Ricci-Flat Metric & Diffeomorphism – Does there Exist a Possibility of a Complete Construction of a Metric if the Surface isn’t a Smooth Manifold? Note Quote.

Using twistors, the Gibbons-Hawking ansatz is generalized to investigate 4n-dimensional hyperkähler metrics admitting an action of the n-torus Tⁿ. The hyperkähler case could further admit a tri-holomorphic free action of the n-torus Tⁿ. It turns out that the metric may be written in coordinates adapted to the torus action, in a form similar to the Gibbons-Hawking ansatz in dimension 4, and such that the non-linear Einstein equations reduce to a set of linear equations (essentially saying that certain functions on Euclidean 3-space are harmonic). In the case of 8-manifolds (n = 2), the solutions can be described geometrically, in terms of arrangements of 3-dimensional linear subspaces in Euclidean 6-space.

There are in fact many explicit examples known of metrics on non-compact manifolds with SU(n) or Sp(2n) holonomy. The other holonomy groups automatically yielding Ricci-flat metrics are the special holonomy groups G₂ in dimension 7 and Spin(7) in dimension 8. Until fairly recently only three explicit examples of complete metrics (in dimension 7) with G₂-holonomy and one explicit example (in dimension 8) with Spin(7)-holonomy were known. The G₂-holonomy examples are asymptotically conical and live on the bundle of self-dual two-forms over S⁴, the bundle of self-dual two-forms over CP², and the spin bundle of S³ (topologically R⁴ × S³), respectively. The metrics are of cohomogeneity one with respect to the Lie groups SO(5), SU(3) and SU(2) × SU(2) × SU(2) respectively. A cohomogeneity-one metric has a Lie group acting via isometries, with general (principal) orbits of real codimension one. In particular, if the metric is complete, then X is the holomorphic cotangent bundle of projective n-space T^∗CPⁿ, and the metric is the Calabi hyperkähler metric.

The G₂-holonomy examples are all examples in which a Lie group G acts with low codimension orbits. This is a general feature of explicit examples of Einstein metrics. The simplest case of such a situation would be when there is a single orbit of a group action, in which case the metric manifold is homogeneous. For metrics on homogeneous manifolds, the Einstein condition may be expressed purely algebraically. Moreover, all homogeneous Ricci-flat manifolds are flat, and so no interesting metrics occur. Then what about cohomogeneity one with respect to G, i.e., the orbits of G are codimension one in general? Here, the Einstein condition reduces to a system of non-linear ordinary differential equations in one variable, namely the parameter on the orbit space. In the Ricci-flat case, Cheeger-Gromoll theorem implies that the manifold has at most one end. In the non-compact case, the orbit space is R₊ and there is just one singular orbit. Geometrically, if the principal orbit is of the form G/K, the singular orbit (the bolt) is G/H for some subgroup H ⊃ K; if G is compact, a necessary and sufficient condition for the space to be a smooth manifold is that H/K is diffeomorphic to a sphere. In many cases, this is impossible because of the form of the group G, and so any metric constructed will not be complete.

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Canonical Actions on Bundles – Philosophizing Identity Over Gauge Transformations.

In physical applications, fiber bundles often come with a preferred group of transformations (usually the symmetry group of the system). The modem attitude of physicists is to regard this group as a fundamental structure which should be implemented from the very beginning enriching bundles with a further structure and defining a new category.

A similar feature appears on manifolds as well: for example, on ℜ² one can restrict to Cartesian coordinates when we regard it just as a vector space endowed with a differentiable structure, but one can allow also translations if the “bigger” affine structure is considered. Moreover, coordinates can be chosen in much bigger sets: for instance one can fix the symplectic form w = dx ∧ dy on ℜ² so that ℜ² is covered by an atlas of canonical coordinates (which include all Cartesian ones). But ℜ² also happens to be identifiable with the cotangent bundle T*ℜ so that we can restrict the previous symplectic atlas to allow only natural fibered coordinates. Finally, ℜ² can be considered as a bare manifold so that general curvilinear coordinates should be allowed accordingly; only if the full (i.e., unrestricted) manifold structure is considered one can use a full maximal atlas. Other choices define instead maximal atlases in suitably restricted sub-classes of allowed charts. As any manifold structure is associated with a maximal atlas, geometric bundles are associated to “maximal trivializations”. However, it may happen that one can restrict (or enlarge) the allowed local trivializations, so that the same geometrical bundle can be trivialized just using the appropriate smaller class of local trivializations. In geometrical terms this corresponds, of course, to impose a further structure on the bare bundle. Of course, this newly structured bundle is defined by the same basic ingredients, i.e. the same base manifold M, the same total space B, the same projection π and the same standard fiber F, but it is characterized by a new maximal trivialization where, however, maximal refers now to a smaller set of local trivializations.

Examples are: vector bundles are characterized by linear local trivializations, affine bundles are characterized by affine local trivializations, principal bundles are characterized by left translations on the fiber group. Further examples come from Physics: gauge transformations are used as transition functions for the configuration bundles of any gauge theory. For these reasons we give the following definition of a fiber bundle with structure group.

A fiber bundle with structure group G is given by a sextuple B = (E, M, π; F ;>.., G) such that:

• (E, M, π; F) is a fiber bundle. The structure group G is a Lie group (possibly a discrete one) and λ : G —–> Diff(F) defines a left action of G on the standard fiber F .

• There is a family of preferred trivializations {(U_α, t_(α)}_α∈I of B such that the following holds: let the transition functions be gˆ_(αβ) : U_αβ —–> Diff(F) and let e_G be the neutral element of G. ∃ a family of maps g_(αβ) : U_αβ —–> G such

  that, for each x ∈ U_αβγ = U_α ∩ U_β ∩ U_γ

  g_(αα)(x) = e_G

  g_(αβ)(x) = [g_(βα)(x)]^-1

  g_(αβ)(x) . g_(βγ)(x) . g_(γα)(x) = e_G

  and

  gˆ_(αβ)(x) = λ(g_(αβ)(x)) ∈ Diff(F)

The maps g_(αβ) : U_αβ —–> G, which depend on the trivialization, are said to form a cocycle with values in G. They are called the transition functions with values in G (or also shortly the transition functions). The preferred trivializations will be said to be compatible with the structure. Whenever dealing with fiber bundles with structure group the choice of a compatible trivialization will be implicitly assumed.

Fiber bundles with structure group provide the suitable framework to deal with bundles with a preferred group of transformations. To see this, let us begin by introducing the notion of structure bundle of a fiber bundle with structure group B = (B, M, π; F; x, G).

Let B = (B, M, π; F; x, G) be a bundle with a structure group; let us fix a trivialization {(U_α, t_(α)}_α∈I and denote by g_(αβ) : U_αβ —–> G its transition functions. By using the canonical left action L : G —–> Diff(G) of G onto itself, let us define gˆ_(αβ) : U_αβ —–> Diff(G) given by gˆ_(αβ)(x) = L (g_(αβ)(x)); they obviously satisfy the cocycle properties. Now by constructing a (unique modulo isomorphisms) principal bundle P_B = P(B) having G as structure group and g_(αβ) as transition functions acting on G by left translation L_g : G —> G.

The principal bundle P(B) = (P, M, p; G) constructed above is called the structure bundle of B = (B, M, π; F; λ, G).

Notice that there is no similar canonical way of associating a structure bundle to a geometric bundle B = (B, M, π; F), since in that case the structure group G is at least partially undetermined.

Each automorphism of P(B) naturally acts over B.

Let, in fact, {σ^(α)}_α∈I be a trivialization of P_B together with its transition functions g_(αβ) : U_αβ —–> G defined by σ^(β) = σ^(α) . g_(αβ). Then any principal morphism Φ = (Φ, φ) over P_B is locally represented by local maps ψ(α) : U_α —> G such that

Φ : [x, h]_α ↦ [φ_(α)(x), ψ_(α)(x).h]_(α)

Since Φ is a global automorphism of P_B for the above local expression, the following property holds true in U_αβ.

φ_(α)(x) = φ_(β)(x) ≡ x’

ψ_(α)(x) = g_(αβ)(x’) . ψ_(β)(x) . g_(βα)(x)

By using the family of maps {(φ_(α), ψ_(α))} one can thence define a family of global automorphisms of B. In fact, using the trivialization {(U_α, t_(α)}_α∈I, one can define local automorphisms of B given by

Φ^(α)_B : (x, y) ↦ (φ_(α)(x), [λ(ψ_(α)(x))](y))

These local maps glue together to give a global automorphism Φ_B of the bundle B, due to the fact that g_(αβ) are also transition functions of B with respect to its trivialization {(U_α, t_(α)}_α∈I.

In this way B is endowed with a preferred group of transformations, namely the group Aut(P_B) of automorphisms of the structure bundle P_B, represented on B by means of the canonical action. These transformations are called (generalized) gauge transformations. Vertical gauge transformations, i.e. gauge transformations projecting over the identity, are also called pure gauge transformations.

Principal Bundles Preserve Structures…

A bundle P = (P, M ,π; G) is a principal bundle if the standard fiber is a Lie group G and ∃ (at least) one trivialization the transition functions of which act on G by left translations Lg : G → G : h ↦ f  g . h (where . denotes here the group multiplication).

The principal bundles are slightly different from affine bundles and vector bundles. In fact, while in affine bundles the fibers π^-1(x) have a canonical structure of affine spaces and in vector bundles the fibers π^-1(x) have a canonical structure of vector spaces, in principal bundles the fibers have no canonical Lie group structure. This is due to the fact that, while in affine bundles transition functions act by means of affine transformations and in vector bundles transition functions act by means of linear transformations, in principal bundles transition functions act by means of left translations which are not group automorphisms. Thus the fibers of a principal bundle do not carry a canonical group structure, but rather many non-canonical (trivialization-depending) group structures. In the fibers of a vector bundle there exists a preferred element (the “zero”) the definition of which does not depend on the local trivialization. On the contrary, in the fibers of a principal bundle there is no preferred point which is fixed by transition functions to be selected as an identity. Thus, while in affine bundles affine morphisms are those which preserve the affine structure of the fibers and in vector bundles linear morphisms are the ones which preserve the linear structure of the fibers, in a principal bundle P = (P, M, π; G) principal morphisms preserve instead a structure, the right action of G on P.

Let P = (P, M, π; G) be a principal bundle and {(U_α, t_(α)}_α∈I a trivialization. We can locally consider the maps

R^(α)_g : π^-1(U_α) → π^-1(U_α) : [x, h]_(α) ↦ [x, h . g]_(α) —– (1)

∃ a (global) right action R_g of G on P which is free, vertical and transitive on fibers; the local expression in the given trivialization of this action is given by R^(α)_g .

Using the local trivialization, we set p = [x, h]_(α) = [x, g_(βα)(x) . h]_β following diagram commutes:

which clearly shows that the local expressions agree on the overlaps U_αβ, to define a right action. This is obviously a vertical action; it is free because of the following:

R_gp = p => [x, h . g]_(α) = [x, h]_(α) => h · g = h => g = e —– (2)

Finally, if p = [x, h₁]_(α) and q = [x, h₂]_(α) are two points in the same fiber of p, one can choose g = h₂^-1 . h₁ (where · denotes the group multiplication) so that p = R_gq. This shows that the right action is also transitive on the fibers.

On the contrary, that a global left action cannot be defined by using the local maps

L^(α)_g : π^-1(U_α) → π^-1(U_α) : [x, h]_(α) ↦ [x, g . h]_(α) —– (3)

since these local maps do not satisfy a compatibility condition analogous to the condition of the commuting diagram.

let P = (P, M, π; G) and P’ = (P’, M’, π’ ; G’ ) be two principal bundles and θ : G → G’ be a homomorphism of Lie groups. A bundle morphism Φ = (Φ, φ) : P → P’ is a principal morphism with respect to θ if the following diagram is commutative:

When G = G’ and θ = id_G we just say that Φ is a principal morphism.

A trivial principal bundle (M x G, M, π; G) naturally admits the global unity section I ∈ Γ(M x G), defined with respect to a global trivialization, I : x ↦ (x, e), e being the unit element of G. Also, principal bundles allow global sections iff they are trivial. In fact, on principal bundles there is a canonical correspondence between local sections and local trivializations, due to the presence of the global right action.

Local Lifts into Period Domains: Holonomies: Philosophies of Conjugacy. Part 2.

Let F = G^C/P be a flag manifold. Then there is a unique inner symmetric space G-space N associated to F together with a finite number of homogeneous fibrations F → N.

Let us emphasise that this construction depends on nothing but the conjugacy class of p ⊂ g^C and the choice of compact real form g. Equivalently, it depends solely on the choice of invariant complex structure on F.

Every flag manifold fibres over an inner symmetric space. Conversely, every inner symmetric space is the target of the canonical fibrations of at least one flag manifold. Let us now see how this story relates to the geometry of J(N).

So let p : F → N be a canonical fibration. By construction, the fibres of p are complex submanifolds of F and this allows us to define a fibre map i_p : F → J(N) as follows: at f ∈ F we have an orthogonal splitting of T_fF into horizontal and vertical subspaces both of which are invariant under the complex structure of F. Then dp restricts to give an isomorphism of the horizontal part with T_p(f)N and therefore induces an almost Hermitian structure on T_p(f)N : this is i_p(f) ∈ J_p(f)N. Such a construction is possible whenever we have a Riemannian submersion of a Hermitian manifold with complex submanifolds as fibres.

i_p : F → J(N) is a G-equivariant holomorphic embedding. This implies that i_p (F) is an almost complex submanifold of J(N) on which J is integrable.

If j ∈ Z ⊂ J(N) then G · j is a flag manifold canonically fibred over N. In fact, G · j = i_p(F ) for some canonical fibration p : F → N of a flag manifold F .

For this, the main observation is the following: at π(j), we have the symmetric decomposition g = k ⊕ q

with q ≅ T_π(j)N. If q⁻ is the (0,1)-space for j then [q⁻, q⁻] ⊕ q⁻

is the nilradical of a parabolic subalgebra p, where G · j is equivariantly biholomorphic to the corresponding flag manifold G^C/P. Each canonical fibration of a flag manifold gives rise to a G-orbit in Z for some inner symmetric G-space N and that all such orbits arise in this way. But, for fixed G, there are only a finite number of biholomorphism types of flag manifold (they are in bijective correspondence with the conjugacy classes of parabolic subalgebras of g^C) and each flag manifold admits but a finite number of canonical fibrations. Thus Z is composed of a finite number of G-orbits all of which are closed. In this way, we obtain a geometric interpretation of the purely algebraic construction of the canonical fibrations: they are just the restrictions of the projection π : J(N) → N to the various realisations of F as an orbit in Z.

For each non-compact real form G_R of a complex semisimple group Lie group G^C, there is a unique Riemannian symmetric space G_R/K of non-compact type. The corresponding involution is called the Cartan involution of G_R. Consider now the orbits of such a G_R on the various flag manifolds F = G^C/P. Those orbits which are open subsets of F are called flag domains: an orbit is a flag domain precisely when the stabilisers contain a compact Cartan subgroup of G_R. It turns out that the presence of this compact Cartan subgroup is precisely what we need to define a canonical element of g_R and thus an involution of g_R just as in the compact case. However the involution is not necessarily a Cartan involution (i.e. the associated symmetric space need not be Riemmanian). In case that the involution is a Cartan involution, the flag domain is a canonical flag domain which is then exponentiated such that the involution gets to a Riemannian symmetric space of non-compact type and a canonical fibration of canonical flag domain over it.

Let X be a flag manifold or canonical flag domain and π : X → N be a homogeneous fibration onto a Riemannian symmetric space. If φ : M → X is a holomorphic map of a Kahler manifold with image tangent to the super-horizontal distribution then π ◦ φ : M → N is harmonic.

Maps φ : M → X satisfying such hypotheses of being super-horizontal are holomorphic maps.

For instance, let X = SU(n + 1)/S(U(1) × ··· × U(1)) = F(1,…,1; C_n+1). There are n+1 homogeneous fibrations π_i : X → CPⁿ, i = 1,…,n, with π₀ holomorphic and π_n anti-holomorphic. Now a super-horizontal holomorphic map φ : S² → X is essentially just the Frenet frame of the holomorphic map π₀ ◦ φ : S² → CPⁿ while each π ◦ φ is a harmonic map S² → CPⁿ. It is the content of the classification theorem for harmonic 2-spheres in CPⁿ that all such harmonic maps arise in this way. On the non-compact side of the fence, the super-horizontal distribution is precisely that which defines the infinitesimal period relation. Thus the (local lifts of) period maps are precisely the super-horizontal holomorphic maps into period domains.

Weyl’s Lagrange Density of General Relativistic Maxwell Theory

Bildnachweis: ETH-Bibliothek Zürich, Bildarchiv

Weyl pondered on the reasons why the structure group of the physical automorphisms still contained the “Euclidean rotation group” (respectively the Lorentz group) in such a prominent role:

The Euclidean group of rotations has survived even such radical changes of our concepts of the physical world as general relativity and quantum theory. What then are the peculiar merits of this group to which it owes its elevation to the basic group pattern of the universe? For what ‘sufficient reasons’ did the Creator choose this group and no other?”

He reminded that Helmholtz had characterized ∆_o ≅ SO (3, ℜ) by the “fact that it gives to a rotating solid what we may call its just degrees of freedom” of a rotating solid body; but this method “breaks down for the Lorentz group that in the four-dimensional world takes the place of the orthogonal group in 3-space”. In the early 1920s he himself had given another characterization living up to the new demands of the theories of relativity in his mathematical analysis of the problem of space.

He mentioned the idea that the Lorentz group might play its prominent role for the physical automorphisms because it expresses deep lying matter structures; but he strongly qualified the idea immediately after having stated it:

Since we have the dualism of invariance with respect to two groups and Ω certainly refers to the manifold of space points, it is a tempting idea to ascribe ∆_o to matter and see in it a characteristic of the localizable elementary particles of matter. I leave it undecided whether this idea, the very meaning of which is somewhat vague, has any real merits.

. . . But instead of analysing the structure of the orthogonal group of transformations ∆_o, it may be wiser to look for a characterization of the group ∆_o as an abstract group. Here we know that the homogeneous n-dimensional orthogonal groups form one of 3 great classes of simple Lie groups. This is at least a partial solution of the problem.

He left it open why it ought to be “wiser” to look for abstract structure properties in order to answer a natural philosophical question. Could it be that he wanted to indicate an open-mindedness toward the more structuralist perspective on automorphism groups, preferred by the young algebraists around him at Princetion in the 1930/40s? Today the classification of simple Lie groups distinguishes 4 series, A_k,B_k,C_k,D_k. Weyl apparently counted the two orthogonal series B_k and D_k as one. The special orthogonal groups in even complex space dimension form the series of simple Lie groups of type D_k, with complex form (SO 2k,C) and real compact form (SO 2k,ℜ). The special orthogonal group in odd space dimension form the series type B_k, with complex form SO(2k + 1, C) and compact real form SO(2k + 1, ℜ).

But even if one accepted such a general structuralist view as a starting point there remained a question for the specification of the space dimension of the group inside the series.

But the number of the dimensions of the world is 4 and not an indeterminate n. It is a fact that the structure of ∆_o is quite different for the various dimensionalities n. Hence the group may serve as a clue by which to discover some cogent reason for the di- mensionality 4 of the world. What must be brought to light, is the distinctive character of one definite group, the four-dimensional Lorentz group, either as a group of linear transformations, or as an abstract group.

The remark that the “structure of ∆_o is quite different for the various dimensionalities n” with regard to even or odd complex space dimensions (type D_k, resp. B_k) strongly qualifies the import of the general structuralist characterization. But already in the 1920s Weyl had used the fact that for the (real) space dimension n “4 the universal covering of the unity component of the Lorentz group SO (1, 3)^o is the realification of SL (2, C). The latter belongs to the first of the A_k series (with complex form SL (k + 1,C). Because of the isomorphism of the initial terms of the series, A₁ ≅ B₁, this does not imply an exception of Weyl’s general statement. We even may tend to interpret Weyl’s otherwise cryptic remark that the structuralist perspective gives a “at least a partial solution of the problem” by the observation that the Lorentz group in dimension n “4 is, in a rather specific way, the realification of the complex form of one of the three most elementary non-commutative simple Lie groups of type A₁ ≅ B₁. Its compact real form is SO (3, ℜ), respectively the latter’s universal cover SU (2, C).

Weyl stated clearly that the answer cannot be expected by structural considerations alone. The problem is only “partly one of pure mathematics”, the other part is “empirical”. But the question itself appeared of utmost importance to him

We can not claim to have understood Nature unless we can establish the uniqueness of the four-dimensional Lorentz group in this sense. It is a fact that many of the known laws of nature can at once be generalized to n dimensions. We must dig deep enough until we hit a layer where this is no longer the case.

In 1918 he had given an argument why, in the framework of his new scale gauge geometry, the “world” had to be of dimension 4. His argument had used the construction of the Lagrange density of general relativistic Maxwell theory L_f = f_μν f^μν √(|detg|), with f_μν the components of curvature of his newly introduced scale/length connection, physically interpreted by him as the electromagnetic field. L_f is scale invariant only in spacetime dimension n = 4. The shift from scale gauge to phase gauge undermined the importance of this argument. Although it remained correct mathematically, it lost its convincing power once the scale gauge transformations were relegated from physics to the mathematical automorphism group of the theory only.

Weyl said:

Our question has this in common with most questions of philosophical nature: it depends on the vague distinction between essential and non-essential. Several competing solutions are thinkable; but it may also happen that, once a good solution of the problem is found, it will be of such cogency as to command general recognition.

Diffeomorphic Lift

Let G be a connected and simply connected Lie group, Γ ⊂ G an arbitrary totally disconnected subgroup. While it is possible to develop a general theory of fibre bundles and covering spaces in the diffeological setting, we shall directly prove some lifting properties for the quotient map π : G → G/Γ.

Lifting of diffeomorphisms: The factor space G/Γ is endowed with the quotient diffeology, that is the collection of plots α: U → G/Γ which locally lift through π to a smooth map U → G.

Proposition: Any differentiable map (in the diffeological sense) φ^– : G/Γ → G/Γ has a sooth lift φ : G → G, that is a C∞ map such that πφ = φ^–π.

Proof. By definition of quotient diffeology, the result is locally true, that is for any x ∈ G there exists an open neighbourhood U_x and a smooth map φ_x : U_x → G such that π ◦ φ_x = φ^– ◦ π on U_x. We can suppose that U_x is a connected open set.

Now we define an integrable distribution D on G × G in the following way. Since an arbitrary point (x, g) ∈ G × G can be written as (x, φ_x(x)h) for some h ∈ G, let

D_(x,g) = {(v,(Rh ◦ φ_x)_∗x(v)): v ∈ T_xG} ⊂ T_(x,g)(G × G).

The distribution D is well defined because two local lifts differ by some translation. In fact, let x, y ∈ G such that U_x ∩ U_y ≠ ∅. Then for any z ∈ U_x ∩ U_y and any connected neighbourhood V_z ⊂ U_x ∩ U_y, the local lifts φ_x, φ_y define the continuous map γ : V_z → Γ given by γ(t) = φ_x(t)⁻¹φ_y(t). Since V_z is connected and the set Γ is totally disconnected, the map γ must be constant, hence φ_y = R_γ ◦ φ_x.

Moreover D has constant rank, and it is integrable, the integral submanifolds being translations of the graphs of the local lifts.

Let us choose some point x₀ ∈ G such that [x₀] = φ^–([e]), and let G^~ be the maximal integral submanifold passing through (e, x₀). We shall prove that the projection of G ⊂ G × G onto the first factor is a covering map. Since G is simply connected, it follows that G is the graph of a global lift.

Lemma: The projection p₁: G^~ ⊂ G × G → G is a covering map.

Proof. Clearly p₁ is a differentiable submersion, hence an open map, so p₁(G) is an open subspace of G. Let us prove that it is closed too; this will show that the map p₁ : G^~ → G is onto, because the Lie group G is connected.

Suppose x ∈ G is in the closure of p₁(G), and let U_x be a connected open neighbourhood where the local lift φ_x is defined. Let y ∈ U_x ∩ p₁(G), then (y, φ_x(y)h) ∈ G for some h ∈ G. This implies that the graph of R_h ◦ φ_x, which is an integral submanifold of D, is contained in G . Hence x ∈ p₁(G).

It remains to prove that any x ∈ G has a neighbourhood U_x such that (p₁)⁻¹(U_x) is a disjoint union of open sets, each one homeomorphic to U_x by p₁. It is clear that we can restrict ourselves to the case x = e. Let φ_e : U_e → G be a connected local lift of φ^–. We can suppose that φ_e(e) = x₀.

Let U^~_e be its graph. Then U^~_e is an open subset of G^~, containing (e, x₀), with p₁(U^~_e) = U_e. Let I be the non-empty set

I = {γ ∈Γ : (e,x₀γ) ∈ G^~} .

Then (p₁)⁻¹(U) is the disjoint union of the sets R_γ(U^~_e), γ ∈ I.

Corollary: Any diffeomorphism of G/Γ can be lifted to a diffeomorphism of G.

Corollary: Let U be a connected simply connected open subset of Rⁿ, n ≥ 0. Any differentiable map (resp. diffeomorphism) U × G/Γ → U × G/Γ can be lifted to a C∞ map (resp. diffeomorphism) U × G → U × G.

Philosophizing Forgetful Functors: This Functor Forgets only Properties: Namely, the Property of Being Abelian + This Functor Forgets Both Structure (the generating set) and Properties (the property of being a free group).

A forgetful functor is a functor which is defined by ‘forgetting’ something. For example, the forgetful functor from Grp to Set forgets the group structure of a group, remembering only the underlying set.

In common parlance, the term ‘forgetful functor’ has no precise definition, being simply used whenever a functor is obviously defined by forgetting something. Many forgetful functors of this sort have left or right adjoints (and many are actually monadic or comonadic), leading to the paradigmatic adjunction “free ⊣⊣ forgetful.”

On the other hand, from the perspective of stuff, structure, property, every functor is regarded as a forgetful functor and classified by how much it forgets (namely, stuff, structure, or properties). From this perspective, the forgetful functor from GrpGrp to SetSet forgets the structure of a group and the property of admitting a group structure, as usual; but its left adjoint (the free group functor) is also forgetful: if you identify SetSet with the category of free groups with specified generators, then it forgets the structure of a set of free generators and the property of being free.

There are many cases in which we want to say that one kind of mathematical object has more structure than another kind of mathematical object. For instance, a topological space has more structure than a set. A Lie group has more structure than a smooth manifold. A ring has more structure than a group. And so on. In each of these cases, there is a sense in which the first sort of object – say, a topological space – results by taking an instance of the second sort – say, a set – and adding something more – in this case, a topology. In other cases, we want to say that two different kinds of mathematical objects have the same amount of structure. For instance, given a Boolean algebra, one can construct a special kind of topological space, known as a Stone space, from which one can uniquely reconstruct the original Boolean algebra; and vice-versa.

These sorts of relationships between mathematical objects are naturally captured in the language of category theory, via the notion of a forgetful functor. For instance, there is a functor F : Top → Set from the category Top, whose objects are topological spaces and whose arrows are continuous maps, to the category Set, whose objects are sets and whose arrows are functions. This functor takes every topological space to its underlying set, and it takes every continuous function to its underlying function. We say this functor is forgetful because, intuitively speaking, it forgets something: namely the choice of topology on a given set.

The idea of a forgetful functor is made precise by a classification of functors due to Baez et al. (2004). This requires some machinery. A functor F : C → D is said to be full if for every pair of objects A, B of C, the map F : hom(A, B) → hom(F (A), F (B)) induced by F is surjective, where hom(A, B) is the collection of arrows from A to B. Likewise, F is faithful if this induced map is injective for every such pair of objects. Finally, a functor is essentially surjective if for every object X of D, there exists some object A of C such that F(A) is isomorphic to X.

If a functor is full, faithful, and essentially surjective, we will say that it forgets nothing. A functor F : C → D is full, faithful, and essentially surjective if and only if it is essentially invertible, i.e., there exists a functor G : D → C such that G ◦ F : C → C is naturally isomorphic to 1_C, the identity functor on C, and F ◦ G : D → D is naturally isomorphic to 1_D. (Note, then, that G is also essentially invertible, and thus G also forgets nothing.) This means that for each object A of C, there is an isomorphism η_A : G ◦ F (A) → A such that for any arrow f : A → B in C, η_B ◦ G ◦ F(f) = f ◦ η_A, and similarly for every object of D. When two categories are related by a functor that forgets nothing, we say the categories are equivalent and that the pair F, G realizes an equivalence of categories.

Conversely, any functor that fails to be full, faithful, and essentially surjective forgets something. But functors can forget in different ways. A functor F : C → D forgets structure if it is not full; properties if it is not essentially surjective; and stuff if it is not faithful. Of course, “structure”, “property”, and “stuff” are technical terms in this context. But they are intended to capture our intuitive ideas about what it means for one kind of object to have more structure (resp., properties, stuff) than another. We can see this by considering some examples.

For instance, the functor F : Top → Set described above is faithful and essentially surjective, but not full, because not every function is continuous. So this functor forgets only structure – which is just the verdict we expected. Likewise, there is a functor G : AbGrp → Grp from the category AbGrp whose objects are Abelian groups and whose arrows are group homomorphisms to the category Grp whose objects are (arbitrary) groups and whose arrows are group homomorphisms. This functor acts as the identity on the objects and arrows of AbGrp. It is full and faithful, but not essentially surjective because not every group is Abelian. So this functor forgets only properties: namely, the property of being Abelian. Finally, consider the unique functor H : Set → 1, where 1 is the category with one object and one arrow. This functor is full and essentially surjective, but it is not faithful, so it forgets only stuff – namely all of the elements of the sets, since we may think of 1 as the category whose only object is the empty set, which has exactly one automorphism.

In what follows, we will say that one sort of object has more structure (resp. properties, stuff) than another if there is a functor from the first category to the second that forgets structure (resp. properties, stuff). It is important to note, however, that comparisons of this sort must be relativized to a choice of functor. In many cases, there is an obvious functor to choose – i.e., a functor that naturally captures the standard of comparison in question. But there may be other ways of comparing mathematical objects that yield different verdicts.

For instance, there is a natural sense in which groups have more structure than sets, since any group may be thought of as a set of elements with some additional structure. This relationship is captured by a forgetful functor F : Grp → Set that takes groups to their underlying sets and group homomorphisms to their underlying functions. But any set also uniquely determines a group, known as the free group generated by that set; likewise, functions generate group homomorphisms between free groups. This relationship is captured by a different functor, G : Set → Grp, that takes every set to the free group generated by it and every function to the corresponding group homomorphism. This functor forgets both structure (the generating set) and properties (the property of being a free group). So there is a sense in which sets may be construed to have more structure than groups.