it.