synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
Models
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
The functor
which sends a smooth manifold (finite dimensional, paracompact, second countable) to (the formal dual of) its -algebra of smooth functions is a full and faithful functor.
In other words, for two smooth manifolds there is a natural bijection between the smooth functions and the -algebra homomorphisms .
In particular, the diffeomorphisms between smooth manifolds are in natural bijection to the isomorphisms between these algebras.
(attribution)
For the case of diffeomorphisms, Thm. was proven by Pursell (1952), following an announcement by Shanks (1951). This is the case that most reviews focus on, e.g. Grabowski (1978), Marsden, Ratiu & Abraham (2002), Grabowski (2005).
For the case that the domain is a point the statement is an an exercise (without reference to Pursell) in Milnor & Stasheff (1974), Problem 1-C (p. 11), sometimes now referred to as “Milnor’s exercise”.
The general statement of Thm. , with a detailed proof, is given in Kolář, Slovák & Michor (1993), lemma 35.8, corollaries 35.9, 35.10.
The statement of theorem 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 may break. For instance allowing uncountably many connected components, then there are counterexamples (MO discussion).
The analogous statement in topology is:
duality between algebra and geometry
in physics:
The case of the category of smooth manifolds with (just) diffeomorphisms between them is proved in
following an announcement in
The statement for domain a point is due to
Expository accounts for the case of isomorphisms are in
Janusz Grabowski, Isomorphisms and ideals of the Lie algebras of vector fields, Inventiones mathematicae volume 50, pages 13–33 (1978) (doi:10.1007/BF01406466)
Jerrold Marsden, J. Ratiu, R. Abraham, Theorem 4.2.36 in: Manifolds, tensor analysis, and applications, Springer 2003 (ISBN:978-1-4612-1029-0)
Janusz Grabowski, Isomorphisms of algebras of smooth functions revisited, Arch. Math. 85 (2005), 190-196 (arXiv:math/0310295)
The general statement and its proof is discussed in:
pdf]
Discussion that takes the dual algebraic formulation as the very definition of smooth functions is in
The analog of the statement for real algebras refined to smooth algebras is theorem 2.8 in
Last revised on March 31, 2023 at 12:09:18. See the history of this page for a list of all contributions to it.