The theory of connections and G-structures. Applications to afﬁne and isometric immersions Paolo Piccione Daniel V. Tausk

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The theory of connections and G-structures.

Applications to affine and isometric immersions

Paolo Piccione Daniel V. Tausk

Dedicated to Prof. Francesco Mercuri on occasion of his 60th birthday

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Preface

This book contains the notes of a short course given by the two authors at the 14th School of Differential Geometry, held at theUniversidade Federal da Bahia, Salvador, Brazil, in July 2006. Our goal is to provide the reader/student with the necessary tools for the understanding of an immersion theorem that holds in the very general context of affine geometry. As most of our colleagues know, there is no better way for learning a topic thanteaching a courseabout it and, even better, writing a bookabout it. This was precisely our original motivation for undertaking this task, that lead uswaybeyond our most optimistic previsions of writing ashort and conciseintroduction to the machinery of fiber bundles and connections, and a self-containedcompactproof of a general immersion theorem.

The original idea was to find a unifying language for several isometric immersion theorems that appear in the classical literature [5] (immersions into Riemann- ian manifolds with constant sectional curvature, immersions into K¨ahler manifolds of constant holomorphic curvature), and also some recent results (see for instance [6, 7]) concerning the existence of isometric immersions in more general Riemann- ian manifolds. The celebrated equations of Gauss, Codazzi and Ricci are well known necessary conditions for the existence of isometric immersions. Additional assumptions are needed in specific situations; the starting point of our theory was precisely the interpretation of such additional assumptions in terms of “structure preserving” maps, that eventually lead to the notion ofG-structure. Giving a G- structure on ann-dimensional manifoldM, whereGis a Lie subgroup ofGL(Rⁿ), means that it is chosen a set of “preferred frames” of the tangent bundle ofM on whichGacts freely and transitively. For instance, giving anO(Rⁿ)structure is the same as giving a Riemannian metric onMby specifying which are the orthonormal frames of the metric.

The central result of the book is an immersion theorem into (infinitesimally) homogeneous affine manifolds endowed with aG-structure. The covariant derivative of theG-structure with respect to the given connection gives a tensor field onM, called the inner torsionof theG-structure, that plays a central role in our theory.Infinitesimally homogeneousmeans that the curvature and the torsion of the connection, as well as the inner torsion of theG-structure, are constant in frames of theG-structure. For instance, consider the case thatM is a Riemannian manifold endowed with the Levi-Civita connection of its metric tensor, Gis the orthogonal group and the G-structure is given by the set of orthonormal frames. Since parallel transport takes orthonormal frames to orthonormal frames, the inner torsion of thisG-structure is zero. The condition that the curvature tensor should be

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constant in orthonormal frames is equivalent to the condition thatM has constant sectional curvature, and we recover in this case the classical “fundamental theorem of isometric immersions in spaces of constant curvature”. Similarly, ifM is a Riemannian manifold endowed with an orthogonal almost complex structure, then one has aG-structure onM, whereGis the unitary group, by considering the set of orthonormal complex frames ofT M. In this case, the inner torsion of theG- structure relatively to the Levi-Civita connection of the Riemannian metric is the covariant derivative of the almost complex structure, which vanishes if and only if M is K¨ahler. Requiring that the curvature tensor be constant in orthonormal complex frames means thatMhas constant holomorphic curvature; in this context, our immersion theorem reproduces the classical result of isometric immersions into K¨ahler manifolds of constant holomorphic curvature. Another interesting example ofG-structure that will be considered in detail in these notes is the case of Rie- mannian manifolds endowed with a distinguished unit vector fieldξ; in this case, we obtain an immersion theorem into Riemannian manifolds with the property that both the curvature tensor and the covariant derivative of the vector field are constant in orthonormal frames whose first vector isξ. This is the case in a number of important examples, like for instance all manifolds that are Riemannian products of a space form with a copy of the real line, as well as all homogeneous, simply- connected3-dimensional manifolds whose isometry group has dimension4. These examples were first considered in [6]. Two more examples will be studied in some detail. First, we will consider isometric immersions into Lie groups endowed with a left invariant semi-Riemannian metric tensor. These manifolds have an obvious 1-structure, given by the choice of a distinguished orthonormal left invariant frame;

clearly, the curvature tensor is constant in this frame. Moreover, the inner torsion of the structure is simply the Christoffel tensor associated to this frame, which is also constant. The second example that will be treated in some detail is the case of isometric immersions into products of manifolds with constant sectional curvature;

in this situation, the G-structure considered is the one consisting of orthonormal frames adapted to a smooth distribution.

The book was written under severe time restrictions. Needless saying that, in its present form, these notes carry a substantial number of lacks, imprecisions, omissions, repetitions, etc. One evident weak point of the book is the total ab- sence of reference to the already existing literature on the topic. Most the material discussed in this book, as well as much of the notations employed, was simply created on the blackboard of our offices, and not much attention has been given to the possibility that different conventions might have been established by previ- ous authors. Also, very little emphasis was given to the applications of the affine immersion theorem, that are presently confined to the very last section of Chap- ter 3, where a few isometric immersion theorems are discussed in the context of semi-Riemannian geometry. Applications to general affine geometry are not even mentioned in this book. Moreover, the reference list cited in the text is extremely reduced, and it does not reflect the intense activity of research produced in the last decades about affine geometry, submanifold theory, etc. In our apology, we must emphasize that the entire material exposed in these three long Chapters and two

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PREFACE vii

Appendices started from zero and was produced in a period of seven months since the beginning of our project.

On the other hand, we are particularly proud of having been able to write a text which is basically self-contained, and in which very little prerequisite is assumed on the reader’s side. Many preliminary topics discussed in these notes, that form the core of the book, have been treated in much detail, with the hope that the text might serve as a reference also for other purposes, beyond the problem of affine immersions. Particular care has been given to the theory of principal fiber bundles and principal connections, which are the basic tools for the study of many topics in differential geometry. The theory of vector bundles is deduced from the theory of principal fiber bundles via the principal bundle of frames. We feel we have done a good job in relating the notions of principal connections and of linear connections on vector bundles, via the notions of associated bundle and contraction map. A certain effort has been made to clarify some points that are sometimes treated without many details in other texts, like for instance the question of inducing connections on vector bundles constructed from a given one by functorial constructions. The question is treated formally in this text with the introduction of the notion ofsmooth natural transformationbetween functors, and with the proof of several results that allow one to give a formal justification for many types of computations using connections that are very useful in many applications. Also, we have tried to make the exposition of the material in such a way that generalizations to the infinite dimensional case should be easy to obtain. Theglobal immersion results in this book have been proven using a general “globalization technique” that is explained in Appendix B in the language of pre-sheafs. An intensive effort has been made in order to maintain the (sometimes heavy) notations and terminology self-consistent throughout the text. The book has been written having in mind an hypothetical reader that would read it sequentially from the beginning to the end. In spite of this, lots of cross references have been added, and complete (and sometimes repet- itive) statements have been chosen for each proposition proved.

Thanks are due to the Scientific Committee of the 14th School of Differential Geometry for giving the authors the opportunity to teach this course. We also want to thank the staff at IMPA for taking care of the publishing of the book, which was done in a very short time. The authors gratefully acknowledge the sponsorship by CNPq and Fapesp.

The two authors wish to dedicate this book to their colleague and friend Francesco Mercuri, in occasion of his 60th birthday. Francohas been to the two authors an example of careful dedication to research, teaching, and supervision of graduate students.

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CHAPTER 1

Principal and associated fiber bundles

1.1. G-structures on sets

A field of mathematics is sometimes characterized by the category it works with. Of central importance among categories are the ones whose objects are sets endowed with some sort of structure and whose morphisms are maps that preserve the given structure. A structure on a setXis often described by a certain number of operations, relations or some distinguished collection of subsets of the setX.

Following the ideas of the Klein program for geometry, a structure on a setXcan also be described along the following lines: one fixes a model spaceX0, which is supposed to be endowed with a canonical version of the structure that is being defined. Then, a collection P of bijective maps p : X0 → X is given in such a way that ifp : X₀ → X, q : X₀ → X belong toP then the transition map p⁻¹◦q :X₀ → X₀ belongs to the groupGof all automorphisms of the structure of the model spaceX0. The setXthus inherits the structure from the model space X₀via the given collection of bijective mapsP. The mapsp∈P can be thought of asparameterizationsofX.

To illustrate the ideas described above in a more concrete way, we consider the following example. We wish to endow a set V with the structure of an n- dimensional real vector space, where n is some fixed natural number. This is usually done by defining on V a pair of operations and by verifying that such operations satisfy a list of properties. Following the ideas explained in the para- graph above, we would instead proceed as follows: letP be a set of bijective maps p:Rⁿ→V such that:

(a) forp, q∈P, the mapp⁻¹◦q :Rⁿ→Rⁿis a linear isomorphism;

(b) for everyp∈Pand every linear isomorphismg:Rⁿ→Rⁿ, the bijective mapp◦g:Rⁿ→V is inP.

The setP can be thought of as being ann-dimensional real vector space structure on the setV. Namely, usingP and the canonical vector space operations ofRⁿ, one can define vector space operations on the setV by setting:

(1.1.1) v+w=p p⁻¹(v) +p⁻¹(w)

, tv=p tp⁻¹(v) ,

for all v, w ∈ V and all t ∈ R, where p ∈ P is fixed. Clearly condition (a) above implies that the operations on V defined by (1.1.1) do not depend on the choice of the bijectionp∈ P. Moreover, the fact that the vector space operations ofRⁿsatisfy the standard vector space axioms implies that the operations defined on V also satisfy the standard vector space axioms. If V is endowed with the

1

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operations defined by (1.1.1) then the bijective mapsp:Rⁿ→ V belonging toP are linear isomorphisms; condition (b) above implies thatPis actually the set ofall linear isomorphisms fromRⁿtoV. Thus every set of bijective mapsP satisfying conditions (a) and (b) defines ann-dimensional real vector space structure on V. Conversely, everyn-dimensional real vector space structure onV defines a set of bijectionsP satisfying conditions (a) and (b); just takeP to be the set of all linear isomorphisms fromRⁿ toV. Using the standard terminology from the theory of group actions, conditions (a) and (b) above say thatPis an orbit of the right action of the general linear groupGL(Rⁿ)on the set of all bijective mapsp:Rⁿ → V. The setP will be thus called aGL(Rⁿ)-structureon the setV.

Let us now present more explicitly the notions that were informally explained in the discussion above. To this aim, we quickly recall the basic terminology of the theory of group actions. LetGbe a group with operation

G×G3(g₁, g₂)7−→g₁g₂ ∈G

and unit element1∈G. Given an elementg∈G, we denote byL_g :G→Gand Rg : G → Grespectively theleft translation mapand theright translation map defined by:

(1.1.2) Lg(x) =gx, Rg(x) =xg,

for allx∈G; we also denote byI_g :G→Gtheinner automorphismofGdefined by:

(1.1.3) I_g =Lg◦R⁻¹_g =R⁻¹_g ◦Lg. Given a setAthen aleft actionofGonAis a map:

G×A3(g, a)7−→g·a∈A

satisfying the conditions1·a=aand(g1g2)·a=g1·(g2·a), for allg1, g2 ∈G, and alla∈A; similarly, aright actionofGonAis a map:

A×G3(a, g)7−→a·g∈A

satisfying the conditionsa·1 =aanda·(g₁g₂) = (a·g₁)·g₂, for allg₁, g₂ ∈G, and alla ∈ A. Given a left action (resp., right action) ofGonA then for every a∈ Awe denote byβa :G → Athe map given byaction on the elementa, i.e., we set:

(1.1.4) βa(g) =g·a,

(resp.,β_a(g) =a·g), for allg∈G. The set:

Ga=β_a⁻¹(a)

is a subgroup of Gand is called the isotropy groupof the element a ∈ A. The G-orbit (or, more simply, the orbit) of the element a ∈ A is the set Ga(resp., aG) given by the image ofGunder the mapβ_a; a subset ofAis called aG-orbit (or, more simply, anorbit) if it is equal to theG-orbit of some element ofA. The set of all orbits constitute a partition of the setA. The mapβ_ainduces a bijection from the setG/G_a of left (resp., right) cosets of the isotropy subgroupG_a onto

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1.1.G-STRUCTURES ON SETS 3

theG-orbit of a. In particular, when the isotropy groupGa is trivial (i.e., when G_a={1}) then the mapβ_ais a bijection fromGonto theG-orbit ofa. The action is said to betransitiveif there is only oneG-orbit, i.e., if the mapβa is surjective for some (and hence for any)a ∈ A. The action is said to befreeif the isotropy groupG_ais trivial for everya∈A. For eachg∈Gwe denote byγ_g :A→Athe bijection ofAcorresponding to theaction of the elementg, i.e., we set:

(1.1.5) γ_g(a) =g·a,

(resp.,γ_g(a) = a·g), for alla∈ A. If Bij(A)denotes the group of all bijective maps ofAendowed with the operation of composition then the map:

(1.1.6) G3g7−→γg ∈Bij(A)

is a homomorphism (resp., a anti-homomorphism¹). Conversely, every homomorphism (resp., every anti-homomorphism) (1.1.6) defines a left action (resp., a right action) ofGon Aby settingg·a = γg(a) (resp.,a·g = γg(a)), for allg ∈ G and alla ∈ A. The action ofGon Ais said to beeffective if the map (1.1.6) is injective, i.e., ifT

a∈AG_a ={1}; more generally, given a subsetA⁰ ofA, we say that the action ofGiseffective onA⁰ ifT

a∈A⁰G_a ={1}. The image of the map (1.1.6) is a subgroup ofGand it will be denoted byGef. Notice that if the action is effective thenGis isomorphic toG_ef; in the general case,G_ef is isomorphic to the quotient ofGby the normal subgroupT

a∈AG_a.

We now proceed to the statement of the main definitions of the section. Given setsX₀ andX, we denote byBij(X₀, X)the set of all bijectionsp : X₀ → X.

The groupBij(X0)of all bijections ofX0acts on the right on the setBij(X0, X) by composition of maps. The action ofBij(X₀)onBij(X₀, X)is clearly free and transitive.

DEFINITION 1.1.1. Let X0 be a set and G a subgroup of Bij(X0). A G- structureon a setXis a subset P ofBij(X₀, X)which is aG-orbit. We say that theG-structureP ismodeledupon the setX0.

More explicitly, aG-structure on a setXis a nonempty subsetPofBij(X₀, X) satisfying the following conditions:

(a) p⁻¹◦q :X₀ →X₀is inG, for allp, q∈P;

(b) p◦g:X0 →Xis inP, for allp∈P and allg∈G.

EXAMPLE1.1.2. Given a natural numbern, denote byI_nthe set:

In={0,1, . . . , n−1}.

LetXbe a set havingnelements. By anorderingof the setXwe mean a bijective map p : I_n → X; notice that an ordering p : I_n → X ofX can be identified with the n-tuple p(0), p(1), . . . , p(n−1)

∈ Xⁿ. Denote by Sn = Bij(In) the symmetric group on n elements. The group S_n acts on the right on the set Bij(I_n, X) of all orderings ofX. If G is a subgroup of S_n then aG-structure on X is a choice of a set of orderings P ⊂ Bij(In, X) which is an orbit of the

1Given groupsG,H, then aanti-homomorphismφ:G→His a map satisfying the condition φ(g1g2) =φ(g2)φ(g1), for allg1, g2∈G.

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action of Gon Bij(In, X). For example, if G = {1} is the trivial group then a G-structure onXis the same as the choice of one particular orderingp:I_n→X ofX. IfG= Snthen there is only oneG-structure onX, which is the entire set Bij(In, X). Ifn ≥ 2andG = An ⊂ Snis the group of even permutations then there are exactly two possible G-structures on X; ifn = 3 and X = {a, b, c}, theseG-structures are:

P =

(a, b, c),(c, a, b),(b, c, a) , and:

P⁰ =

(a, c, b),(c, b, a),(b, a, c) .

IfGis an arbitrary subgroup ofSnthen the number of possibleG-structures onX is equal to the index ofGonS_n(see Exercise 1.3). IfXis the set of vertices of an (n−1)-dimensional affine simplex andG=Anthen the two possibleG-structures ofXare usually known as the twoorientationsof the given affine simplex.

IfX0andXare arbitrary sets having the same cardinality, then bijective maps p:X₀ →Xwill also be calledX₀-orderingsof the setX. We remark that, when this terminology is used, it is not assumed that the setX₀ is endowed with some order relation.

EXAMPLE 1.1.3. LetV be an n-dimensional real vector space. A frameof V is a linear isomorphismp : Rⁿ → V. Notice thatpcan be identified with the basis ofV obtained as image underpof the canonical basis ofRⁿ; given a vector v ∈ V, then-tuple p⁻¹(v) ∈ Rⁿ contains the coordinatesofv with respect to the framep. LetFR(V)denote the set of all frames ofV and letGL(Rⁿ)denote thegeneral linear groupofRⁿ, i.e., the group of all linear isomorphisms ofRⁿ. ThenGL(Rⁿ)is a subgroup ofBij(Rⁿ)andFR(V)is aGL(Rⁿ)-structure onV modeled upon Rⁿ. Notice that given a set V and a GL(Rⁿ)-structure P on V then there exists a uniquen-dimensional real vector space structure onV such that P = FR(V). AGL(Rⁿ)-structure on a set can thus be thought of as being the same as ann-dimensional real vector space structure on that set.

LetV₀ andV be arbitrary vector spaces having the same dimension and the same field of scalars; a linear isomorphismp :V0 → V will be called aV0-frame ofV. LetGL(V₀)denote thegeneral linear groupofV₀, i.e., the group of all linear isomorphisms ofV₀. ThenGL(V₀)is a subgroup ofBij(V₀)and the setFR_V₀(V) of allV0-frames ofV is aGL(V0)-structure on the setV modeled uponV0. Given a setV and aGL(V₀)-structure onV then there exists a unique vector space structure onV such thatP = FRV0(V).

EXAMPLE 1.1.4. Let M₀ and M be diffeomorphic differentiable manifolds and denote byDiff(M0)⊂Bij(M0)the group of all diffeomorphisms ofM0. The setDiff(M₀, M)of all diffeomorphismsp:M₀ →M is aDiff(M₀)-structure on M modeled uponM0. Conversely, given aDiff(M0)-structureP on a setM then there exists a unique differentiable manifold structure on M such that P equals Diff(M₀, M).

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In Exercise 1.4 the reader is asked to explore more examples like 1.1.3 and 1.1.4.

EXAMPLE1.1.5. IfX₀is a set andGis a subgroup ofBij(X₀)then the setG itself is aG-structure onX0; namely,Gis theG-orbit of the identity map ofX0. The setGis called thecanonicalG-structure of the model spaceX₀. Notice that the canonicalGL(Rⁿ)-structure ofRⁿis identified with the canonical real vector space structure ofRⁿ.

LetX₀be a set,Gbe a subgroup ofBij(X₀)andHbe a subgroup ofG. IfPis aG-structure on a setXthenP is a union ofH-orbits; any one of thisH-orbits is anH-structure onX. On the other hand, ifGis a subgroup ofBij(X₀)containing G then there exists exactly one G-orbit containing P (see Exercise 1.2); it’s the onlyG-structure onXcontainingP. We state the following:

DEFINITION 1.1.6. IfP is a G-structure on a setX andH is a subgroup of Gthen anH-structureQon Xcontained inP is said to be astrengtheningof the G-structureP. We also say thatP is aweakeningof theH-structureQ.

Thus ifHis a proper subgroup ofGthere are several ways of strengthening a G-structureP into anH-structure (it follows from the result of Exercise 1.3 that the number of such strengthenings is precisely the index ofH inG); on the other hand, there is only one way of weakening an H-structure into aG-structure. In order to strengthen a structure one has to introduce new information; in order to weaken a structure, one has just to “forget” about something. In this sense, G- structures are stronger when the groupGis smaller. The largest possible groupG, which isBij(X₀), givesno structure at all; namely, ifXhas the same cardinality asX0 then there exists exactly one Bij(X0)-structure on X, which is the entire setBij(X₀, X). On the other extreme, ifG = {1}is the trivial group containing only the identity map of X0 then a G-structure onX is the same as a bijection p:X₀ →X; it allows one to identify the setXwith the model setX₀.

The following particularization of Definition 1.1.1 is the one that we will be more interested in.

DEFINITION1.1.7. LetV₀,V be vector spaces having the same dimension and the same field of scalars. Given a subgroupGofGL(V0)then by aG-structure on the vector space V we mean a G-structure P on the set V that strengthens the GL(V₀)-structureFR_V₀(V)ofV.

In other words, ifGis a subgroup ofGL(V0), aG-structure on a vector space V is aG-structureP on the setV such that everyp ∈P is a linear isomorphism fromV₀ toV.

EXAMPLE1.1.8. LetV be ann-dimensional real vector space endowed with an inner producth·,·i_V, i.e., a positive definite symmetric bilinear form. A frame p:Rⁿ→V is calledorthonormalif it is a linear isometry, i.e., if:

hp(x), p(y)i_V =hx, yi,

for allx, y∈Rⁿ, whereh·,·idenotes the canonical (positive definite) inner product h·,·iofRⁿ. Equivalently,pis orthonormal if it carries the canonical basis ofRⁿto

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an orthonormal basis ofV. LetO(Rⁿ)denote theorthogonal groupofRⁿ, i.e., the subgroup ofGL(Rⁿ)consisting of all linear isometries ofRⁿ. The setFR^o(V)of all orthonormal frames ofV is anO(Rⁿ)-structure on the vector spaceV modeled uponRⁿ. Conversely, given ann-dimensional real vector spaceV and anO(Rⁿ)- structure P on V then there exists a unique inner producth·,·i_V onV such that P = FR^o(V).

Let V₀ and V be finite-dimensional real vector spaces having the same dimension, endowed with inner productsh·,·i_V₀ andh·,·i_V, respectively; aV0-frame p:V₀ →V ofV is calledorthonormalifpis a linear isometry. LetO(V₀,h·,·i_V₀) denote the orthogonal group of V0, i.e., the subgroup of GL(V0) consisting of all linear isometries. The setFR^o_V₀(V) of all orthonormalV0-frames ofV is an O(V0,h·,·i_V₀)-structure onV modeled uponV0. Conversely, given a real vector space V and an O(V₀,h·,·i_V₀)-structureP on V then there exists a unique inner producth·,·i_V onV such thatP = FR^o_V₀(V).

EXAMPLE1.1.9. LetV be a real vector space. A bilinear form:

V ×V 3(v, w)7−→ hv, wi_V ∈R

onV is said to benondegenerateifhv, wi_V = 0for allw∈V impliesv= 0. The indexof a symmetric bilinear formh·,·i_V onV is defined by:

n− h·,·i_V

= sup

dim(W) :W is a subspace ofV and

h·,·i_V is negative definite onW . Anindefinite inner producth·,·i_V onV is a nondegenerate symmetric bilinear form onV. For instance, theMinkowski bilinear formof indexrinRⁿ, defined by:

hx, yi=

n−r

X

i=1

xiyi−

n

X

i=n−r+1

xiyi,

for all x, y ∈ Rⁿ, is an indefinite inner product of index r in Rⁿ. If h·,·i_V is an indefinite inner product onV then we denote byO(V,h·,·i_V)the subgroup of GL(V)consisting of all linear isometriesT :V →V, i.e.:

O(V,h·,·i_V) =

T ∈GL(V) :hT(v), T(w)i_V =hv, wi_V,

for allv, w∈V . We callO(V,h·,·i_V)theorthogonal groupofV; when the indefinite inner product h·,·i_V is given by the context, we will write simplyO(V). Ifh·,·iis the Minkowski bilinear form of indexrinRⁿthen the orthogonal groupO(Rⁿ,h·,·i)will also be denoted byOr(Rⁿ).

Let V0 and V be finite-dimensional real vector spaces having the same dimension, endowed with indefinite inner productsh·,·i_V₀ andh·,·i_V, respectively;

assume that h·,·i_V₀ and h·,·i_V have the same index. A V0-frame p : V0 → V ofV is calledorthonormalif pis a linear isometry. The setFR^o_V₀(V) of all or- thonormalV₀-frames ofV is an O(V₀,h·,·i_V₀)-structure on V modeled uponV₀. Conversely, given a real vector space V and an O(V₀,h·,·i_V₀)-structure P on V

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then there exists a unique indefinite inner product h·,·i_V on V, having the same index as h·,·i_V₀, such that P = FRô_V₀(V). If h·,·i_V has index r, V₀ = Rⁿ and h·,·i_V₀ is the Minkowski bilinear form of indexrthen we writeFRô(V)instead of FRô_V₀(V).

EXAMPLE 1.1.10. LetV0, V be finite dimensional vector spaces having the same dimension and the same field of scalars; let W₀ be a subspace of V₀ and W be a subspace of V such that W₀ has the same dimension as W. A V₀- frame p ∈ FRV0(V) of V is said to be adapted to (W0, W) if p(W0) = W. The setFR_V₀(V;W₀, W)of allV₀-frames ofV that are adapted to(W₀, W)is a GL(V0;W0)-structure on the vector spaceV modeled uponV0, whereGL(V0;W0) denotes the subgroup of the general linear group GL(V₀)consisting of all linear isomorphismsT : V0 → V0 such thatT(W0) = W0. IfV0 andV are endowed with positive definite or indefinite inner products, we set:

FR^o_V₀(V;W0, W) = FRV0(V;W0, W)∩FR^o_V₀(V), O(V0;W0) = GL(V0;W0)∩O(V0).

If the setFR^o_V₀(V;W₀, W)is nonempty then it is anO(V₀;W₀)-structure on the vector spaceV modeled uponV0.

EXAMPLE1.1.11. LetV0,V be vector spaces having the same dimension and the same field of scalars. Letv₀ ∈ V₀,v ∈ V be fixed nonzero vectors. AV₀- framep ∈ FRV0(V) ofV is said to beadaptedto(v0, v) ifp(v0) = v. The set FR_V₀(V;v₀, v)of allV₀-frames of V that are adapted to(v₀, v)is aGL(V₀;v₀)- structure on the vector spaceV modeled uponV0, whereGL(V0;v0)denotes the subgroup ofGL(V₀)consisting of all linear isomorphismsT :V₀ → V₀ such that T(v₀) =v₀. IfV₀ andV are real vector spaces endowed with positive definite or indefinite inner products, we set:

FR^o_V₀(V;v₀, v) = FR_V₀(V;v₀, v)∩FR^o_V₀(V), O(V0;v0) = GL(V0;v0)∩O(V0).

If the setFR^o_V₀(V;v₀, v)is nonempty then it is anO(V₀;v₀)-structure on the vector spaceV modeled uponV₀.

EXAMPLE1.1.12. LetV be a real vector space endowed with acomplex struc- tureJ, i.e.,Jis a linear endomorphism ofV such thatJ²equals minus the identity map ofV. Thecanonical complex structureJ0 ofR²ⁿis defined by:

J₀(x, y) = (−y, x),

for allx, y∈Rⁿ. IfV0,V are real vector spaces with the same dimension endowed with complex structuresJ₀andJ, respectively then the set:

FR^c_V₀(V) =

p∈FRV0(V) :p◦J0=J◦p

is aGL(V₀, J₀)-structure on the vector spaceV modeled uponV₀, whereGL(V₀, J₀) denotes the subgroup ofGL(V0)consisting of all linear isomorphisms of V0 that commute withJ₀. Conversely, ifP is aGL(V₀, J₀)-structure on the vector space V then there exists a unique complex structureJ onV such that P = FR^c_V

0(V).

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An elementp ofFR^c_V₀(V) is called acomplex frame ofV. WhenV0 is equal to R²ⁿ endowed with its canonical complex structure, we writeFR^c(V) instead of FR^c_V₀(V).

Leth·,·i_V be a positive definite or indefinite inner product onV. Assume that J is anti-symmetric with respect toh·,·i_V, i.e.:

hJ(v), wi_V +hv, J(w)i_V = 0,

for allv, w ∈ V. Theunitary groupofV with respect toJ andh·,·i_V is defined by:

U(V, J,h·,·i_V) = O(V,h·,·i_V)∩GL(V, J).

We write alsoU(V)whenJ andh·,·i_V are fixed by the context. IfR²ⁿis endowed with the canonical complex structureJ0and the indefinite inner product:

(1.1.7) h(x, y),(x⁰, y⁰)i=

n−r

X

i=1

(x_ix⁰_i+y_iy⁰_i)

−

n

X

i=n−r+1

(x_ix⁰_i+y_iy⁰_i), x, y, x⁰, y⁰ ∈Rⁿ, of index2rthen the unitary groupU(R²ⁿ, J₀,h·,·i)will be denoted byU_r(R²ⁿ).

Given finite dimensional real vector spacesV0,V having the same dimension, and endowed respectively with indefinite inner productsh·,·i_V₀,h·,·i_V having the same index and complex structures J0 : V0 → V0, J : V → V anti-symmetric with respect toh·,·i_V₀,h·,·i_V respectively then we set:

FR^u_V₀(V) =

p∈FR^o_V₀(V) :p◦J0 =J ◦p .

The set FRû_V₀(V) is a U(V0, J0,h·,·i_V₀)-structure on the vector space V. Con- versely, ifP is aU(V0, J0,h·,·i_V₀)-structure on the vector spaceV then there exists a unique indefinite inner producth·,·i_V onV and a unique complex structure J :V →V anti-symmetric with respect toh·,·i_V such thatP = FRû_V₀(V). When V0 isR²ⁿendowed with its canonical complex structure and the indefinite inner product (1.1.7) we writeFRû(V)instead ofFRû_V₀(V).

Let us now define the natural morphisms of the category of sets endowed with G-structure.

DEFINITION1.1.13. LetX0be a set,Gbe a subgroup ofBij(X0)and letX, Y be sets endowed withG-structuresP andQrespectively. A mapf :X →Y is said to beG-structure preservingiff ◦pis inQ, for allp∈P.

REMARK1.1.14. Notice that iff ◦pis inQ forsomep ∈ P then the map f :X → Y isG-structure preserving; namely, every other element ofP is of the formp◦gwithg∈Gandf ◦(p◦g) = (f◦p)◦gis also inQ.

The composite of G-structure preserving maps is again a G-structure preserving map; moreover, everyG-structure preserving map is bijective and its inverse is also a G-structure preserving map (see Exercise 1.5). We denote by

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1.2. PRINCIPAL SPACES AND FIBER PRODUCTS 9

IsoG(X, Y⁰) the set of allG-structure preserving maps from X toY and we set Iso_G(X) = Iso_G(X, X).

EXAMPLE1.1.15. LetV₀,V,W be vector spaces having the same dimension and the same field of scalars. IfV andW are regarded respectively as sets endowed with theGL(V₀)-structuresFR_V₀(V)andFR_V₀(W) then a mapf : V → W is GL(V₀)-structure preserving if and only if f is a linear isomorphism. Assume thatV0, V and W are finite-dimensional real vector spaces endowed with inner products. If V and W are regarded as sets endowed with the O(V₀)-structures FR^o_V₀(V)andFR^o_V₀(W)respectively then a mapf :V → W isO(V0)-structure preserving if and only iff is a linear isometry.

Notice that ifV₀,V,W are vector spaces,Gis a subgroup ofGL(V₀) and if P ⊂ FRV0(V) and Q ⊂ FRV0(W) areG-structures on V andW respectively then every G-structure preserving map f : V → W is automatically a linear isomorphism.

EXAMPLE1.1.16. LetM0,M,N be differentiable manifolds withM andN both smoothly diffeomorphic toM0. If the setsM andNare endowed respectively with theDiff(M₀)-structuresDiff(M₀, M)andDiff(M₀, N)then a mapf :M → N isDiff(M0)-structure preserving if and only if it is a smooth diffeomorphism.

See Exercise 1.6 for more examples like 1.1.15 and 1.1.16.

EXAMPLE1.1.17. LetX0,Xbe sets,Gbe a subgroup ofGL(X0)andPbe a G-structure on the setX. If the model spaceX₀ is endowed with its canonicalG- structure (recall Example 1.1.5) then theG-structure preserving mapsf :X0→X are precisely the elements of theG-structureP, i.e.:

(1.1.8) Iso_G(X₀, X) =P.

EXAMPLE1.1.18. LetX₀ be a set,G,G⁰ be subgroups ofBij(X₀)such that G ⊂ G⁰, P, Q be G-structures on sets X, Y respectively and P⁰, Q⁰ be G⁰- structures that weaken respectivelyP andQ. If a mapf :X →Y isG-structure preserving then it is alsoG⁰-structure preserving, i.e.,Iso_G(X, Y)⊂Iso_G⁰(X, Y).

1.2. Principal spaces and fiber products

Principal spaces are the algebraic structures that will play the role of the fibers of the principal bundles, to be introduced later on Section 1.3. Principal spaces bare the same relation to groups as affine spaces bare to vector spaces. Recall that anaffine spaceis a nonempty setAendowed with a free and transitive action of the additive abelian group of a vector spaceV. The affine spaceA can be identified with the vector spaceV once a point ofA(aorigin) is chosen. In a similar way, a principal space is, roughly speaking, an object that becomes a group once the position of the unit element is chosen.

The name “principal space” comes from the idea that any set withG-structure can be obtained from a principal space through a natural construction that we call thefiber product. Fiber products will be studied in Subsection 1.2.1.

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DEFINITION 1.2.1. A principal space consists of a nonempty set P and a groupGacting freely and transitively onP on the right. We callGthestructural groupof the principal spaceP.

Observe that the condition that the action of G on P be free and transitive means that for everyp, p⁰ ∈Pthere exists a unique elementg∈Gwithp⁰ =p·g;

we say thatg carries p to p⁰. The unique element g ofGthat carries p top⁰ is denoted byp⁻¹p⁰. The operation:

P×P 3(p, p⁰)7−→p⁻¹p⁰ ∈G

is analogous to the operation ofdifference of pointsin the theory of affine spaces.

Notice that it’s thewhole expressionp⁻¹p⁰that has a meaning; for a general principal space, we cannot write justp⁻¹, although in most concrete examples the object p⁻¹is indeed defined (but it’snotan element of the principal spaceP).

EXAMPLE1.2.2. Any groupGis a principal space with structural groupG, if we letGact on itself by right translations.

EXAMPLE1.2.3. LetGbe a group andHbe a subgroup ofG. For anyg∈G, the left cosetgHis a principal space with structural groupH.

EXAMPLE1.2.4. Given a natural numbernand a setXwithnelements then the setBij(In, X)of all orderings ofX(recall Example 1.1.2) is a principal space with structural group Sn. More generally, ifX0 andX are sets having the same cardinality then the setBij(X₀, X)of allX₀-orderings ofX is a principal space with structural groupBij(X0).

EXAMPLE1.2.5. LetV be ann-dimensional real vector space. The setFR(V) of all frames ofV (recall Example 1.1.3) is a principal space with structural group GL(Rⁿ). More generally, ifV₀ andV are arbitrary vector spaces having the same dimension and the same field of scalars then the setFRV0(V)of allV0-frames of V is a principal space with structural groupGL(V₀).

In Exercise 1.8 the reader is asked to generalize Examples 1.2.4 and 1.2.5.

EXAMPLE1.2.6. LetX₀ be a set,Gbe a subgroup ofBij(X₀)andP be aG- structure on a setX. ThenP is a principal space with structural groupG. Notice that, sinceP = Iso_G(X₀, X) (see Example 1.1.17), we are again dealing with a particular case of the situation described in Exercise 1.8.

EXAMPLE 1.2.7. LetP andQ be principal spaces with structural groupsG andH respectively. The cartesian productP ×Qcan be naturally regarded as a principal space with structural groupG×H; the right action ofG×HonP×Q is defined by:

(p, q)·(g, h) = (p·g, q·h), for all(p, q)∈P×Qand all(g, h)∈G×H.

DEFINITION1.2.8. LetPbe a principal space with structural groupGand let H be a subgroup ofG. IfQ ⊂P is anH-orbit thenQis itself a principal space with structural groupH; we callQaprincipal subspaceofP.

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The result of Exercise 1.3 implies that the number of principal subspaces ofP with structural groupHis equal to the index ofHinG.

EXAMPLE1.2.9. LetX0 andXbe sets having the same cardinality and letG be a subgroup ofBij(X₀). The setBij(X₀, X)is a principal space with structural groupBij(X0); the principal subspaces ofBij(X0, X)with structural groupGare precisely theG-structures of X. IfP is aG-structure onXandHis a subgroup ofGthen theH-structures onXthat strengthenP are precisely the principal subspaces ofPwith structural groupH.

DEFINITION 1.2.10. Let P, Q be principal spaces with the same structural groupG. A mapt:P →Qis called aleft translationif

t(p·g) =t(p)·g,

for allp ∈ P and all g ∈ G. The set of all left translationst : P → Qwill be denoted byLeft(P, Q).

If we think of the structural groupGas being the group ofright translationsof a principal space, then left translations are precisely the maps that commute with right translations.

Notice that the composite of left translations is again a left translation; moreover, a left translations is always bijective and its inverse is also a left translation (see Exercise 1.9). Ift : P → P is a left translation from a principal spaceP to itself, we say simply thattis aleft translation ofP. The setLeft(P, P)of all left translations of P is a group under composition and it will be denoted simply by Left(P).

EXAMPLE 1.2.11. IfP is a principal space with structural groupGthen for allp∈P the mapβp :G→ P of action on the elementp(recall (1.1.4)) is a left translation that carries the unit element1∈Gtop∈P.

We think informally of βp as being the identification between the principal spaceP and the structural groupGthat arises by declaringp ∈ P to be the unit element; this is analogous to the identification between an affine space and the corresponding vector space that arises by declaring a point of the affine space to be the origin.

Let us compare the identifications βp and β_p⁰ of G with P that arise from different choices of pointsp, p⁰ ∈P. Ifg=p⁻¹p⁰ is the element ofGthat carries ptop⁰ then we have the following commutative diagram:

(1.2.1)

G _β

p

$$H

HH HH H

L_g−1

P G ^β^p⁰

::v

vv vv v

Diagram 1.2.1 says that two different identifications of a principal space P with its structural groupGdiffer by a left translation ofG. This is the same that happens in the theory of affine spaces: two different choices of origin for an affine

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spaceAgive identifications with the corresponding vector spaceV that differ by a translation ofV. Obviously, since the additive group of a vector space is abelian, there is no distinction between right and left translations in the theory of affine spaces and vector spaces.

A left translation is uniquely determined by its value at a point of its domain.

More explicitly, we have the following:

LEMMA1.2.12. LetP,Qbe principal spaces with the same structural group G. Givenp∈P,q ∈Qthen there exists a unique left translationt∈Left(P, Q) witht(p) =q.

PROOF. Clearly t = βq ◦ β_p⁻¹ is a left translation from P to Q such that t(p) = q (see Example 1.2.11). To prove uniqueness, lett1, t2 ∈ Left(P, Q) be given witht₁(p) =t₂(p); then:

t1(p·g) =t1(p)·g=t2(p)·g=t2(p·g),

for allg∈G, so thatt1 =t2.

Lemma 1.2.12 implies that the canonical left action of the group of left trans- lationsLeft(P)on P is free and transitive. Givenp, p⁰ ∈ P then the unique left translationt∈Left(P)witht(p) =p⁰is denoted byp⁰p⁻¹.

EXAMPLE 1.2.13. If G is a group then the left translations of the principal spaceG(recall Example 1.2.2) are just the left translations of the groupG, i.e., the mapsL_g :G→Gwithg∈G. Namely, the associativity of the multiplication ofG implies that the mapsLgare left translations of the principal spaceG; conversely, if t : G → G is a left translation of the principal space Gthen Lemma 1.2.12 implies thatt=Lg, withg=t(1). Thus:

Left(G) =

Lg :g∈G .

Obviously the mapg7→Lggives an isomorphism from the groupGonto the group Left(G)of left translations ofG.

EXAMPLE 1.2.14. We have seen in Example 1.2.11 that if P is a principal space with structural groupGthen the mapsβ_p :G→ P,p∈ P are left translations. It follows from Lemma 1.2.12 that these are in fact theonlyleft translations fromGtoP, i.e.:

Left(G, P) =

βp:p∈P .

EXAMPLE1.2.15. LetP,Qbe principal spaces with the same structural group G and letp ∈ P, q ∈ Q be fixed. Ift : P → Qis a left translation then the composition β_q⁻¹ ◦t◦β_p : G → G is also a left translation and therefore, by Example 1.2.13, there exists a uniqueg∈Gwithβ_q⁻¹◦t◦βp =Lg. This situation is illustrated by the following commutative diagram:

(1.2.2)

P ^t ^//Q

G _L

g

//

βp ∼=

OO

G

βq

∼=

OO

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We see that a choice of elementsp ∈ P, q ∈ Qinduces a bijection between the setLeft(P, Q)and the groupG; such bijection associates to eacht ∈Left(P, Q) the elementg ∈ Gthat makes diagram (1.2.2) commutative. WhenP = Q, the bijection just described betweenLeft(P)andGis an isomorphism of groups and we will denote it byI_p. More explicitly, for eachp∈P we define the mapI_p by:

(1.2.3) I_p :G3g7−→βp◦Lg◦β_p⁻¹ ∈Left(P).

We see that the group of left translationsLeft(P) is isomorphic to the structural groupG(the group of right translations ofP), but the isomorphism is in general not canonical: it depends on the choice of an elementp ∈ P. Forp, p⁰ ∈ P, the group isomorphismsI_pandI_p⁰ differ by an inner automorphism ofG; namely, the following diagram commutes:

(1.2.4)

G Ip

((P

PP PP P

I_g−1

Left(P) G ^I^p⁰

66n

nn nn n

whereg=p⁻¹p⁰is the element ofGthat carriesptop⁰.

REMARK1.2.16. LetP be a principal space with structural groupGand let g ∈ Gbe fixed. If P is identified withGby means of the mapβ_p : G → P for some choice ofp∈ P, then the mapγg :P →P given by the action ofg(recall (1.1.5)) is identified with the mapR_g : G → Gof right translation byg; more explicitly, we have a commutative diagram:

P ^γ^g ^//P

G _R

g

//

βp ∼=

OO

G

∼= βp

OO

We could also identify the domain ofγ_gwithGviaβ_pand the counter-domain ofγg withGvia βp·g; this yields an identification ofγg with the inner automor- phismI_g−1 ofG, which is illustrated by the commutative diagram:

(1.2.5)

P ^γ^g ^//P

G _I

g−1

//

βp ∼=

OO

G

∼= βp·g

OO

EXAMPLE1.2.17. LetV0,V,W be vector spaces having the same dimension and the same field of scalars. Given a linear isomorphismT : V → W then the map

T∗ : FR_V₀(V)−→FR_V₀(W)

given by composition with T on the left is a left translation. Moreover, every left translation t : FR_V₀(V) → FR_V₀(W) is equal to T∗ for a unique linear

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isomorphism T : V → W. To prove this, choose any p ∈ FRV0(V) and let T :V → W be the unique linear isomorphism such thatT ◦p =t(p); it follows from Lemma 1.2.12 that t = T∗. We conclude that the rule T 7→ T∗ defines a bijection from the set of linear isomorphisms T : V → W onto the set of left translationst: FR_V₀(V)→FR_V₀(W). IfV =W, we obtain a bijection:

(1.2.6) GL(V)3T 7−→T∗∈Left FRV0(V)

;

such bijection is in fact a group isomorphism. We will therefore, from now on,always identify the groupsGL(V)andLeft FR_V₀(V)

via the isomorphism(1.2.6).

Notice that, under such identification, for any givenp ∈ FR_V₀(V), the isomor- phismI_p : GL(V0)→Left FRV0(V)∼= GL(V)is given by:

(1.2.7) I_p(g) =p◦g◦p⁻¹∈GL(V), g∈GL(V₀).

It may be instructive to solve Exercise 1.15 now.

EXAMPLE1.2.18. LetX₀ be a set,Gbe a subgroup ofBij(X₀)andP,Qbe respectively aG-structure on a setX and aG-structure on a setY. ThenP and Qare principal spaces with structural groupG. Iff : X → Y is aG-structure preserving map then the mapf∗ :P →Qgiven by composition withf on the left is a left translation. Arguing as in Example 1.2.17, we see that every left translation fromPtoQis of the formf∗for a uniqueG-structure preserving mapf :X→Y; in other words, the map:

IsoG(X, Y)3f 7−→f∗∈Left(P, Q) is a bijection. Moreover, forX=Y,P =Q, the map:

(1.2.8) Iso_G(X)3f 7−→f∗ ∈Left(P)

is a group isomorphism. We will from now onalways identify the groupsIsoG(X) andLeft(P)via the isomorphism(1.2.8).

In Exercises 1.11 and 1.12 the reader is asked to generalize the idea of Exam- ples 1.2.17 and 1.2.18 to a more abstract context.

IfP andQ areG-structures on setsX andY respectively, H is a subgroup ofGand P⁰,Q⁰ are H-structures that strengthen respectively P andQthen the set Left(P, Q) is identified with the set Iso_G(X, Y) and the set Left(P⁰, Q⁰) is identified with the setIsoH(X, Y). SinceIsoH(X, Y)is a subset ofIsoG(X, Y), we should have an identification ofLeft(P⁰, Q⁰)with a subset ofLeft(P, Q). This is the objective of our next lemma.

LEMMA1.2.19. LetP,Qbe principal spaces with structural groupGand let P⁰ ⊂ P, Q⁰ ⊂ Qbe principal subspaces with structural groupH ⊂ G. Then every left translationt:P⁰ →Q⁰extends uniquely to a left translation¯t:P →Q.

The map:

(1.2.9) Left(P⁰, Q⁰)3t7−→¯t∈Left(P, Q) is injective and its image is the set:

(1.2.10)

s∈Left(P, Q) :s(P⁰)⊂Q⁰ .

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Moreover, if P = Q and P⁰ = Q⁰ then the map (1.2.9) is an injective group homomorphism and therefore its image(1.2.10)is a subgroup of Left(P).

PROOF. Let t ∈ Left(P⁰, Q⁰) be given and choose any p ∈ P⁰; then, by Lemma 1.2.12, there exists a unique left translation¯t:P →Qwith¯t(p) =t(p).

For anyg ∈ H we havet(p¯ ·g) = ¯t(p)·g = t(p)·g = t(p·g), which proves that¯tis an extension oft; clearly,¯tis the unique left translation that extendst. We have thus established that the map (1.2.9) is well-defined; obviously, such map is injective and its image is contained in (1.2.10). Given anys ∈ Left(P, Q) with s(P⁰)⊂Q⁰ then the mapt:P⁰ →Q⁰obtained by restrictingsis a left translation and thuss= ¯t. This proves that the image of (1.2.9) is equal to (1.2.10). Finally, if P =Q,P⁰ =Q⁰andt1, t2 ∈Left(P⁰)then¯t1◦¯t2is a left translation that extends t₁◦t₂; hencet₁◦t₂= ¯t₁◦¯t₂and (1.2.9) is a group homomorphism.

Under the conditions of the statement of Lemma 1.2.19,we will from now on always identify the setLeft(P⁰, Q⁰)with the subset(1.2.10)ofLeft(P, Q)via the map (1.2.9). In particular, the group Left(P⁰) is identified with a subgroup of Left(P). Under such identification, the canonical left action ofLeft(P⁰)onP⁰ is identified with the restriction of the canonical left action ofLeft(P)onP. Observe also that the identification we have made here is consistent with the identifications made in Example 1.2.18. More explicitly, if P and Q are G-structures on sets X and Y respectively, H is a subgroup of G and P⁰, Q⁰ are H-structures that strengthen respectivelyP andQthen the following diagram commutes:

(1.2.11)

IsoG(X, Y) ^f^7→f∼= ^∗ //Left(P, Q)

IsoH(X, Y)

inclusion

OO

f7→f∗

∼= //Left(P⁰, Q⁰)

(1.2.9)

OO

In Exercise 1.24 the reader is asked to generalize Lemma 1.2.19.

REMARK1.2.20. LetP be a principal space with structural groupG; for each p ∈P, we have an isomorphismI_p :G→ Left(P)(recall (1.2.3)). For the sake of this discussion, let us writeI_p^P instead of justI_p. IfQis a principal subspace of P with structural groupH⊂Gthen for eachp∈Qwe also have an isomorphism I_p^Q : H → Left(Q). For a fixed p ∈ Q, we have the following commutative diagram:

(1.2.12)

Left(Q) ^t7→¯^t ^//Left(P)

H

I^Q_p ^∼=

OO

inclusion //G

∼= I_p^P

OO

This means that, identifyingLeft(Q)with a subgroup ofLeft(P)then the isomor- phismI_p^Qis just a restriction of the isomorphismI_p^P.

In Exercise 1.25 the reader is asked to generalize this.

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DEFINITION1.2.21. LetG,Hbe groups,P be aG-principal space andQbe an H-principal space. A mapφ : P → Qis said to be amorphism of principal spacesif there exists a group homomorphismφ0:G→Hsuch that:

(1.2.13) φ(p·g) =φ(p)·φ₀(g),

for allp∈P and allg∈G. We callφ₀thegroup homomorphism subjacent to the morphismφ.

The fact that the action ofHonQis free implies that mapφ0 :G→ Hsuch that equality (1.2.13) holds for allp∈P,g∈Gis unique.

The compositionψ◦φof morphisms of principal spacesφandψwith subjacent group homomorphismsφ₀andψ₀is a morphism of principal spaces with subjacent group homomorphismψ₀◦φ₀(see Exercise 1.16). A morphism of principal spaces φis bijective if and only if its subjacent group homomorphismφ0is bijective (see Exercise 1.17). A bijective morphism of principal spaces is called anisomorphism of principal spaces. If φis an isomorphism of principal spaces with subjacent group homomorphismφ₀thenφ⁻¹is also an isomorphism of principal spaces with subjacent group homomorphismφ⁻¹₀ (see Exercise 1.18).

EXAMPLE1.2.22. IfPis a principal space with structural groupGandQ⊂P is a principal subspace with structural groupH ⊂Gthen the inclusion map from QtoP is a morphism of principal spaces whose subjacent group homomorphism is the inclusion map fromHtoG.

There is a natural notion of quotient of a principal space and the quotient map is another example of a morphism of principal spaces. See Exercise 1.21 for the details.

EXAMPLE1.2.23. IfP,Qare principal spaces with the same structural group Gthen the left translationst : P → Qare precisely the morphisms of principal spaces whose subjacent group homomorphism is the identity map ofG.

1.2.1. Fiber products. IfXis a set endowed with aG-structure then the set of allG-structure preserving maps from the model spaceX0toXis a principal space with structural group G (recall Example 1.2.6). Thus, to each set X endowed with a G-structure there corresponds a principal space with structural group G.

The notion of fiber product that we study in this subsection provides us with a construction that goes in the opposite direction.

Before we give the definition of fiber product, we need the following:

DEFINITION 1.2.24. LetGbe a group. By aG-spacewe mean a setN endowed with a left action ofG. The subgroupG_efofBij(N)given by the image of the homomorphismG3g7→γg∈Bij(N)corresponding to the action ofGonN is called theeffective groupof theG-spaceN.

LetGbe a group,P be a principal space with structural groupGandN be a G-space. We have a left-action ofGon the cartesian productP×N defined by:

(1.2.14) g·(p, n) = (p·g⁻¹, g·n),

The theory of connections and G-structures. Applications to afﬁne and isometric immersions Paolo Piccione Daniel V. Tausk