Natural Operations in Differential Geometry

437 Pages·1993·2.81 MB·English

by  Ivan Kolar, Peter W. Michor, Jan Slovak

NATURAL
OPERATIONS
IN
DIFFERENTIAL
GEOMETRY
Ivan Kol´aˇr
Peter W. Michor
Jan Slov´ak
Mailing address: Peter W. Michor,
Institut fu¨r Mathematik der Universit¨at Wien,
Strudlhofgasse 4, A-1090 Wien, Austria.
Ivan Kol´aˇr, Jan Slov´ak,
Department of Algebra and Geometry
Faculty of Science, Masaryk University
Jana´ˇckovo n´am 2a, CS-662 95 Brno, Czechoslovakia
Electronicedition. OriginallypublishedbySpringer-Verlag,BerlinHeidelberg
1993, ISBN 3-540-56235-4, ISBN 0-387-56235-4.
TypesetbyAMS-TEX
v
TABLE OF CONTENTS
PREFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
CHAPTER I.
MANIFOLDS AND LIE GROUPS . . . . . . . . . . . . . . . . 4
1. Differentiable manifolds . . . . . . . . . . . . . . . . . . . . . 4
2. Submersions and immersions . . . . . . . . . . . . . . . . . . 11
3. Vector fields and flows . . . . . . . . . . . . . . . . . . . . . 16
4. Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5. Lie subgroups and homogeneous spaces . . . . . . . . . . . . . 41
CHAPTER II.
DIFFERENTIAL FORMS . . . . . . . . . . . . . . . . . . . 49
6. Vector bundles . . . . . . . . . . . . . . . . . . . . . . . . 49
7. Differential forms . . . . . . . . . . . . . . . . . . . . . . . 61
8. Derivations on the algebra of differential forms
and the Fr¨olicher-Nijenhuis bracket . . . . . . . . . . . . . . . 67
CHAPTER III.
BUNDLES AND CONNECTIONS . . . . . . . . . . . . . . . 76
9. General fiber bundles and connections . . . . . . . . . . . . . . 76
10. Principal fiber bundles and G-bundles . . . . . . . . . . . . . . 86
11. Principal and induced connections . . . . . . . . . . . . . . . 99
CHAPTER IV.
JETS AND NATURAL BUNDLES . . . . . . . . . . . . . . . 116
12. Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
13. Jet groups . . . . . . . . . . . . . . . . . . . . . . . . . . 128
14. Natural bundles and operators . . . . . . . . . . . . . . . . . 138
15. Prolongations of principal fiber bundles . . . . . . . . . . . . . 149
16. Canonical differential forms . . . . . . . . . . . . . . . . . . 154
17. Connections and the absolute differentiation . . . . . . . . . . . 158
CHAPTER V.
FINITE ORDER THEOREMS . . . . . . . . . . . . . . . . . 168
18. Bundle functors and natural operators . . . . . . . . . . . . . . 169
19. Peetre-like theorems . . . . . . . . . . . . . . . . . . . . . . 176
20. The regularity of bundle functors . . . . . . . . . . . . . . . . 185
21. Actions of jet groups . . . . . . . . . . . . . . . . . . . . . . 192
22. The order of bundle functors . . . . . . . . . . . . . . . . . . 202
23. The order of natural operators . . . . . . . . . . . . . . . . . 205
CHAPTER VI.
METHODS FOR FINDING NATURAL OPERATORS . . . . . . 212
24. Polynomial GL(V)-equivariant maps . . . . . . . . . . . . . . 213
25. Natural operators on linear connections, the exterior differential . . 220
26. The tensor evaluation theorem . . . . . . . . . . . . . . . . . 223
27. Generalized invariant tensors . . . . . . . . . . . . . . . . . . 230
28. The orbit reduction . . . . . . . . . . . . . . . . . . . . . . 233
29. The method of differential equations . . . . . . . . . . . . . . 245
Electroniceditionof: NaturalOperationsinDifferentialGeometry,Springer-Verlag,1993
vi
CHAPTER VII.
FURTHER APPLICATIONS . . . . . . . . . . . . . . . . . . 249
30. The Fr¨olicher-Nijenhuis bracket . . . . . . . . . . . . . . . . . 250
31. Two problems on general connections . . . . . . . . . . . . . . 255
32. Jet functors . . . . . . . . . . . . . . . . . . . . . . . . . . 259
33. Topics from Riemannian geometry . . . . . . . . . . . . . . . . 265
34. Multilinear natural operators . . . . . . . . . . . . . . . . . . 280
CHAPTER VIII.
PRODUCT PRESERVING FUNCTORS . . . . . . . . . . . . 296
35. Weil algebras and Weil functors . . . . . . . . . . . . . . . . . 297
36. Product preserving functors . . . . . . . . . . . . . . . . . . 308
37. Examples and applications . . . . . . . . . . . . . . . . . . . 318
CHAPTER IX.
BUNDLE FUNCTORS ON MANIFOLDS . . . . . . . . . . . . 329
38. The point property . . . . . . . . . . . . . . . . . . . . . . 329
39. The flow-natural transformation . . . . . . . . . . . . . . . . 336
40. Natural transformations . . . . . . . . . . . . . . . . . . . . 341
41. Star bundle functors . . . . . . . . . . . . . . . . . . . . . 345
CHAPTER X.
PROLONGATION OF VECTOR FIELDS AND CONNECTIONS . 350
42. Prolongations of vector fields to Weil bundles . . . . . . . . . . . 351
43. The case of the second order tangent vectors . . . . . . . . . . . 357
44. Induced vector fields on jet bundles . . . . . . . . . . . . . . . 360
45. Prolongations of connections to FY →M . . . . . . . . . . . . 363
46. The cases FY →FM and FY →Y . . . . . . . . . . . . . . . 369
CHAPTER XI.
GENERAL THEORY OF LIE DERIVATIVES . . . . . . . . . . 376
47. The general geometric approach . . . . . . . . . . . . . . . . 376
48. Commuting with natural operators . . . . . . . . . . . . . . . 381
49. Lie derivatives of morphisms of fibered manifolds . . . . . . . . . 387
50. The general bracket formula . . . . . . . . . . . . . . . . . . 390
CHAPTER XII.
GAUGE NATURAL BUNDLES AND OPERATORS . . . . . . . 394
51. Gauge natural bundles . . . . . . . . . . . . . . . . . . . . 394
52. The Utiyama theorem . . . . . . . . . . . . . . . . . . . . . 399
53. Base extending gauge natural operators . . . . . . . . . . . . . 405
54. Induced linear connections on the total space
of vector and principal bundles . . . . . . . . . . . . . . . . . 409
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
List of symbols . . . . . . . . . . . . . . . . . . . . . . . . . . 428
Author index . . . . . . . . . . . . . . . . . . . . . . . . . . 429
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431
Electroniceditionof: NaturalOperationsinDifferentialGeometry,Springer-Verlag,1993
1
PREFACE
The aim of this work is threefold:
First it should be a monographical work on natural bundles and natural op-
erators in differential geometry. This is a field which every differential geometer
has met several times, but which is not treated in detail in one place. Let us
explain a little, what we mean by naturality.
Exterior derivative commutes with the pullback of differential forms. In the
background of this statement are the following general concepts. The vector
bundle ΛkT∗M is in fact the value of a functor, which associates a bundle over
M to each manifold M and a vector bundle homomorphism over f to each local
diffeomorphism f between manifolds of the same dimension. This is a simple
exampleoftheconceptofanaturalbundle. Thefactthattheexteriorderivative
dtransformssectionsofΛkT∗M intosectionsofΛk+1T∗M foreverymanifoldM
can be expressed by saying that d is an operator from ΛkT∗M into Λk+1T∗M.
Thattheexteriorderivativedcommuteswithlocaldiffeomorphismsnowmeans,
thatdisanaturaloperatorfromthefunctorΛkT∗intofunctorΛk+1T∗. Ifk >0,
one can show that d is the unique natural operator between these two natural
bundles up to a constant. So even linearity is a consequence of naturality. This
resultisarchetypicalforthefieldwearediscussinghere. Asystematictreatment
ofnaturalityindifferentialgeometryrequirestodescribeallnaturalbundles,and
this is also one of the undertakings of this book.
Second this book tries to be a rather comprehensive textbook on all basic
structures from the theory of jets which appear in different branches of dif-
ferential geometry. Even though Ehresmann in his original papers from 1951
underlined the conceptual meaning of the notion of an r-jet for differential ge-
ometry,jetshavebeenmostlyusedasapurelytechnicaltoolincertainproblems
in the theory of systems of partial differential equations, in singularity theory,
in variational calculus and in higher order mechanics. But the theory of nat-
ural bundles and natural operators clarifies once again that jets are one of the
fundamental concepts in differential geometry, so that a thorough treatment of
theirbasicpropertiesplaysanimportantroleinthisbook. Wealsodemonstrate
that the central concepts from the theory of connections can very conveniently
be formulated in terms of jets, and that this formulation gives a very clear and
geometric picture of their properties.
This book also intends to serve as a self-contained introduction to the theory
of Weil bundles. These were introduced under the name ‘les espaces des points
proches’ by A. Weil in 1953 and the interest in them has been renewed by the
recent description of all product preserving functors on manifolds in terms of
products of Weil bundles. And it seems that this technique can lead to further
interesting results as well.
Third in the beginning of this book we try to give an introduction to the
fundamentals of differential geometry (manifolds, flows, Lie groups, differential
forms,bundlesandconnections)whichstressesnaturalityandfunctorialityfrom
thebeginningandisascoordinatefreeaspossible. HerewepresenttheFr¨olicher-
Nijenhuis bracket (a natural extension of the Lie bracket from vector fields to
Electroniceditionof: NaturalOperationsinDifferentialGeometry,Springer-Verlag,1993
2 Preface
vector valued differential forms) as one of the basic structures of differential
geometry, and we base nearly all treatment of curvature and Bianchi identities
on it. This allows us to present the concept of a connection first on general
fiber bundles (without structure group), with curvature, parallel transport and
Bianchi identity, and only then add G-equivariance as a further property for
principal fiber bundles. We think, that in this way the underlying geometric
ideasaremoreeasilyunderstoodbythenovicethaninthetraditionalapproach,
where too much structure at the same time is rather confusing. This approach
was tested in lecture courses in Brno and Vienna with success.
A specific feature of the book is that the authors are interested in general
pointsofviewtowardsdifferentstructuresindifferentialgeometry. Themodern
development of global differential geometry clarified that differential geomet-
ric objects form fiber bundles over manifolds as a rule. Nijenhuis revisited the
classical theory of geometric objects from this point of view. Each type of geo-
metric objects can be interpreted as a rule F transforming every m-dimensional
manifold M into a fibered manifold FM → M over M and every local diffeo-
morphism f :M →N into a fibered manifold morphism Ff :FM →FN over
f. ThegeometriccharacterofF isthenexpressedbythefunctorialitycondition
F(g◦f) = Fg◦Ff. Hence the classical bundles of geometric objects are now
studied in the form of the so called lifting functors or (which is the same) natu-
ralbundlesonthecategoryMf ofallm-dimensionalmanifoldsandtheirlocal
m
diffeomorphisms. An important result by Palais and Terng, completed by Ep-
stein and Thurston, reads that every lifting functor has finite order. This gives
a full description of all natural bundles as the fiber bundles associated with the
r-th order frame bundles, which is useful in many problems. However in several
cases it is not sufficient to study the bundle functors defined on the category
Mf . For example, if we have a Lie group G, its multiplication is a smooth
m
map µ : G×G → G. To construct an induced map Fµ : F(G×G) → FG,
we need a functor F defined on the whole category Mf of all manifolds and
all smooth maps. In particular, if F preserves products, then it is easy to see
that Fµ endows FG with the structure of a Lie group. A fundamental result
in the theory of the bundle functors on Mf is the complete description of all
product preserving functors in terms of the Weil bundles. This was deduced by
KainzandMichor,andindependentlybyEckandLuciano,anditispresentedin
chapterVIIIofthisbook. Atseveralotherplaceswethencompareandcontrast
the properties of the product preserving bundle functors and the non-product-
preserving ones, which leads us to interesting geometric results. Further, some
functors of modern differential geometry are defined on the category of fibered
manifolds and their local isomorphisms, the bundle of general connections be-
ing the simplest example. Last but not least we remark that Eck has recently
introduced the general concepts of gauge naturalbundlesand gauge naturalop-
erators. Taking into account the present role of gauge theories in theoretical
physicsandmathematics,wedevotethelastchapterofthebooktothissubject.
Ifweinterpretgeometricobjectsasbundlefunctorsdefinedonasuitablecat-
egoryovermanifolds,thensomegeometricconstructionshavetheroleofnatural
transformations. Several others represent natural operators, i.e. they map sec-
Electroniceditionof: NaturalOperationsinDifferentialGeometry,Springer-Verlag,1993
Preface 3
tions of certain fiber bundles to sections of other ones and commute with the
action of local isomorphisms. So geometric means natural in such situations.
That is why we develop a rather general theory of bundle functors and natural
operatorsinthisbook. Theprincipaladvantageofinterpretinggeometricasnat-
ural is that we obtain a well-defined concept. Then we can pose, and sometimes
evensolve,theproblemofdeterminingallnaturaloperatorsofaprescribedtype.
Thisgivesusthecompletelistofallpossiblegeometricconstructionsofthetype
inquestion. Insomecasesweevendiscovernewgeometricoperatorsinthisway.
Our practical experience taught us that the most effective way how to treat
natural operators is to reduce the question to a finite order problem, in which
thecorrespondingjetspacesarefinitedimensional. Sincethefiniteordernatural
operatorsareinasimplebijectionwiththeequivariantmapsbetweenthecorre-
spondingstandardfibers,wecanapplythenseveralpowerfultoolsfromclassical
algebraandanalysis,whichcanleadratherquicklytoacompletesolutionofthe
problem. Such a passing to a finite order situation has been of great profit in
other branches of mathematics as well. Historically, the starting point for the
reductiontothejetspacesisthefamousPeetretheoremsayingthateverylinear
supportnon-increasingoperatorhaslocallyfiniteorder. Wedevelopanessential
generalization of this technique and we present a unified approach to the finite
order results for both natural bundles and natural operators in chapter V.
The primary purpose of chapter VI is to explain some general procedures,
whichcanhelpusinfindingalltheequivariantmaps,i.e. allnaturaloperatorsof
a given type. Nevertheless, the greater part of the geometric results is original.
Chapter VII is devoted to some further examples and applications, including
Gilkey’s theorem that all differential forms depending naturally on Riemannian
metrics and satisfying certain homogeneity conditions are in fact Pontryagin
forms. This is essential in the recent heat kernel proofs of the Atiyah Singer
Index theorem. We also characterize the Chern forms as the only natural forms
onlinearsymmetricconnections. Inaspecialsectionwecommentontheresults
ofKirillovandhiscolleagueswhoinvestigatedmultilinearnaturaloperatorswith
the help of representation theory of infinite dimensional Lie algebras of vector
fields. InchapterXwestudysystematicallythenaturaloperatorsonvectorfields
and connections. Chapter XI is devoted to a general theory of Lie derivatives,
in which the geometric approach clarifies, among other things, the relations to
natural operators.
ThematerialforchaptersVI,Xandsections12, 30–32, 47, 49, 50, 52–54was
preparedbythefirstauthor(I.K.), forchaptersI,II,III,VIIIbythesecondau-
thor(P.M.)andforchaptersV,IXandsections13–17,33,34,48,51bythethird
author (J.S.). The authors acknowledge A. Cap, M. Doupovec, and J. Janyˇska,
for reading the manuscript and for several critical remarks and comments and
A. A. Kirillov for commenting section 34.
The joint work of the authors on the book has originated in the seminar of
the first two authors and has been based on the common cultural heritage of
Middle Europe. The authors will be pleased if the reader realizes a reflection of
those traditions in the book.
Electroniceditionof: NaturalOperationsinDifferentialGeometry,Springer-Verlag,1993
4
CHAPTER I.
MANIFOLDS AND LIE GROUPS
Inthischapterwepresentanintroductiontothebasicstructuresofdifferential
geometrywhichstressesglobalstructuresandcategoricalthinking. Thematerial
presented is standard - but some parts are not so easily found in text books:
wetreatinitialsubmanifoldsandtheFrobeniustheoremfordistributionsofnon
constantrank,andwegiveaveryquickproofoftheCampbell-Baker-Hausdorff
formula for Lie groups. We also prove that closed subgroups of Lie groups are
Lie subgroups.
1. Differentiable manifolds
1.1. A topological manifold is a separable Hausdorff space M which is locally
homeomorphic to Rn. So for any x∈M there is some homeomorphism u:U →
u(U) ⊆ Rn, where U is an open neighborhood of x in M and u(U) is an open
subset in Rn. The pair (U,u) is called a chart on M.
From topology it follows that the number n is locally constant on M; if n is
constant, M is sometimes called a pure manifold. We will only consider pure
manifolds and consequently we will omit the prefix pure.
A family (Uα,uα)α∈A of charts on M such that the Uα form a cover of M is
called an atlas. The mappings u :=u ◦u−1 :u (U )→u (U ) are called
αβ α β β αβ α αβ
the chart changings for the atlas (U ), where U :=U ∩U .
α αβ α β
An atlas (Uα,uα)α∈A for a manifold M is said to be a Ck-atlas, if all chart
changings u : u (U ) → u (U ) are differentiable of class Ck. Two Ck-
αβ β αβ α αβ
atlases are called Ck-equivalent, if their union is again a Ck-atlas for M. An
equivalence class of Ck-atlases is called a Ck-structure on M. From differential
topologyweknowthatifM hasaC1-structure,thenitalsohasaC1-equivalent
C∞-structure and even a C1-equivalent Cω-structure, where Cω is shorthand
for real analytic. By a Ck-manifold M we mean a topological manifold together
with a Ck-structure and a chart on M will be a chart belonging to some atlas
of the Ck-structure.
But there are topological manifolds which do not admit differentiable struc-
tures. For example, every 4-dimensional manifold is smooth off some point, but
there are such which are not smooth, see [Quinn, 82], [Freedman, 82]. There
are also topological manifolds which admit several inequivalent smooth struc-
tures. The spheres from dimension 7 on have finitely many, see [Milnor, 56].
But the most surprising result is that on R4 there are uncountably many pair-
wiseinequivalent(exotic)differentiablestructures. Thisfollowsfromtheresults
Electroniceditionof: NaturalOperationsinDifferentialGeometry,Springer-Verlag,1993
1. Differentiablemanifolds 5
of [Donaldson, 83] and [Freedman, 82], see [Gompf, 83] or [Freedman-Feng Luo,
89] for an overview.
Note that for a Hausdorff C∞-manifold in a more general sense the following
properties are equivalent:
(1) It is paracompact.
(2) It is metrizable.
(3) It admits a Riemannian metric.
(4) Each connected component is separable.
In this book a manifold will usually mean a C∞-manifold, and smooth is
used synonymously for C∞, it will be Hausdorff, separable, finite dimensional,
to state it precisely.
NotefinallythatanymanifoldM admitsafiniteatlasconsistingofdimM+1
(not connected) charts. This is a consequence of topological dimension theory
[Nagata, 65], a proof for manifolds may be found in [Greub-Halperin-Vanstone,
Vol. I, 72].
1.2. A mapping f : M → N between manifolds is said to be Ck if for each
x ∈ M and each chart (V,v) on N with f(x) ∈ V there is a chart (U,u) on M
with x∈U, f(U)⊆V, and v◦f ◦u−1 is Ck. We will denote by Ck(M,N) the
space of all Ck-mappings from M to N.
A Ck-mapping f : M → N is called a Ck-diffeomorphism if f−1 : N → M
existsandisalsoCk. Twomanifoldsarecalleddiffeomorphicifthereexistsadif-
feomorphismbetweenthem. Fromdifferentialtopologyweknowthatifthereisa
C1-diffeomorphism between M and N, then there is also a C∞-diffeomorphism.
All smooth manifolds together with the C∞-mappings form a category, which
will be denoted by Mf. One can admit non pure manifolds even in Mf, but
we will not stress this point of view.
A mapping f : M → N between manifolds of the same dimension is called
a local diffeomorphism, if each x ∈ M has an open neighborhood U such that
f|U : U → f(U) ⊂ N is a diffeomorphism. Note that a local diffeomorphism
need not be surjective or injective.
1.3. The set of smooth real valued functions on a manifold M will be denoted
by C∞(M,R), in order to distinguish it clearly from spaces of sections which
will appear later. C∞(M,R) is a real commutative algebra.
The support of a smooth function f is the closure of the set, where it does
not vanish, supp(f)={x∈M :f(x)(cid:54)=0}. The zero set of f is the set where f
vanishes, Z(f)={x∈M :f(x)=0}.
Any manifold admits smooth partitions of unity: Let (Uα)α∈A be an open
cover of M. Then there is a family (ϕα)α∈A of smooth functions on M, such
that supp(ϕ ) ⊂ U , (supp(ϕ )) is a locally finite family, and ϕ = 1
α α α (cid:80)α α
(locally this is a finite sum).
1.4. Germs. Let M and N be manifolds and x∈M. We consider all smooth
mappings f :U →N, where U is some open neighborhood of x in M, and we
f f
putf ∼g ifthereissomeopenneighborhoodV ofxwithf|V =g|V. Thisisan
x
equivalence relation onthe set ofmappings considered. The equivalence class of
Electroniceditionof: NaturalOperationsinDifferentialGeometry,Springer-Verlag,1993
6 ChapterI.ManifoldsandLiegroups
a mapping f is called the germ of f at x, sometimes denoted by germ f. The
x
space of all germs at x of mappings M → N will be denoted by C∞(M,N).
x
This construction works also for other types of mappings like real analytic or
holomorphic ones, if M and N have real analytic or complex structures.
If N = R we may add and multiply germs, so we get the real commutative
algebra C∞(M,R) of germs of smooth functions at x.
x
Using smooth partitions of unity (see 1.3) it is easily seen that each germ of
a smooth function has a representative which is defined on the whole of M. For
germs of real analytic or holomorphic functions this is not true. So C∞(M,R)
x
is the quotient of the algebra C∞(M,R) by the ideal of all smooth functions
f :M →R which vanish on some neighborhood (depending on f) of x.
1.5. The tangent space of Rn. Let a ∈ Rn. A tangent vector with foot
point a is simply a pair (a,X) with X ∈ Rn, also denoted by X . It induces
a
a derivation X : C∞(Rn,R) → R by X (f) = df(a)(X ). The value depends
a a a
only on the germ of f at a and we have X (f ·g)=X (f)·g(a)+f(a)·X (g)
a a a
(the derivation property).
If conversely D : C∞(Rn,R) → R is linear and satisfies D(f ·g) = D(f)·
g(a)+f(a)·D(g) (a derivation at a), then D is given by the action of a tangent
vector with foot point a. This can be seen as follows. For f ∈ C∞(Rn,R) we
have
1
f(x)=f(a)+(cid:90) df(a+t(x−a))dt
dt
0
n 1
=f(a)+ (cid:90) ∂f (a+t(x−a))dt(xi−ai)
(cid:88) ∂xi
i=1 0
n
=f(a)+ h (x)(xi−ai).
(cid:88) i
i=1
D(1)=D(1·1)=2D(1), so D(constant)=0. Thus
n
D(f)=D(f(a)+ h (x)(xi−ai))
(cid:88) i
i=1
n n
=0+ D(h )(ai−ai)+ h (a)(D(xi)−0)
(cid:88) i (cid:88) i
i=1 i=1
n
= ∂f (a)D(xi),
(cid:88)∂xi
i=1
where xi is the i-th coordinate function on Rn. So we have the expression
n n
D(f)= D(xi) ∂ | (f), D = D(xi) ∂ | .
(cid:88) ∂xi a (cid:88) ∂xi a
i=1 i=1
Thus D is induced by the tangent vector (a, n D(xi)e ), where (e ) is the
standard basis of Rn. (cid:80)i=1 i i
Electroniceditionof: NaturalOperationsinDifferentialGeometry,Springer-Verlag,1993
1. Differentiablemanifolds 7
1.6. The tangent space of a manifold. Let M be a manifold and let x ∈
M and dimM = n. Let T M be the vector space of all derivations at x of
x
C∞(M,R), the algebra of germs of smooth functions on M at x. (Using 1.3 it
x
may easily be seen that a derivation of C∞(M,R) at x factors to a derivation of
C∞(M,R).)
x
SoT M consistsofalllinearmappingsX :C∞(M,R)→RsatisfyingX (f·
x x x
g)=X (f)·g(x)+f(x)·X (g). The space T M is called the tangent space of
x x x
M at x.
If (U,u) is a chart on M with x ∈ U, then u∗ : f (cid:55)→ f ◦u induces an iso-
morphism of algebras C∞ (Rn,R)∼=C∞(M,R), and thus also an isomorphism
u(x) x
T u : T M → T Rn, given by (T u.X )(f) = X (f ◦u). So T M is an n-
x x u(x) x x x x
dimensional vector space. The dot in T u.X means that we apply the linear
x x
mapping T u to the vector X — a dot will frequently denote an application of
x x
a linear or fiber linear mapping.
We will use the following notation: u = (u1,... ,un), so ui denotes the i-th
coordinate function on U, and
∂ | :=(T u)−1( ∂ | )=(T u)−1(u(x),e ).
∂ui x x ∂xi u(x) x i
So ∂ | ∈T M is the derivation given by
∂ui x x
∂(f ◦u−1)
∂ | (f)= (u(x)).
∂ui x ∂xi
From 1.5 we have now
n
T u.X = (T u.X )(xi) ∂ | =
x x (cid:88) x x ∂xi u(x)
i=1
n n
= X (xi◦u) ∂ | = X (ui) ∂ | .
(cid:88) x ∂xi u(x) (cid:88) x ∂xi u(x)
i=1 i=1
1.7. The tangent bundle. For a manifold M of dimension n we put TM :=
T M, the disjoint union of all tangent spaces. This is a family of vec-
(cid:70)toxr∈sMpacxes parameterized by M, with projection π : TM → M given by
M
π (T M)=x.
M x
For any chart (U ,u ) of M consider the chart (π−1(U ),Tu ) on TM,
α α M α α
where Tu : π−1(U ) → u (U ) × Rn is given by the formula Tu .X =
α M α α α α
(u (π (X)),T u .X). Then the chart changings look as follows:
α M πM(X) α
Tu ◦(Tu )−1 :Tu (π−1(U ))=u (U )×Rn →
β α α M αβ α αβ
→u (U )×Rn =Tu (π−1(U )),
β αβ β M αβ
((Tu ◦(Tu )−1)(y,Y))(f)=((Tu )−1(y,Y))(f ◦u )
β α α β
=(y,Y)(f ◦u ◦u−1)=d(f ◦u ◦u−1)(y).Y
β α β α
=df(u ◦u−1(y)).d(u ◦u−1)(y).Y
β α β α
=(u ◦u−1(y),d(u ◦u−1)(y).Y)(f).
β α β α
Electroniceditionof: NaturalOperationsinDifferentialGeometry,Springer-Verlag,1993