Differential Geometric Structures

By Walter A. Poor

Useful for independent study and as a reference work, this introduction to differential geometry features many examples and exercises. It defines geometric structure by specifying the parallel transport in an appropriate fiber bundle, focusing on the simplest cases of linear parallel transport in a vector bundle.
The treatment opens with an introductory chapter on fiber bundles that proceeds to examinations of connection theory for vector bundles and Riemannian vector bundles. Additional topics include the role of harmonic theory, geometric vector fields on Riemannian manifolds, Lie groups, symmetric spaces, and symplectic and Hermitian vector bundles. A consideration of other differential geometric structures concludes the text, including surveys of characteristic classes of principal bundles, Cartan connections, and spin structures.

• Language: English
• Publisher: Dover Publications
• Release date: Apr 27, 2015
• ISBN: 9780486151915

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Index

PREFACE

This text is intended to serve as an introduction to differential geometry. The prerequisites are the basic facts about manifolds as presented in the books by Auslander and Mackenzie, Boothby, Brickell and Clark, Hu, Lang, Matsushima, Milnor, Singer and Thorpe, or Warner; with extra work, the material from some modern advanced calculus books (for example, Goffman, Loomis and Sternberg, or Spivak) would also be sufficient. For the sake of uniformity almost all references to the elementary concepts of manifolds will be to Warner.

For example, a sample first course in differential geometry could cover the first two chapters of Warner and the first part of this book.

Later sections of the book make heavier demands on manifold theory, for which the reader is again usually referred to Warner. For example, a second geometry course could proceed through the third chapter of this book, Chapters 3 and 5 from Warner, and Chapter 4 of this book (with Warner’s last chapter for reference). Similarly, a course on the basic geometry of Lie groups and symmetric spaces could use Chapter 4 of Warner and Chapters 6 and 7 of this book. Other courses are also possible.

The book should also be useful for independent study and as a reference work for mathematicians and physicists who need the basic ideas from differential geometry.

The book is in no way intended to be encyclopedic; for example, variational theory and submanifolds are not included, although they are fairly standard topics in first-year courses. For such topics the reader is referred to [BC], [CE], [GKM], [Hi], [KN], [Spi], or [Sb] in the bibliography.

Some comments on the more unusual features of the book follow.

In 1917 Levi-Civita interpreted the classical Ricci tensor calculus as an analytic description of a geometric concept which he called parallel transport. É. Cartan reversed the process in his study of affine, conformal, and projective differential geometry by postulating the existence of some system of parallel transport between nearby fibers of an appropriate fiber bundle, and then deriving the differential equations for parallel transport; these were written in terms of local connection forms. Since fiber bundles were still to be defined, Cartan’s initial description of parallel transport was necessarily heuristic, and seems to have been taken by most people as an intuitive guideline, rather than as a rigorous starting point. For this reason, subsequent research either continued with the classical tensor calculus, or else emphasized the connection forms as the fundamental concept.

Once a satisfactory theory of fiber bundles had been developed, everything was reinterpreted in terms of connections on fiber bundles, global connection forms on principal fiber bundles, index-free covariant differentiation, and sprays.

In this book Cartan’s viewpoint is taken: a geometric structure is defined by specifying the parallel transport in an appropriate fiber bundle (for a comparison of this approach with that of Riemann and Klein, see [Ca5]). Parallel transport is defined axiomatically, and everything else is then derived from it. Most of the book is devoted to the simplest case—linear parallel transport in a vector bundle. Only after the reader has come to grips with this case is parallel transport considered in more general fiber bundles.

The axioms for parallel transport in this book are my attempt to state carefully the heuristic comments in Section 2.1 of the book Riemannsche Geometrie im Grossen, by Gromoll, Klingenberg, and Meyer.†

The book begins with a study of fiber bundles. In Chapter 2, parallel transport in vector bundles is considered as an abstraction of parallelism of vectors in Euclidean space. After the statement of the axioms and some examples, holonomy is looked at briefly. The connection (as a horizontal distribution) is then defined, and connections and parallel transport are shown to be equivalent concepts. More study of holonomy leads to the curvature tensor. Only then is parallel transport rephrased in terms of covariant differentiation, which is the main computational tool used throughout the book; it is used to study the space of connections on a vector bundle, and for an introduction to characteristic classes using the exterior covariant derivative operator. The chapter closes with the special case of the tangent bundle and Cartan’s generalization, the affine tangent bundle.

Chapter 3 covers Riemannian vector bundles, the Levi-Civita connection, curvature, and the metric space structure of a Riemannian manifold; it concludes with Chern’s proof of the Gauss-Bonnet theorem.

The fourth chapter covers the Laplacian and basic harmonic theory. Weitzenböck’s formula for the Laplacian is followed by Chern’s much less familiar formula; so far as I know this has not appeared in any other differential geometry book (in fact the only references I know for it are Chern’s original paper, Weil’s Séminaire Bourbaki lecture notes, and my Proceedings note). Chern’s formula is used to give a simple proof of D. Meyer’s positive curvature operator theorem.

Chapter 5 covers harmonic, Killing, conformal, affine, and projective vector fields; the conformal vector fields are described on the sphere, thought of as the so-called Möbius space.

Lie groups appear often in the early chapters as examples; their basic geometry is explored further in Chapter 6. The difference between positive and negative curvature in the study of Lie groups is exemplified by the fact that although S¹ and S³ are the only spheres which are Lie groups (a geometric proof is presented), hyperbolic space is a Lie group in each dimension. The classical simple Lie groups are studied in some detail, as are the spinor groups. The chapter concludes with the basic geometry of homogeneous spaces.

Chapter 7 introduces affine symmetric spaces, which are then described via symmetric pairs and Loos’ axioms for the affine symmetries; the geodesic spray is derived directly from these axioms. This is followed by a discussion of locally affine symmetric spaces, symmetric Lie algebras, and Riemannian symmetric spaces.

Chapter 8 starts with symplectic vector bundles, shows that they are related to Hermitian vector bundles, and then covers enough complex manifold theory for an introduction to Kähler manifolds.

The last chapter deals with parallel transport, first in principal fiber bundles, and then in associated bundles for which the fiber has a prescribed geometric structure. Cartan connections are used to reinterpret affine and conformal differential geometry. The final topic is spin structures, culminating in Lichnerowicz’ harmonic spinors theorem.

There are many examples and exercises scattered throughout the text; the reader is encouraged at least to read through each exercise before proceeding further. Because of space considerations, some of the examples must also be considered as exercises.

The book is based in part on a course which I taught at the University of Bonn in 1975 to 1976; I would like to thank the Sonder-forschungsbereich 40 at the University of Bonn for its support. Thanks are also due to Professors Hirzebruch and Klingenberg for inviting me to Bonn, and to the students at Bonn; among other things, the students showed me the geometric proof given in Chapter 5 that S¹ and S³ are the only spheres which are Lie groups.

The book was begun while I was supported by grant NSF No. MCS72 05055 A04 at the Institute for Advanced Study; my thanks to the National Science Foundation for its support, and to Professor Milnor for inviting me to the Institute. Further support was provided by a Mellon grant while I was at Skidmore College; for this I thank the Mellon Foundation and Dean Weller. The book was completed during a visit to Rensselaer Polytechnic Institute; I would like to thank Professor DiPrima for the invitation, and Rensselaer for the congenial working atmosphere.

It is a pleasure to express my deep gratitude to Jean-Pierre Bourguignon and Wolfgang Ziller, who read preliminary versions of the typescript and made innumerable penetrating, helpful suggestions (and corrections).

Others who are to be thanked for their comments on parts of the typescript are Glen Castore, Jeff Cheeger, David Elliot, Robert Greene, I. M. James, Michio Kuga, Robert Maltz, Charles Marshall, John Milnor, Katsumi Nomizu, Phillip Parker, John Thorpe, Frank Warner, and Steve Wilson.

Fred Cohen and Ravindra Kulkarni deserve special thanks; the former convinced me to write the book, and the latter read the finished product.

It is appropriate to thank Leonard Charlap for telling me to study parallel transport rather than covariant derivatives, Takushiro Ochiai and Tsunero Takahashi for explaining to me what I was doing on principal bundles, Ernst Heintze for helping me with Lie algebras, Seiki Nishikawa for explaining Cartan connections to me, and Nigel Hitchin and Jacques Tits for teaching me about spinors. Allen Adler and Edward Spitznagel have answered many questions on all sorts of topics.

Especially important to me has been the general encouragement I have received from Marcel Berger and Shiing-Shen Chern, and also from my wife Ellen Sara.

Finally I would like to thank my teachers—John Brillhart, Jeff Cheeger, David Ebin, Detlef Gromoll, Irwin Kra, Jim Simons, and John Thorpe.

Irene Abaganale and Peggy Murray typed the first draft for me, Rosanne Hammond and JJ Williams helped on the second draft, and my daughter Nureet helped me type the final draft of the book. Carolyn Boyce proofread it with me. To all these people I say thank you.

Most of the basic notation is the same as in Warner’s book; one difference the reader should be aware of at the beginning has to do with the differential of a C∞ map. If f : M → N is a C∞ map of manifolds, then f*: TM → TN denotes the induced tangent map (or differential), and f*: T*N → T*M denotes the induced cotangent map. For each p ∈ M, the tangent space at p is denoted by Mp, and the tangent map induced by f at p is f* |p : Mp → Nf(p).

on the real line is often denoted by D : f′ = Df for f ∈ C.

n are usually written as row vectors except when they must be written as column vectors for matrix multiplication. The transpose of a matrix A is denoted by tA.

Walter A. Poor

† Gromoll later told me that those comments were based on axioms developed by Dombrowski, who in turn kindly informed me of prior axioms given by Rinow.

CHAPTER

ONE

AN INTRODUCTION TO FIBER BUNDLES

Fiber bundles constitute an important generalization of the product of two topological spaces; locally a fiber bundle is the product of two given topological spaces. For differential geometric purposes, the appropriate definition must be in the category of C∞ manifolds.

THE DEFINITION OF A FIBER BUNDLE

1.1Notation Given C∞ manifolds M1 and M2, the topological product M1 × M2 is naturally a C∞ manifold [W: 1.5(g); 1, exercise 24]. Denote the C∞ projection maps of M1 × M2 onto M1 and M2 by pr1 and pr2, respectively.

1.2Definition A C∞ fiber bundle consists of:

1.3Convention A precisely specified collection of bundle charts on a fiber bundle is not nearly so important as the fact that bundle charts exist relative to some open covering of M. For example, suppose that V is an open subset of M, and that α is a diffeomorphism from π–1V to V × F such that the diagram

. For all practical purposes, the pair (V, (π, ψ)) might as well be a bundle chart of E over M. Any collection of bundle charts on E over M relative to an open covering of M will be called a bundle atlas on E. In particular, the union of two bundle atlases on E over M is a new bundle atlas on E over M; in this case, the two original fiber bundles on E over M will be identified with the fiber bundle determined by the union of the two original bundle atlases.

Alternatively, we could require that a bundle atlas be maximal, but this is not necessary for most purposes.

1.4Definition For each p in Mof the total space E of a fiber bundle over M is called the fiber of π (or the fiber of E) over p; the projection π is sometimes called a C∞ fibration of E. Very often the adjective C∞ will be deleted.

1.5Proposition The projection map π: E → M of a C∞ fiber bundle is a submersion, that is, for each point ξ in E, the induced tangent map π*|ξ: Εξ → Μπ(ξ) is surjective; furthermore, for each p ∈ M, the fiber Ep = π–1p in E over p is an embedded submanifold diffeomorphic to the standard fiber F of the bundle.

PROOF Let (π, ψ) be a bundle chart on E over an open set U in M. For each (p, υ) ∈ U × F, pr1*|(p, υ) maps the tangent space (U × F)(p, υ) onto Mp = Up, so pr1 is a submersion of U × F onto U. Since (π, ψ)* maps Τ(π–1U) diffeomorphically onto T(U × F), π* = pr1* ∘ (π, ψ)* is then surjective at each ξ ∈ π–1U; thus π is a submersion. By the implicit function theorem [W: 1.38], each fiber of π is naturally an embedded submanifold of E. For each p ∈ U, (π, ψ) maps Ep diffeomorphically onto the embedded submanifold {p} × F of U × F, and pr2 is a diffeomorphism from this onto F.

If (π, ψ) is a bundle chart on E over U in M, and if p ∈ U, then (ψ|Ep)–1 is a diffeomorphism from F to Ep in E; therefore the bundle E over M with fiber F . More often, E is used by itself.

1.6Let Diff(F) denote the diffeomorphism group of F under composition.

Assume that (π, φ) and (π, ψ) are bundle charts on E over overlapping open sets U and V in M, respectively. The restrictions of (π, φ) and (π, ψ) to π–1(U ∩ V) are also bundle charts on E by 1.3; a priori there is no reason for them to agree, but their difference is easily measured in Diff(F). For each p ∈ U ∩ V, φ and ψ map the fiber Ep diffeomorphically to F; thus φ ∘ (ψ|Ep)–1 is a diffeomorphism of F; the result is a map

satisfying the relations

where · denotes the action of Diff(F) on F. Furthermore, if (π, η) is a bundle chart on E over W in M with U ∩ V ∩ W nonempty, then

Definition Given bundle charts (π, φ) and (π, ψ) over open sets U and V in M with U ∩ V ≠ Ø, the map fφ, ψ: U ∩ V → Diff(F) is called the transition function from ψ to φ.

1.7Comments Let (π, ψ) be a bundle chart on E over U in M. For each C∞ manifold chart x on V ⊂ U in M and each C∞ chart y on W in F, (x ∘ π, y ∘ ψ) is a C∞ manifold chart on an open set in E; in fact, by using charts on M and F this way, one obtains an atlas of manifold charts on E. Thus a bundle atlas on E generates the C∞ structure on the manifold E. In practice, this means that one often starts with a surjective map π from a set E to a manifold M, then defines a bundle atlas on E, and finally defines a C∞ structure on E by checking that the bundle charts on E are C∞-compatible; only then is E a manifold, and therefore eligible for use as the total space of a C∞ fiber bundle over M (see also 1.12i and j).

1.8A fiber bundle is locally a product manifold; a bundle chart on E exhibits this property by inducing a local C∞ product structure on E over an open subset of M. The local product structures induced on E over an open set in M by different bundle charts can differ greatly because the group Diff(F) is so big. For a fiber bundle to be of use in elementary differential geometry, one usually restricts severely the choice of possible transition functions; these restrictions are the content of the next three definitions. Not all fiber bundles allow these restrictions [Om].

1.9Definition A Lie group G is said to be a Lie transformation group on a manifold F if there is given a C∞ left action of G on F, that is [W: 3.44], a C∞ map μ: G × F → F such that

for all g, h ∈ G, where e is the identity element of G. Such an action μ can be interpreted as a homomorphism from G to Diff(F): for each g ∈ G, μg is the element of Diff(Ffor all ξ ∈ F.

1.10Definition Let G be a Lie group acting on F on the left. Bundle charts (π, φ) and (π, ψ) on E over U and V in M, respectively, will be said to be G-compatible if U ∩ V = Ø, or if U ∩ V ≠ Ø and there exists a C∞ map g from U ∩ V to G such that fφ,ψ(p) = μg(p) ∈ Diff(F) for all p ∈ U ∩ V. In this case we shall identify fφ,ψ with g, and consider fφ,ψ as a C∞ map from U ∩ V to G; this yields the identity fφ,ψ(p) · ξ = μ(fφ,ψ(p), ξ), p ∈ U ∩ V ξ ∈ F, where μ is the action of G on F.

1.11Definition Let G be a Lie group. A Cis called a C∞ fiber bundle with structure group G if G is a Lie transformation group on Fon E are G-compatible. A C∞ fiber bundle chart is admissible as a bundle chart on the bundle E with structure group G if and only if it is G-compatible with every element of the given G-bundle atlas.

Although G will be referred to as the structure group of the bundle, it must be emphasized that G is not the only possible choice of structure group. For example, let a Lie group H act trivially on F; the Lie group G × H is then a structure group for E in a natural way. More important, often a subgroup of G can be chosen as structure group of E, just as G itself is homomorphic to a subgroup of Diff(F); there will be many examples of this phenomenon throughout the book. If a subgroup of G is specified as the structure group of E, the choice of admissible bundle charts is reduced because of the new restrictions on the transition functions. This process is called reducing the group of the bundle.

1.12Examples

(a)Trivial bundles The product M × N of manifolds M and N . The trivial group G = {idN} can be chosen as structure group, so the bundle is called trivial Similarly, M × N . These bundles are naturally trivialized. A bundle chart on an arbitrary fiber bundle trivializes the bundle locally; in general, there is no natural local trivialization.

(b)The Möbius strip The circle S/~, where for x and n , x ~ x + 2πn× (–1, 1) by (x, t(x + 2πn, (–1)nt), n . The result is the familiar Möbius strip. Define Π: E → S, where the brackets denote the equivalence classes. Bundle charts (Π, φ) and (Π, ψ) are defined on E . The group of E 2. Exercise: Exhibit the Klein bottle as a bundle over S¹ with fiber S¹.

(c)The tangent bundle of a manifold [W: 1.25] Let M be a C∞ manifold. The union over all p ∈ M of the tangent space Mp on M at p is the total space TM of the tangent bundle of M; the projection map sends Mp to p. It follows from the definition of tangent vectors that if U in M is open, then Up can be canonically identified with M p for all p ∈ U. Hence if π is the projection of TM to M, then n–1U and TU are canonically identified.

The abbreviation Mp will be used for the fiber in TM at p, rather than the symbol TMp. Caution: if E is the total space of a fiber bundle over M, then the symbol Εξ must be read carefully. If ξ ∈ M, then Εξ is the fiber in E over ξ, while if ξ ∈ E, then Εξ is the tangent space on E at ξ.

The standard fiber of TM n, n = dim M.

There is a slight discrepancy between the usual bundle charts on TM [W: 1.25] and bundle charts in general, as defined in 1.2. Given a chart x = (x¹, …, xn): U n on M, the associated tangent bundle chart on π–1 U = TU is the induced tangent map of x:

Replacing x* by (π, dx): TU → U n we obtain a bundle chart according to the definition in 1.2. Since x is a diffeomorphism between U and x(Un, the difference between the two versions of the bundle chart is purely formal.

As the structure group of TM we can choose GL(n). Proof: Given charts x and y near p ∈ M, the value at p of the transition function from y* to x* is fx,y(p) = dx ∘ (dy|Mp)–1 = [∂xi∂yj](p), which is the usual Jacobian matrix of x and y at p. Therefore fx,y is a C∞ map from a neighborhood of p to GL(n). QED.

(d)Subbundles Let M and N be manifolds; call (M, N) a manifold pair if N is a submanifold of M. Suppose that (F1, F2), (E1, E2) and (M1, Mis a fiber bundle, i = 1, 2. Call E2 a subbundle of E1 if the following condition is satisfied for each bundle chart (π2, φ) on E2 over an open set U in M2: given p ∈ U, there exists an open neighborhood V of p in M1 and a bundle chart (π1, ψ) on E1 over V such that

Not all bundle charts on E1 restrict to bundle charts on E2 if F1 ≠ F2, for there are diffeomorphisms of F1 which do not map F2 to F2.

for each submanifold X of M; if E is trivial over a neighborhood of X, then π–1X is a trivial bundle.

(e)The universal line bundle over a projective space Let , and fix n > 0. The projective space of dimension n modulo the equivalence relation ~ such that u ~ υ such that u = cυ . For j = 0, …, n, define a chart zj on the open set

by

where ˆ indicates an entry to be deleted. The charts (zj, Uj) generate the C(see [W: 1.5(c)] for the C∞ structure on Cn); the differentiability of the transition function from a chart zk to a chart zj,

The denominators are nonzero because ξ ∈ zk(Uj). This map is obviously C∞.

will be bundle charts once it is checked that they generate a C.

Let p ∈ Uj ∩ Uk, j ≠ k; by the definition of Uk, υk ≠ 0. For all

so

Thus, fψj, ψk (pfurthermore, the map

determine a C).

,

is a C[W: 1.6].

Now that the Chas been shown to be C. Since each transition function fψj,ψk maps Uj ∩ Uk .

is canonically a subbundle of a bundle E ; calling the new point (0, p) instead of just 0 is a technical device to guarantee that distinct fibers in the new bundle have distinct zero points. Set E of Epand ψj to E .

Exercise are bundle charts on E over the sets Uj is a subbundle of E.

is called the universal line bundle (see [Ch5: 6, exercise 2], [Ws: 1.2.6]). Topologists call this the canonical line bundle, but in geometry that term usually refers to something else.

Exercises to the universal k-plane bundle over the Grassmann manifold of k.

(f)Product bundles If Fj → Ej → Mj is a fiber bundle, j = 1, 2, then F1 × F2 → E1 × E2 → M1 × M2 is also a fiber bundle. For example, TM1 × TM2 is the tangent bundle of M1 × M2. The structure group of E1 × E2 is the product of the structure groups of the Ej.

(g)The pullback of a bundle Suppose that h: N → M is a Cbe a fiber bundle. By example (a), N × E is a fiber bundle over N with fiber E. The map h now determines a subbundle of N × E, which in most cases is far more interesting than the original bundle N × E. Set

project h*E onto N by the map pr1|h*E, which will also be denoted by pr1. Similarly, denote the restriction of pr2 to h*E by pr2.

The fiber of h*E at p ∈ N is (h*E)p = {p} × Eh(p), which is diffeomorphic to Eh(p) under pr2 ; thus the standard fiber of h*E is F.

Assume that (π, ψ) is a local trivialization of E over an open set U in M; the subset (h ∘ pr1)–1U of h*E is a trivial bundle over the open set h–1U in N; in fact, a bundle chart on h*E over h–1U is the map (pr1, ψ ∘ pr2). The bundle h*E is called the pullback of E by the map h. It will appear frequently throughout the book.

Exercises Prove that h*E over N is a subbundle of N × E over N. If the structure group of E is a Lie group G, show that the structure group of h*E is a Lie subgroup of G. Since h*E is a submanifold of N × E, its tangent bundle is a submanifold of T(N × F); prove that T(h*E) = {(u, υ) ∈ TN × TE|h*u = π*υ}.

(h)Composite bundles Assume are fiber bundles. Composition of the projection maps yields the composite fiber bundle . If (π, φ) is a bundle chart on E over U ⊂ M and (π1, ψ) is a bundle chart on E1 over V ⊂ E, then the map (π ∘ π1, φ ∘ π1, ψ) is a bundle chart on E1 over W. The transition function from the map β = (φ ∘ π1, ψ) to the map α = (η ∘ π1, μ) satisfies the rule

for (ζ, ξ) ∈ F × F1 and appropriate p ∈ M.

For example, the composite of TM and TTM is the bundle TTM over M ³n. The composite of the universal line bundle E Pn and the tangent bundle TE is the bundle TE Pn n + ².

(ibe an open covering of a manifold Mwith U ∩ V ≠ Ø there is given a C∞ map fU, V from U ∩ V to a fixed Lie group G. Suppose given a C∞ action of G on a manifold F. If the maps fU, V satisfy the relations stated in 1.6, that is, fU,V(p)–1 = fV,U(p) for all p ∈ U ∩ V, and fU,V(p) = fU, W(p)·fW, V(p) for p ∈ U ∩ V ∩ W and the given maps fU, V can be used to construct a fiber bundle over M with fiber F and group G. This procedure will be outlined first, and then applied to a special famous example in (j).

Suppose U ∩ V ≠ Ø. Over the open sets U and V in M there are the product bundles U × F and V × F. If p ∈ U ∩ V, identify the point (p, ξ) in U × F with the point (p, fU, V(p) · ξ) in V × F. By the first of the two relations stated above, it follows that (p, fU, V(p) · ξ) in V × F is identified with (p, fV, U(p)fU, V(p)ξ) = (p, ξ) in U × F. Let E be the disjoint union of U × F and V × F modulo the identification just described. Define π from E to U V to be projection onto the first factor. (See p. 10.) There are natural bundle charts on E over U and V; if υ ∈ E comes from a point (p, ζ) ∈ U × F, while if υ ∈ E comes from a point (p, ξ) ∈ V × F. The maps fU, V and fV, U are the transition functions between the bundle charts.

The second relation stated above for the collection of fU, V is the necessary consistency requirement if E is extended to U V W, where U V W ≠ Ø.

(j)Milnor’s exotic spheres Let N be the north pole (1, 0, …, 0) in Sn n+1, and S the south pole (–1, 0, ...,0) ∈ Sn. The stereographic projection of Sn onto the equatorial plane from N is the map

The stereographic projection of Sn from S is the map

For each u ∈ U ∩ V, y(u) = x(u)/||x(u)||², and x(u) = y(u)/||y(u)||², where ||υ||² denotes the square of the norm of υ nn+1. The maps x and y are charts on Sn which generate the C∞ structure.

The result is that Sn n modulo the identification of each nonzero vector u in the first copy with the vector u/||u, where u ~ u/||u||², u ≠ 0.

The quaternions with real basis {1, i, j, k} such that multiplication satisfies i² = j² = k² = ijk = jki = kij = – 1. Conjugation is the real involution