nLab
smooth manifold

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differential geometry

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Contents

Idea

A smooth manifold is a space that is locally isomorphic to a Cartesian space n\mathbb{R}^n equipped with its canonical smooth structure.

Definition

Traditional definition

Traditionally, a smooth manifold is defined as follows.

As special topological manifold

Definition

A manifold is a smooth manifold if its transition functions are smooth functions n n\mathbb{R}^n \to \mathbb{R}^n, or in other words a GG-manifold over the pseudogroup GG of C C^\infty diffeomorphisms between open sets of a Euclidean space.

So a smooth manifold is a C kC^k-differentiable manifold for all kk.

A homomorphism of smooth manifolds is a smooth function. Smooth manifolds and smooth functions form the category Diff.

As special locally ringed space

Proposition

A smooth manifold is equivalently a locally ringed space (X,𝒪 X)(X,\mathcal{O}_X) which is locally isomorphic to the ringed space ( n,C ())(\mathbb{R}^n, C^\infty(-) ).

(…)

Patching as idempotent splitting

In an exercise of his 1973 Perugia lectures F. William Lawvere 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 DiffDiff be the category of smooth manifolds and smooth maps, where by a “smooth manifold”, we mean a finite-dimensional, second-countable, Hausdorff, C C^\infty manifold without boundary. Let i:OpenDiffi: Open \hookrightarrow Diff be the full subcategory whose objects are the open subspaces of finite-dimensional Cartesian spaces.

Theorem

The subcategory i:OpenDiffi: Open \hookrightarrow Diff exhibits DiffDiff as an idempotent-splitting completion of OpenOpen.

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;

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

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

For the second statement, assume that the origin 00 is a fixed point of pp, and let T 0(U) nT_0(U) \cong \mathbb{R}^n be its tangent space (observe the presence of a canonical isomorphism to n\mathbb{R}^n). Thus we have idempotent linear maps dp(0),Iddp(0):T 0(U)T 0(U)d p(0), Id-d p(0): T_0(U) \to T_0(U) where the latter factors through the inclusion kerdp(0)T 0(U)\ker \; d p(0) \hookrightarrow T_0(U) via a projection map π:T 0(U)kerdp(0)\pi: T_0(U) \to \ker \; d p(0). We have a map f:U nf: U \to \mathbb{R}^n that takes xUx \in U to xp(x)x - p(x); let gg denote the composite

Uf nT 0(U)πkerdp(0).U \stackrel{f}{\to} \mathbb{R}^n \cong T_0(U) \stackrel{\pi}{\to} \ker\; d p(0).

Now we make some easy observations:

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

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

  3. The derivative dg(0):T 0(U)T 0(kerdp(0))kerdp(0)d g(0): T_0(U) \to T_0(\ker \; d p(0)) \cong \ker \; d p(0) is π\pi again since Iddp(0)Id - d p(0) is idempotent. Thus dg(0)d g(0) has full rank (mm say), and so the restriction of gg to some neighborhood VV has 00 as a regular value, and g 1(0)Vg^{-1}(0) \cap V is a manifold of dimension mm by the implicit function theorem. The tangent space T 0(g 1(0)V)T_0(g^{-1}(0) \cap V) is canonically identified with im(dp(0))im(d p(0)).

  4. There are smaller neighborhoods VVVV'' \subseteq V' \subseteq V so that pp restricts to maps p 1,p 2p_1, p_2 as in the following diagram (i,i,ii, i', i'' are inclusion maps, all taking a domain element xx to itself):

    g 1(0)V i g 1(0) p 2 p g 1(0)V i g 1(0) p 1 p g 1(0)V i g 1(0)\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 2p_1, p_2 are diffeomorphisms by the inverse function theorem (noting here that dp i(0):im(dp(0))im(dp(0))d p_i(0): im(d p(0)) \to im(d p(0)) is the identity map, by idempotence of pp).

  5. Letting q:g 1(0)Vg 1(0)Vq: g^{-1}(0) \cap V' \to g^{-1}(0) \cap V'' denote the smooth inverse to p 2p_2, we calculate i=piqi' = p \circ i'' \circ q, and

    ip 1=pi=ppiq=piq=i,i p_1 = p i' = p p i''q = p i'' q = i',

    so that p 1(x)=xp_1(x) = x for every xg 1(0)Vx \in g^{-1}(0) \cap V'. Hence g 1(0)VFix(p)g^{-1}(0) \cap V' \subseteq Fix(p).

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

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 AA in 3-space EE which contains it but not its center (taken to be 00) and the smooth idempotent endomap of AA given by e(x)=x/|x|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 ee' on the tangent bundle A×VA \times V of AA (VV being the vector space of translations of EE) which is obtained by differentiating ee. 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 𝒢\mathcal{G} a geometry (for structured (∞,1)-toposes) and then plenty of geometric notions are defined canonically in terms of 𝒢\mathcal{G}. The theory of smooth manifolds appears if one takes 𝒢=\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 CartSpCartSp

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

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

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

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

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

𝒪:𝒢jPSh(𝒢)LSh(𝒢). \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(𝒢)Sh(\mathcal{G}) as equipped with this local 𝒢\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)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)Sh(CartSp) regarded as a CartSpCartSp-structured topos, smooth manifolds are precisely its locally representable objects.

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

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

Indeed, every object UCartSpU \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 UCartSpU \in CartSp is canonically incarnated as the CartSpCartSp-structured (∞,1)-topos

(𝒳,𝒪 𝒳):=(Sh (,1)(CartSp)/U,𝒪 U:CartSpjSh (,1)(CartSp)U *Sh (,1)(CartSp)/U) (\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 CartSpCartSp and the structure sheaf given by the composite of the (∞,1)-Yoneda embedding and the inverse image of the etale geometric morphism induced by UU.

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

Definition

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

iU iX \coprod_i U_i \to X

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

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

Proposition

There is an equivalence of categories

DiffLocRep(CartSp) Diff \simeq LocRep(CartSp)

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

Proof

Define a functor DiffLocRep(CartSp)Diff \to LocRep(CartSp) by sending each smooth manifold to the sheaf over CartSpCartSp that it naturally represents. By definition of manifold there is an open cover {U iX}\{U_i \hookrightarrow X\}. We claim that iU iX\coprod_i U_i \to X is an effective epimorphism, so that this functor indeed lands in LocRep(CartSp)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

i,jU i× XU j iU iX \coprod_{i, j} U_i \times_X U_j \stackrel{\to}{\to} \coprod_i U_i \to X

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

i,jU i× XU j iU iS({U i}) \coprod_{i, j} U_i \times_X U_j \stackrel{\to}{\to} \coprod_i U_i \to S(\{U_i\})

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

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

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

i,jU i× XU j iU iX \coprod_{i, j} U_i \times_X U_j \stackrel{\to}{\to} \coprod_{i} U_i \to X

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

It follows that in the pullback diagrams

U i× XU j U j U i X \array{ U_i \times_X U_j &\to& U_j \\ \downarrow && \downarrow \\ U_i &\to& X }

the object U iU jU_i \cap U_j is the diffeological space whose underlying topological space is the intersection of U iU_i and U jU_j in the topological space underlying XX. In particular the inclusions U i× XU jU iU_i \times_X U_j \hookrightarrow U_i are open embeddings.

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

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

For every choice of geometry (for structured (∞,1)-toposes) there is a notion of 𝒢\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 XX may be identified with an ∞-stack on CartSp (an ∞-Lie groupoid) which is represented by a CartSp-structured (∞,1)-topos (𝒳,𝒪 𝒳)(\mathcal{X}, \mathcal{O}_{\mathcal{X}}) such that

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

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

  3. For every iIi \in I there is an equivalence

(𝒳/U i,𝒪 𝒳|U i)t i(Sh (,1)( n),𝒪( n)). (\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 (𝒳,𝒪 𝒳)(\mathcal{X}, \mathcal{O}_{\mathcal{X}}) is locally equivalent to the ordinary cannonically CartSp-locally ringed space n\mathbb{R}^n (for nn \in \mathbb{N} the dimension. The first condition then says that these local identifications cover 𝒳\mathcal{X}.

(…)

(…)

References

A textbook reference is

An algebraic approach to classical smooth manifolds with an eye toward physics is in

  • Jet Nestruev, Smooth Manifolds and Observables , Springer LNM 220 Heidelberg 2003.

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

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

The proof that idempotents split in the category of smooth manifolds was adapted from this MO answer:

  • Zack (http://mathoverflow.net/users/300/zack), Idempotents split in category of smooth manifolds?, URL (version: 2014-04-06): http://mathoverflow.net/q/162556 (web)

Which provides a solution to exercise 3.21 in

  • F. W. Lawvere, Perugia Notes - Theory of Categories over a Base Topos , Ms. Università di Perugia 1973.

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)

Revised on August 11, 2014 23:27:33 by Thomas Holder (89.204.130.80)