nLab smooth manifold

Idea

A differentiable manifold is a topological space which is locally homeomorphic to a Euclidean space (a topological manifold) and such that the gluing functions which relate these Euclidean local charts to each other are differentiable functions, for a fixed degree of differentiability. If one considers arbitrary differentiablity, then one speaks of smooth manifolds. For a general discussion, see at manifold.

Differential and smooth manifolds are the basis for much of differential geometry. They are the analogs in differential geometry of what schemes are in algebraic geometry.

If one relaxes the condition from being locally isomorphic to a Euclidean space to admitting local smooth maps from a Euclidean space, then one obtains the concept of diffeological spaces or even smooth sets; see at generalized smooth space for more on this.

The generalization of differentiable manifolds to higher differential geometry are orbifolds and more generally differentiable stacks. If one combines this with the generalization to smooth sets then one obtains the concept of smooth stacks and eventually smooth infinity-stacks.

Smooth manifolds form a category, SmoothManifolds.

Definition

Traditional definition

For the traditional definition, see at differentiable manifold.

Patching as idempotent splitting

The lecture notes Lawvere 1973, Ex. 3.21 reported a somewhat surprising observation:

In the case of smooth manifolds, the process of piecing together the local data can be elegantly summed up as splitting of idempotents in a category of open subsets of Euclidean spaces. More precisely:

Let $Diff$ be the category of smooth manifolds and smooth maps, where by a “smooth manifold”, we mean a finite-dimensional, second-countable, Hausdorff, $C^\infty$ manifold without boundary.

Let $i \colon Open \hookrightarrow Diff$ be the full subcategory whose objects are the open subspaces of finite-dimensional Cartesian spaces.

Theorem

The subcategory $i \colon Open \hookrightarrow Diff$ exhibits $Diff$ as an idempotent-splitting completion of $Open$.

Proof

By a general lemma for idempotent splittings, it suffices to prove that

• every smooth manifold is a smooth retract of an open set in Euclidean space; and

• if $p : U \to U$ is a smooth idempotent on an open set $U \subseteq \mathbb{R}^n$, then the subset $Fix(p) \hookrightarrow U$ is an embedded submanifold.

For the first statement, we use the fact that any manifold $M$ can be realized as a closed submanifold of some $\mathbb{R}^n$, and every closed submanifold has a tubular neighborhood $U \subseteq \mathbb{R}^n$. In this case, $U$ carries the structure of a vector bundle over $M$ in such a way that the inclusion $M \hookrightarrow U$ is identified with the zero section, so that the bundle projection $U \to M$ provides a retraction, with right inverse given by the zero section.

For the second statement (cf. Zack 2014), assume that the origin $0$ is a fixed point of $p$, and let $T_0(U) \cong \mathbb{R}^n$ be its tangent space (observe the presence of a canonical isomorphism to $\mathbb{R}^n$). Thus, we have idempotent linear maps $d p(0), Id-d p(0): T_0(U) \to T_0(U)$, where the latter factors through the inclusion $\ker \; d p(0) \hookrightarrow T_0(U)$ via a projection map $\pi: T_0(U) \to \ker \; d p(0)$. We have a map $f: U \to \mathbb{R}^n$ that takes each $x \in U$ to $x - p(x)$; let $g$ denote the composite

$$U \stackrel{f}{\longrightarrow} \mathbb{R}^n \cong T_0(U) \stackrel{\pi}{\longrightarrow} \ker\; d p(0).$$

Now we make some easy observations:

1. $Fix(p) \subseteq g^{-1}(0)$.

2. The map $p: U \to U$ restricts to a map $p: g^{-1}(0) \to g^{-1}(0)$, by idempotence of $p$.

3. The derivative $d g(0): T_0(U) \to T_0(\ker \; d p(0)) \cong \ker \; d p(0)$ is $\pi$ again, since $Id - d p(0)$ is idempotent. Thus, $d g(0)$ has full rank ($m$ say), and so the restriction of $g$ to some neighborhood $V$ has $0$ as a regular value, and $g^{-1}(0) \cap V$ is a manifold of dimension $m$, by the implicit function theorem. The tangent space $T_0(g^{-1}(0) \cap V)$ is canonically identified with $im(d p(0))$.

4. There are smaller neighborhoods $V'' \subseteq V' \subseteq V$ so that $p$ restricts to maps $p_1, p_2$ as in the following diagram ($i, i', i''$ are inclusion maps, all taking a domain element $x$ to itself):

   $$\array{ g^{-1}(0) \cap V'' & \stackrel{i''}{\hookrightarrow} & g^{-1}(0) \\ _\mathllap{p_2} \downarrow & & \downarrow _\mathrlap{p} \\ g^{-1}(0) \cap V' & \stackrel{i'}{\hookrightarrow} & g^{-1}(0) \\ _\mathllap{p_1} \downarrow & & \downarrow _\mathrlap{p} \\ g^{-1}(0) \cap V & \stackrel{i}{\hookrightarrow} & g^{-1}(0) }$$

   and such that $p_1, p_2$ are diffeomorphisms by the inverse function theorem (noting here that $d p_i(0): im(d p(0)) \to im(d p(0))$ is the identity map, by idempotence of $p$).

5. Letting $q: g^{-1}(0) \cap V' \to g^{-1}(0) \cap V''$ denote the smooth inverse to $p_2$, we calculate $i' = p \circ i'' \circ q$, and

   $$i p_1 = p i' = p p i''q = p i'' q = i',$$

   so that $p_1(x) = x$ for every $x \in g^{-1}(0) \cap V'$. Hence $g^{-1}(0) \cap V' \subseteq Fix(p)$.

From all this, it follows that $Fix(p) \cap V' = g^{-1}(0) \cap V'$, meaning $Fix(p)$ is locally diffeomorphic to $\mathbb{R}^m$, and so $Fix(p)$ is an embedded submanifold of $\mathbb{R}^n$.

Remark

Another proof of this result may be found here.

Lawvere comments on this fact as follows:

“This powerful theorem justifies bypassing the complicated considerations of charts, coordinate transformations, and atlases commonly offered as a ”basic“ definition of the concept of manifold. For example the 2-sphere, a manifold but not an open set of any Euclidean space, may be fully specified with its smooth structure by considering any open set $A$ in 3-space $E$ which contains it but not its center (taken to be $0$) and the smooth idempotent endomap of $A$ given by $e(x) = x/{|x|}$. All general constructions (i.e., functors into categories which are Cauchy complete) on manifolds now follow easily (without any need to check whether they are compatible with coverings, etc.) provided they are known on the opens of Euclidean spaces: for example, the tangent bundle on the sphere is obtained by splitting the idempotent $e'$ on the tangent bundle $A \times V$ of $A$ ($V$ being the vector space of translations of $E$) which is obtained by differentiating $e$. The same for cohomology groups, etc.” (Lawvere 1989, p.267)

General abstract geometric definition

There is a fundamental and general abstract way to think of smooth manifolds, which realizes their theory as a special case of general constructions in higher geometry.

In this context, one specifies for instance $\mathcal{G}$ a geometry (for structured (∞,1)-toposes) and then plenty of geometric notions are defined canonically in terms of $\mathcal{G}$. The theory of smooth manifolds appears if one takes $\mathcal{G} =$ CartSp.

Alternatively, one can specify differential cohesion and proceed as discussed at differential cohesion – structures - Cohesive manifolds (separated).

This is discussed in The geometry CartSp below.

The geometry $CartSp$

Let CartSp be the category of Cartesian spaces and smooth functions between them. This has finite products and is in fact (the syntactic category) of a Lawvere theory: the theory of smooth algebras.

Moreover, $CartSp$ is naturally equipped with the good open cover coverage that makes it a site.

Both properties together make it a pregeometry (for structured (∞,1)-toposes) (if the notion of Grothendieck topology is relaxed to that of coverage in StrSp).

For $\mathcal{X}$ a topos, a product-preserving functor

$$\mathcal{O} : \mathcal{G} \to \mathcal{X}$$

is a $\mathcal{G}$-algebra in $\mathcal{X}$. This makes $\mathcal{X}$ a $\mathcal{G}$-ringed topos. For $\mathcal{G} =$ CartSp, this algebra is a smooth algebra in $\mathcal{X}$. If $\mathcal{X}$ has a site of definition $X$, then this is a sheaf of smooth algebras on $X$.

If $\mathcal{O}$ sends covering families $\{U_i \to U\}$ in $\mathcal{G}$ to effective epimorphism $\coprod_i \mathcal{O}(U_i) \to \mathcal{O}(U)$ we say that it is a local $\mathcal{G}$-algebra in $\mathcal{X}$, making $\mathcal{X}$ a $\mathcal{G}$-locally ringed topos.

The big topos $Sh(\mathcal{G})$ itself is canonically equipped with such a local $\mathcal{G}$-algebra, given by the Yoneda embedding $j$ followed by sheafification $L$

$$\mathcal{O} : \mathcal{G} \stackrel{j}{\to} PSh(\mathcal{G}) \stackrel{L}{\to} Sh(\mathcal{G}) \,.$$

It is important in the context of locally representable locally ringed toposes that we regard $Sh(\mathcal{G})$ as equipped with this local $\mathcal{G}$-algebra. This is what remembers the site and gives a notion of local representability in the first place.

The big topos $Sh(CartSp)$ is a cohesive topos of generalized smooth spaces. Its concrete sheaves are precisely the diffeological spaces. See there for more details. We now discuss how with $Sh(CartSp)$ regarded as a $CartSp$-structured topos, smooth manifolds are precisely its locally representable objects.

Cartesian spaces as representable objects of $Sh(CartSp)$

The representables themselves should evidently be locally representable and canonically have the structure of $CartSp$-structured toposes.

Indeed, every object $U \in \mathrm{CartSp}$ is canonically a CartSp-ringed space, meaning a topological space equipped with a local sheaf of smooth algebras. More generally: every object $U \in CartSp$ is canonically incarnated as the $CartSp$-structured (∞,1)-topos

$$(\mathcal{X}, \mathcal{O}_{\mathcal{X}}) := (Sh_{(\infty,1)}(CartSp)/U , \;\;\; \mathcal{O}_U : CartSp \stackrel{j}{\to} Sh_{(\infty,1)}(CartSp) \stackrel{U^*}{\to} Sh_{(\infty,1)}(CartSp)/U)$$

given by the over-(∞,1)-topos of the big (∞,1)-sheaf (∞,1)-topos over $CartSp$ and the structure sheaf given by the composite of the (∞,1)-Yoneda embedding and the inverse image of the etale geometric morphism induced by $U$.

Smooth manifolds as locally representable objects of $Sh(CartSp)$

Definition

Say a concrete object $X$ in the sheaf topos $Sh(CartSp)$ – a diffeological space – is locally representable if there exists a family of open embeddings $\{U_i \hookrightarrow X\}_{i \in X}$ with $U_i \in CartSp \stackrel{j}{\hookrightarrow} Sh(CartSp)$ such that the canonical morphism out of the coproduct

$$\coprod_i U_i \to X$$

is an effective epimorphism in $Sh(CartSp)$.

Let $LocRep(CartSp) \hookrightarrow Sh(CartSp)$ be the full subcategory on locally representable sheaves.

Proposition

There is an equivalence of categories

$$Diff \simeq LocRep(CartSp)$$

of the category Diff of smooth manifolds with that of locally representable sheaves for the pre-geometry $CartSp$.

Proof

Define a functor $Diff \to LocRep(CartSp)$ by sending each smooth manifold to the sheaf over $CartSp$ that it naturally represents. By definition of manifold there is an open cover $\{U_i \hookrightarrow X\}$. We claim that $\coprod_i U_i \to X$ is an effective epimorphism, so that this functor indeed lands in $LocRep(CartSp)$. (This is a standard argument of sheaf theory in Diff, we really only need to observe that it goes through over CartSp, too.)

For that we need to show that

$$\coprod_{i, j} U_i \times_X U_j \stackrel{\longrightarrow}{\longrightarrow} \coprod_i U_i \to X$$

is a coequalizer diagram in $Sh(CartSp)$ (that the Cech groupoid of the cover is equivalent to $X$.). Notice that the fiber product here is just the intersection in $X$ $U_i \times_X U_j \simeq U_i \cap U_j$. By the fact that the sheaf topos $Sh(CartSp)$ is by definition a reflective subcategory of the presheaf topos $PSh(CartSp)$ we have that colimits in $Sh(CartSp)$ are computed as the sheafification of the corresponding colimit in $PSh(CartSp)$. The colimit in $PSh(CartSp)$ in turn is computed objectwise. Using this, we see that that we have a coequalizer diagram

$$\coprod_{i, j} U_i \times_X U_j \stackrel{\longrightarrow}{\longrightarrow} \coprod_i U_i \to S(\{U_i\})$$

in $PSh(CartSp)$, where $S(\{U_i\})$ is the sieve corresponding to the cover: the subfunctor $S(\{U_i\}) \hookrightarrow X$ of the functor $X : CartSp^{op} \to Set$ which assigns to $V \in CartSp$ the set of smooth functions $V \to X$ that have the property that they factor through any one of the $U_i$.

Essentially by the definition of the coverage on $CartSp$, it follows that sheafification takes this subfunctor inclusion to an isomorphism. This shows that $X$ is indeed the tip of the coequalizer in $Sh(CartSp)$ as above, and hence that it is a locally representable sheaf.

Conversely, suppose that for $X \in Conc(Sh(CartSp)) \hookrightarrow Sh(CartSp)$ there is a family of open embeddings $\{U_i \hookrightarrow X\}$ such that we have a coequalizer diagram

$$\coprod_{i, j} U_i \times_X U_j \stackrel{\longrightarrow}{\longrightarrow} \coprod_{i} U_i \to X$$

in $Sh(CartSp)$, which is the sheafification of the corresponding coequalizer in $PSh(CartSp)$. By evaluating this on the point, we find that the underlying set of $X$ is the coequalizer of the underlying set of the $U_i$ in $Set$. Since every plot of $X$ factors locally through one of the $U_i$ it follows that $X$ is a diffeological space.

It follows that in the pullback diagrams

$$\array{ U_i \times_X U_j &\to& U_j \\ \downarrow && \downarrow \\ U_i &\to& X }$$

the object $U_i \cap U_j$ is the diffeological space whose underlying topological space is the intersection of $U_i$ and $U_j$ in the topological space underlying $X$. In particular the inclusions $U_i \times_X U_j \hookrightarrow U_i$ are open embeddings.

As locally representable $CartSp$-structured $(\infty,1)$-toposes

We may switch from regarding smooth manifolds as objects in the big topos $X \in Sh(CartSp)$ to regrading them as toposes themselves, by passing to the over-topos $Sh(CartSp)/X$. This remembers the extra (smooth) structure on the topological space $X$ by being canonically a locally ringed topos with the structure sheaf of smooth functions on $X$: a CartSp-structured (∞,1)-toposes

For every choice of geometry (for structured (∞,1)-toposes) there is a notion of $\mathcal{G}$-locally representable structured (∞,1)-topos (StrSp).

Claim

Smooth manifolds are equivalently the 0-localic CartSp- generalized schemes of locally finite presentation.

Sketch of proof

The statement says that a smooth manifold $X$ may be identified with an ∞-stack on CartSp (an ∞-Lie groupoid) which is represented by a CartSp-structured (∞,1)-topos $(\mathcal{X}, \mathcal{O}_{\mathcal{X}})$ such that

1. $\mathcal{X}$ is a 0-localic (∞,1)-topos;

2. There exists a family of objects $\{U_i \in \mathcal{X}\}$ such that the canonical morphism $\coprod_i U_i \to *_{\mathcal{X}}$ to the terminal object in $\mathcal{X}$ is a regular epimorphism;

3. For every $i \in I$ there is an equivalence

$$(\mathcal{X}/U_i, \mathcal{O}_{\mathcal{X}|U_i}) \underoverset{\simeq}{t_i}{\to} (Sh_{(\infty,1)}(\mathbb{R}^n), \mathcal{O}(\mathbb{R}^n)) \,.$$

The second and third condition say in words that $(\mathcal{X}, \mathcal{O}_{\mathcal{X}})$ is locally equivalent to the ordinary cannonically CartSp-locally ringed space $\mathbb{R}^n$ (for $n \in \mathbb{N}$ the dimension. The first condition then says that these local identifications cover $\mathcal{X}$.

(…)

(…)

Properties

Faithful embedding into $cAlg^{op}$

Proposition

(Milnor’s exercise)

The functor

$$C^\infty(-) \colon SmthMfd \hookrightarrow cAlg_{\mathbb{R}}^{op}$$

(from the category of smooth manifolds to the opposite category of commutative algebras over the real numbers) that sends a smooth manifold $X$ to its commutative $\mathbb{R}$-algebra of smooth functions $X \to \mathbb{R}$ is a fully faithful functor, hence exhibits $SmthMfd$ as a full subcategory of $cAlg^{op}$.

(Kolar-Slovak-Michor 93, lemma 35.8, corollaries 35.9, 35.10)

For more see at embedding of smooth manifolds into formal duals of R-algebras.

Further properties

• Borel's theorem

• Tietze extension theorem

• Whitney extension theorem

• Steenrod-Wockel approximation theorem

• derivations of smooth functions are vector fields

Classes of examples

• 2-manifolds

• 3-manifolds

• 4-manifolds

• 8-manifolds

With extra structure:

• complex manifold

• symplectic manifold

• analytic manifold

• CR manifold

• etc. …

Related concepts

• embedding of smooth manifolds

• infinite dimensional smooth manifold

• pro-manifold

• diffeological space, smooth set

• orbifold

• differentiable stack, smooth stack, smooth infinity-stack

• differential cohesion – structures - Cohesive manifolds (separated)

References

The vision of the modern notion of manifolds is attributed to:

• Bernhard Riemann, Über die Hypothesen, welche der Geometrie zu Grunde liegen, Göttingen (1845) [doi:10.1007/978-3-642-35121-1]

  Engl. transl: William Clifford: On the hypotheses which underlie geometry, Nature VIII (1873) 183-184 [doi:10.1007/978-3-319-26042-6]

Further historical contributions:

• Oswald Veblen and J.H.C. Whitehead, A set of axioms for differential geometry, Proc. Natl. Acad. Sci. USA 17, 551-561 (1931)

• Oswald Veblen and J.H.C. Whitehead, The foundations of differential geometry, Camb. Tracts Math. Math. Phys. 29, Cambridge University Press, Cambridge (1932)

• Hassler Whitney, Differentiable manifolds, Ann. Math. (2) 37, 645-680 (1936)

Early account with an eye towards cobordism theory and the Pontrjagin-Thom isomorphism:

• Lev Pontrjagin, Smooth manifolds and their applications in Homotopy theory, Trudy Mat. Inst. im Steklov, No 45, Izdat. Akad. Nauk. USSR, Moscow, 1955 (AMS Translation Series 2, Vol. 11, 1959) (pdf, doi:10.1142/9789812772107_0001)

Textbook accounts:

• Georges de Rham, Chapter I of: Differentiable Manifolds – Forms, Currents, Harmonic Forms, Grundlehren 266, Springer (1984) [doi:10.1007/978-3-642-61752-2]

• John Milnor, James Stasheff, Section 1 of: Characteristic Classes, Annals of Mathematics Studies, Princeton University Press 1974 (ISBN:9780691081229)

• Yvonne Choquet-Bruhat, Cécile DeWitt-Morette, Analysis, manifolds and physics, North Holland (1982, 2001) $[$ISBN:9780444860170$]$

  (with an eye towards mathematical physics)

• Ivan Kolar, Jan Slovak and Peter Michor, Natural operations in differential geometry, 1993, 1999 (web)

• Antoni Kosinski, Differential manifolds, Academic Press 1993 (pdf)

• Jeffrey Lee, Manifolds and differential geometry, Graduate Studies in Mathematics 107 (2009) (ISBN: 978-0-8218-4815-9, doi:10.1090/gsm/107)

• John Lee, Introduction to Smooth Manifolds, Springer (2012) [doi:10.1007/978-1-4419-9982-5, book webpage, pdf]

• Theodore Frankel, chapter 1 of The Geometry of Physics - An Introduction, Cambridge University Press (1997, 2004, 2012)

• Jet Nestruev, Smooth Manifolds and Observables , Springer LNM 220 Heidelberg 2003 (doi:10.1007/b98871)

With an eye towards application in mathematical physics:

• Mikio Nakahara, Chapter 3 of Geometry, Topology and Physics, IOP 2003 (doi:10.1201/9781315275826, pdf)

Smooth manifolds are defined as locally ringed spaces in

• Juan A. Navarro González, and Juan B. Sancho de Salas, $C^\infty$-Differentiable Spaces, Springer LNM 1824 2003.

Discussion of smooth manifolds as colimits of the Cech nerves of their good open covers is also at

• MathOverflow, Gluing of manifolds and the Hausdorff condition

The general abstract framework of higher geometry referred to above is discussed in

• Jacob Lurie, Structured Spaces

The proof of Thm. above makes use of the observation in

• Zack, Idempotents split in category of smooth manifolds? (2014) [MO:q/162556]

and provides a solution to exercise 3.21 in:

• William Lawvere, Perugia Notes - Theory of Categories over a Base Topos , Ms. Università di Perugia 1973 [pdf, pdf]

The above comment by Lawvere is taken from

• F. William Lawvere, Qualitative distinctions between some toposes of generalized graphs, Contemporary Mathematics 92 (1989), 261-299. (pdf)