An Introduction to GEOMETRICAL PHYSICS docx

An Introduction to
GEOMETRICAL PHYSICS
R. Aldrovandi & J.G. Pereira
Instituto de F´ısica Te´orica
State Univers ity of S˜ao Paulo – UNESP
S˜ao Paulo — Brazil
To our parents
Nice, Dina, Jos´e and Tito
i
ii
PREAMBLE: SPACE AND GEOMETRY
What stuff’tis made of, whereof it is born,
I am to learn.
Merchant of Venice
The simplest geometrical setting used — consciously or not — by physi-
cists in their everyday work is the 3-dimensional euclidean space E
3
. It con-
sists of the set R
3
of ordered triples of real numbers such as p = (p
1
, p
2
, p
3
), q
= (q
1
, q
2

, q
3
), etc, and is endowed with a very special characteristic, a metric
defined by the distance function
d(p, q) =

3

i=1
(p
i
− q
i
)
2

1/2
.
It is the space of ordinary human experience and the starting point of our
geometric intuition. Studied for two-and-a-half millenia, it has been the
object of celebrated controversies, the most famous concerning the minimum
number of properties necessary to define it completely.
From Aristotle to Newton, through Galileo and Descartes, the very word
space has been reserved to E
3
. Only in the 19-th century has it become clear
that other, different spaces could be thought of, and mathematicians have
since greatly amused themselves by inventing all kinds of them. For physi-
cists, the age-long debate shifted to another question: how can we recognize,
amongst such innumerable possible spaces, that real space chosen by Nature

as the stage-set of its processes? For example, suppose the space of our ev-
eryday experience consists of the same set R
3
of triples above, but with a
different distance function, such as
d(p, q) =
3

i=1
|p
i
− q
i
|.
This would define a different metric space, in principle as good as that
given above. Were it only a matter of principle, it would be as good as
iii
iv
any other space given by any distance function with R
3
as set point. It so
happens, however, that Nature has chosen the former and not the latter space
for us to live in. To know which one is the real space is not a simple question
of principle — something else is needed. What else? The answer may seem
rather trivial in the case of our home space, though less so in other spaces
singled out by Nature in the many different situations which are objects of
physical study. It was given by Riemann in his famous Inaugural Address
1
:
“ those properties which distinguish Space from other con-

ceivable triply extended quantities can only be deduced from expe-
rience.”
Thus, from experience! It is experiment which tells us in which space we
actually live in. When we measure distances we find them to be independent
of the direction of the straight lines joining the points. And this isotropy
property rules out the second proposed distance function, while admitting
the metric of the euclidean space.
In reality, Riemann’s statement implies an epistemological limitation: it
will never be possible to ascertain exactly which space is the real one. Other
isotropic distance functions are, in principle, admissible and more experi-
ments are necessary to decide between them. In Riemann’s time already
other geometries were known (those found by Lobachevsky and Boliyai) that
could be as similar to the euclidean geometry as we might wish in the re-
stricted regions experience is confined to. In honesty, all we can say is that
E
3
, as a model for our ambient space, is strongly favored by present day
experimental evidence in scales ranging from (say) human dimensions down
to about 10
−15
cm. Our knowledge on smaller scales is limited by our ca-
pacity to probe them. For larger scales, according to General Relativity, the
validity of this model depends on the presence and strength of gravitational
fields: E
3
is good only as long as gravitational fields are very weak.
“ These data are — like all data — not logically necessary,
but only of empirical certainty . . . one can therefore investigate
their likelihood, which is certainly very great within the bounds of
observation, and afterwards decide upon the legitimacy of extend-

ing them beyond the bounds of observation, both in the direction of
the immeasurably large and in the direction of the immeasurably
small.”
1
A translation of Riemann’s Address can be found in Spivak 1970, vol. II. Clifford’s
translation (Nature, 8 (1873), 14-17, 36-37), as well as the original transcribed by David
R. Wilkins, can be found in the site />v
The only remark we could add to these words, pronounced in 1854, is
that the “bounds of observation” have greatly receded with respect to the
values of Riemann times.
“ . . . geometry presupposes the concept of space, as well as
assuming the basic principles for constructions in space .”
In our ambient space, we use in reality a lot more of structure than
the simple metric model: we take for granted a vector space structure, or
an affine structure; we transport vectors in such a way that they remain
parallel to themselves, thereby assuming a connection. Which one is the
minimum structure, the irreducible set of assumptions really necessary to
the introduction of each concept? Physics should endeavour to establish on
empirical data not only the basic space to be chosen but also the structures
to be added to it. At present, we know for example that an electron moving
in E
3
under the influence of a magnetic field “feels” an extra connection (the
electromagnetic potential), to which neutral particles may be insensitive.
Experimental science keeps a very special relationship with Mathemat-
ics. Experience counts and measures. But Science requires that the results
be inserted in some logically ordered picture. Mathematics is expected to
provide the notion of number, so as to make countings and measurements
meaningful. But Mathematics is also expected to provide notions of a more
qualitative character, to allow for the modeling of Nature. Thus, concerning

numbers, there seems to be no result comforting the widespread prejudice
by which we measure real numbers. We work with integers, or with rational
numbers, which is fundamentally the same. No direct measurement will sort
out a Dedekind cut. We must suppose, however, that real numbers exist:
even from the strict experimental point of view, it does not matter whether
objects like “π” or “e” are simple names or are endowed with some kind of an
sich reality: we cannot afford to do science without them. This is to say that
even pure experience needs more than its direct results, presupposes a wider
background for the insertion of such results. Real numbers are a minimum
background. Experience, and “logical necessity”, will say whether they are
sufficient.
From the most ancient extant treatise going under the name of Physics
2
:
“When the objects of investigation, in any subject, have first
principles, foundational conditions, or basic constituents, it is
through acquaintance with these that knowledge, scientific knowl-
edge, is attained. For we cannot say that we know an object before
2
Aristotle, Physics I.1.
vi
we are acquainted with its conditions or principles, and have car-
ried our analysis as far as its most elementary constituents.”
“The natural way of attaining such a knowledge is to start
from the things which are more knowable and obvious to us and
proceed towards those which are clearer and more knowable by
themselves . . .”
Euclidean spaces have been the starting spaces from which the basic geo-
metrical and analytical concepts have been isolated by successive, tentative,
progressive abstractions. It has been a long and hard process to remove the

unessential from each notion. Most of all, as will be repeatedly emphasized,
it was a hard thing to put the idea of metric in its due position.
Structure is thus to be added step by step, under the control of experi-
ment. Only once experiment has established the basic ground will internal
coherence, or logical necessity, impose its own conditions.
Contents
I MANIFOLDS 1
1 GENE RAL TOPOLOGY 3
1.0 INTRODUCTORY COMMENTS . . . . . . . . . . . . . . . . . 3
1.1 TOPOLOGICAL SPACES . . . . . . . . . . . . . . . . . . . . 5
1.2 KINDS OF TEXTURE . . . . . . . . . . . . . . . . . . . . . . 15
1.3 FUNCTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.4 QUOTIENTS AND GROUPS . . . . . . . . . . . . . . . . . . . 36
1.4.1 Quotient spaces . . . . . . . . . . . . . . . . . . . . . . 36
1.4.2 Topological groups . . . . . . . . . . . . . . . . . . . . 41
2 HOMOLOGY 49
2.1 GRAPHS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.1.1 Graphs, first way . . . . . . . . . . . . . . . . . . . . . 50
2.1.2 Graphs, second way . . . . . . . . . . . . . . . . . . . . 52
2.2 THE FIRST TOPOLOGICAL INVARIANTS . . . . . . . . . . . 57
2.2.1 Simplexes, complexes & all that . . . . . . . . . . . . . 57
2.2.2 Topological numbers . . . . . . . . . . . . . . . . . . . 64
3 HOMOTOPY 73
3.0 GENERAL HOMOTOPY . . . . . . . . . . . . . . . . . . . . . 73
3.1 PATH HOMOTOPY . . . . . . . . . . . . . . . . . . . . . . . . 78
3.1.1 Homotopy of curves . . . . . . . . . . . . . . . . . . . . 78
3.1.2 The Fundamental group . . . . . . . . . . . . . . . . . 85
3.1.3 Some Calculations . . . . . . . . . . . . . . . . . . . . 92
3.2 COVERING SPACES . . . . . . . . . . . . . . . . . . . . . . 98
3.2.1 Multiply-connected Spaces . . . . . . . . . . . . . . . . 98

3.2.2 Covering Spaces . . . . . . . . . . . . . . . . . . . . . . 105
3.3 HIGHER HOMOTOPY . . . . . . . . . . . . . . . . . . . . . 115
vii
viii CONTENTS
4 MANIFOLDS & CHARTS 121
4.1 MANIFOLDS . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
4.1.1 Topological manifolds . . . . . . . . . . . . . . . . . . . 121
4.1.2 Dimensions, integer and other . . . . . . . . . . . . . . 123
4.2 CHARTS AND COORDINATES . . . . . . . . . . . . . . . . 125
5 DIFFE RENTIABLE MANIFOLDS 133
5.1 DEFINITION AND OVERLOOK . . . . . . . . . . . . . . . . . 133
5.2 SMOOTH FUNCTIONS . . . . . . . . . . . . . . . . . . . . . . 135
5.3 DIFFERENTIABLE SUBMANIFOLDS . . . . . . . . . . . . . . 137
II DIFFERENTIABLE STRUCTURE 141
6 TANGENT STRUCTURE 143
6.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . 143
6.2 TANGENT SPACES . . . . . . . . . . . . . . . . . . . . . . . . 145
6.3 TENSORS ON MANIFOLDS . . . . . . . . . . . . . . . . . . . 154
6.4 FIELDS & TRANSFORMATIONS . . . . . . . . . . . . . . . . 161
6.4.1 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
6.4.2 Transformations . . . . . . . . . . . . . . . . . . . . . . 167
6.5 FRAMES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
6.6 METRIC & RIEMANNIAN MANIFOLDS . . . . . . . . . . . . 180
7 DIFFE RENTIAL FORMS 189
7.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . 189
7.2 EXTERIOR DERIVATIVE . . . . . . . . . . . . . . . . . . . 197
7.3 VECTOR-VALUE D FORMS . . . . . . . . . . . . . . . . . . 210
7.4 DUALITY AND CODERIVATION . . . . . . . . . . . . . . . 217
7.5 INTEGRATION AND HOMOLOGY . . . . . . . . . . . . . . 225
7.5.1 Integration . . . . . . . . . . . . . . . . . . . . . . . . 225

7.5.2 Cohomology of differential forms . . . . . . . . . . . . 232
7.6 ALGEBRAS, ENDOMORPHISMS AND DERIVATIVES . . . . . 239
8 SYMMETR IES 247
8.1 LIE GROUPS . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
8.2 TRANSFORMATIONS ON MANIFOLDS . . . . . . . . . . . . . 252
8.3 LIE ALGEBRA OF A LIE GROUP . . . . . . . . . . . . . . . 259
8.4 THE ADJOINT REPRESENTATION . . . . . . . . . . . . . 265
CONTENTS ix
9 FIBER BUNDLES 273
9.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . 273
9.2 VECTOR BUNDLES . . . . . . . . . . . . . . . . . . . . . . . 275
9.3 THE BUNDLE OF LINEAR FRAMES . . . . . . . . . . . . . . 277
9.4 LINEAR CONNECTIONS . . . . . . . . . . . . . . . . . . . . 284
9.5 PRINCIPAL BUNDLES . . . . . . . . . . . . . . . . . . . . . 297
9.6 GENERAL CONNECTIONS . . . . . . . . . . . . . . . . . . 303
9.7 BUNDLE CLASSIFICATION . . . . . . . . . . . . . . . . . . 316
III FINAL TOUCH 321
10 NONCOMMUTATIV E GEOMETRY 323
10.1 QUANTUM GROUPS — A PEDESTRIAN OUTLINE . . . . . . 323
10.2 QUANTUM GEOMETRY . . . . . . . . . . . . . . . . . . . . 326
IV MATHEMATICAL TOPICS 331
1 THE BASIC ALGEBRAIC STRUCTURES 333
1.1 Groups and lesser structures . . . . . . . . . . . . . . . . . . . . 334
1.2 Rings and fields . . . . . . . . . . . . . . . . . . . . . . . . . . 338
1.3 Module s and vector spaces . . . . . . . . . . . . . . . . . . . . . 341
1.4 Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344
1.5 Coalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348
2 DISCRETE GROUPS. BRAIDS AND KNOTS 351
2.1 A Discrete groups . . . . . . . . . . . . . . . . . . . . . . . . . 351
2.2 B Braids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356

2.3 C Knots and links . . . . . . . . . . . . . . . . . . . . . . . . . 363
3 SETS AND MEASURES 371
3.1 MEASURE SPACES . . . . . . . . . . . . . . . . . . . . . . . . 371
3.2 ERGODISM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375
4 TOPOLOGICAL LINEAR SPACES 379
4.1 Inner product space . . . . . . . . . . . . . . . . . . . . . . . 379
4.2 Norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380
4.3 Normed vector spaces . . . . . . . . . . . . . . . . . . . . . . . 380
4.4 Hilbert space . . . . . . . . . . . . . . . . . . . . . . . . . . . 380
4.5 Banach space . . . . . . . . . . . . . . . . . . . . . . . . . . . 382
4.6 Topological vector spaces . . . . . . . . . . . . . . . . . . . . . 382
x CONTENTS
4.7 Function spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 383
5 BANACH ALGEBRAS 385
5.1 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . 385
5.2 Banach algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 387
5.3 *-algebras and C*-algebras . . . . . . . . . . . . . . . . . . . . 389
5.4 From Geometry to Algebra . . . . . . . . . . . . . . . . . . . . 390
5.5 Von Neumann algebras . . . . . . . . . . . . . . . . . . . . . . 393
5.6 The Jones polynomials . . . . . . . . . . . . . . . . . . . . . . 397
6 REPRESENTATIONS 403
6.1 A Linear representations . . . . . . . . . . . . . . . . . . . . . 404
6.2 B Regular representation . . . . . . . . . . . . . . . . . . . . . 408
6.3 C Fourier expansions . . . . . . . . . . . . . . . . . . . . . . . 409
7 VARIATIONS & FUNCTIONALS 415
7.1 A Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415
7.1.1 Variation of a curve . . . . . . . . . . . . . . . . . . . . 415
7.1.2 Variation fields . . . . . . . . . . . . . . . . . . . . . . 416
7.1.3 Path functionals . . . . . . . . . . . . . . . . . . . . . . 417
7.1.4 Functional differentials . . . . . . . . . . . . . . . . . . 418

7.1.5 Second-variation . . . . . . . . . . . . . . . . . . . . . 420
7.2 B General functionals . . . . . . . . . . . . . . . . . . . . . . . 421
7.2.1 Functionals . . . . . . . . . . . . . . . . . . . . . . . . 421
7.2.2 Linear functionals . . . . . . . . . . . . . . . . . . . . 422
7.2.3 Operators . . . . . . . . . . . . . . . . . . . . . . . . . 423
7.2.4 Derivatives – Fr´echet and Gateaux . . . . . . . . . . . 423
8 FUNCTIONAL FORMS 425
8.1 A Exterior variational calculus . . . . . . . . . . . . . . . . . . . 426
8.1.1 Lagrangian density . . . . . . . . . . . . . . . . . . . . 426
8.1.2 Variations and differentials . . . . . . . . . . . . . . . . 427
8.1.3 The action functional . . . . . . . . . . . . . . . . . . 428
8.1.4 Variational derivative . . . . . . . . . . . . . . . . . . . 428
8.1.5 Euler Forms . . . . . . . . . . . . . . . . . . . . . . . . 429
8.1.6 Higher order Forms . . . . . . . . . . . . . . . . . . . 429
8.1.7 Relation to operators . . . . . . . . . . . . . . . . . . 429
8.2 B Existence of a lagrangian . . . . . . . . . . . . . . . . . . . . 430
8.2.1 Inverse problem of variational calculus . . . . . . . . . 430
8.2.2 Helmholtz-Vainberg theorem . . . . . . . . . . . . . . . 430
8.2.3 Equations with no lagrangian . . . . . . . . . . . . . . 431
CONTENTS xi
8.3 C Building lagrangians . . . . . . . . . . . . . . . . . . . . . . 432
8.3.1 The homotopy formula . . . . . . . . . . . . . . . . . . 432
8.3.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 434
8.3.3 Symmetries of equations . . . . . . . . . . . . . . . . . 436
9 SINGULAR POINTS 439
9.1 Index of a curve . . . . . . . . . . . . . . . . . . . . . . . . . . 439
9.2 Index of a singular point . . . . . . . . . . . . . . . . . . . . . 442
9.3 Relation to topology . . . . . . . . . . . . . . . . . . . . . . . 443
9.4 Basic two-dimensional singularities . . . . . . . . . . . . . . . 443
9.5 Critical points . . . . . . . . . . . . . . . . . . . . . . . . . . . 444

9.6 Morse lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . 445
9.7 Morse indices and topology . . . . . . . . . . . . . . . . . . . 446
9.8 Catastrophes . . . . . . . . . . . . . . . . . . . . . . . . . . . 447
10 EUCLIDEAN SPACES AND SUBSPACES 449
10.1 A Structure equations . . . . . . . . . . . . . . . . . . . . . . 450
10.1.1 Moving frames . . . . . . . . . . . . . . . . . . . . . . 450
10.1.2 The Cartan lemma . . . . . . . . . . . . . . . . . . . . 450
10.1.3 Adapted frames . . . . . . . . . . . . . . . . . . . . . . 450
10.1.4 Second quadratic form . . . . . . . . . . . . . . . . . . 451
10.1.5 First quadratic form . . . . . . . . . . . . . . . . . . . 451
10.2 B Riemannian structure . . . . . . . . . . . . . . . . . . . . . 452
10.2.1 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . 452
10.2.2 Connection . . . . . . . . . . . . . . . . . . . . . . . . 452
10.2.3 Gauss, Ricci and Codazzi equations . . . . . . . . . . . 453
10.2.4 Riemann tensor . . . . . . . . . . . . . . . . . . . . . . 453
10.3 C Geometry of surfaces . . . . . . . . . . . . . . . . . . . . . . 455
10.3.1 Gauss Theorem . . . . . . . . . . . . . . . . . . . . . . 455
10.4 D Relation to topology . . . . . . . . . . . . . . . . . . . . . . 457
10.4.1 The Gauss-Bonnet theorem . . . . . . . . . . . . . . . 457
10.4.2 The Chern theorem . . . . . . . . . . . . . . . . . . . . 458
11 NON-EUCLIDEAN GEOMETRIES 459
11.1 The old controversy . . . . . . . . . . . . . . . . . . . . . . . . 459
11.2 The curvature of a metric space . . . . . . . . . . . . . . . . . 460
11.3 The spherical case . . . . . . . . . . . . . . . . . . . . . . . . . 461
11.4 The Boliyai-Lobachevsky case . . . . . . . . . . . . . . . . . . 464
11.5 On the geodesic curves . . . . . . . . . . . . . . . . . . . . . . 466
11.6 The Poincar´e space . . . . . . . . . . . . . . . . . . . . . . . . 467
xii CONTENTS
12 GEODESICS 471
12.1 Self–parallel curves . . . . . . . . . . . . . . . . . . . . . . . . 472

12.1.1 In General Relativity . . . . . . . . . . . . . . . . . . . 472
12.1.2 The absolute derivative . . . . . . . . . . . . . . . . . . 473
12.1.3 Self–parallelism . . . . . . . . . . . . . . . . . . . . . . 474
12.1.4 Complete spaces . . . . . . . . . . . . . . . . . . . . . 475
12.1.5 Fermi transport . . . . . . . . . . . . . . . . . . . . . . 475
12.1.6 In Optics . . . . . . . . . . . . . . . . . . . . . . . . . 476
12.2 Congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . 476
12.2.1 Jacobi equation . . . . . . . . . . . . . . . . . . . . . . 476
12.2.2 Vorticity, shear and expansion . . . . . . . . . . . . . . 480
12.2.3 Landau–Raychaudhury equation . . . . . . . . . . . . . 483
V PHYSICAL TOPICS 485
1 HAMILTONIAN MECHANICS 487
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487
1.2 Symplectic structure . . . . . . . . . . . . . . . . . . . . . . . 488
1.3 Time evolution . . . . . . . . . . . . . . . . . . . . . . . . . . 490
1.4 Canonical transformations . . . . . . . . . . . . . . . . . . . . 491
1.5 Phase spaces as bundles . . . . . . . . . . . . . . . . . . . . . 494
1.6 The algebraic structure . . . . . . . . . . . . . . . . . . . . . . 496
1.7 Relations between Lie algebras . . . . . . . . . . . . . . . . . . 498
1.8 Liouville integrability . . . . . . . . . . . . . . . . . . . . . . . 501
2 MORE MECHANICS 503
2.1 Hamilton–Jacobi . . . . . . . . . . . . . . . . . . . . . . . . . 503
2.1.1 Hamiltonian structure . . . . . . . . . . . . . . . . . . 503
2.1.2 Hamilton-Jacobi equation . . . . . . . . . . . . . . . . 505
2.2 The Lagrange derivative . . . . . . . . . . . . . . . . . . . . . 507
2.2.1 The Lagrange derivative as a covariant derivative . . . 507
2.3 The rigid body . . . . . . . . . . . . . . . . . . . . . . . . . . 510
2.3.1 Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . 510
2.3.2 The configuration space . . . . . . . . . . . . . . . . . 511
2.3.3 The phase space . . . . . . . . . . . . . . . . . . . . . . 511

2.3.4 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 512
2.3.5 The “space” and the “body” derivatives . . . . . . . . 513
2.3.6 The reduced phase space . . . . . . . . . . . . . . . . . 513
2.3.7 Moving frames . . . . . . . . . . . . . . . . . . . . . . 514
2.3.8 The rotation group . . . . . . . . . . . . . . . . . . . . 515
CONTENTS xiii
2.3.9 Left– and right–invariant fields . . . . . . . . . . . . . 515
2.3.10 The Poinsot construction . . . . . . . . . . . . . . . . . 518
3 STATISTICS AND ELASTICITY 521
3.1 A Statistical Mechanics . . . . . . . . . . . . . . . . . . . . . . 521
3.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 521
3.1.2 General overview . . . . . . . . . . . . . . . . . . . . . 522
3.2 B Lattice models . . . . . . . . . . . . . . . . . . . . . . . . . 526
3.2.1 The Ising model . . . . . . . . . . . . . . . . . . . . . . 526
3.2.2 Spontaneous breakdown of symmetry . . . . . . . . . 529
3.2.3 The Potts model . . . . . . . . . . . . . . . . . . . . . 531
3.2.4 Cayley trees and Bethe lattices . . . . . . . . . . . . . 535
3.2.5 The four-color problem . . . . . . . . . . . . . . . . . . 536
3.3 C Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537
3.3.1 Regularity and defects . . . . . . . . . . . . . . . . . . 537
3.3.2 Classical elasticity . . . . . . . . . . . . . . . . . . . . 542
3.3.3 Nematic systems . . . . . . . . . . . . . . . . . . . . . 547
3.3.4 The Franck index . . . . . . . . . . . . . . . . . . . . . 550
4 PROPAGATION OF DISCONTINUITIES 553
4.1 Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . 553
4.2 Partial differential equations . . . . . . . . . . . . . . . . . . . 554
4.3 Maxwell’s equations in a medium . . . . . . . . . . . . . . . . 558
4.4 The eikonal equation . . . . . . . . . . . . . . . . . . . . . . . 561
5 GEOMETRICAL OPTICS 565
5.0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565

5.1 The light-ray equation . . . . . . . . . . . . . . . . . . . . . . 566
5.2 Hamilton’s point of view . . . . . . . . . . . . . . . . . . . . . 567
5.3 Relation to geodesics . . . . . . . . . . . . . . . . . . . . . . . 568
5.4 The Fermat principle . . . . . . . . . . . . . . . . . . . . . . . 570
5.5 Maxwell’s fish-eye . . . . . . . . . . . . . . . . . . . . . . . . . 571
5.6 Fresnel’s ellipsoid . . . . . . . . . . . . . . . . . . . . . . . . . 572
6 CLASSICAL RELATIVISTIC FIELDS 575
6.1 A The fundamental fields . . . . . . . . . . . . . . . . . . . . . 575
6.2 B Spacetime transformations . . . . . . . . . . . . . . . . . . . 576
6.3 C Internal transformations . . . . . . . . . . . . . . . . . . . . 579
6.4 D Lagrangian formalism . . . . . . . . . . . . . . . . . . . . . 579
xiv CONTENTS
7 GAUGE FIELDS 589
7.1 A The gauge tenets . . . . . . . . . . . . . . . . . . . . . . . . 590
7.1.1 Electromagnetism . . . . . . . . . . . . . . . . . . . . . 590
7.1.2 Nonabelian theories . . . . . . . . . . . . . . . . . . . . 591
7.1.3 The gauge prescription . . . . . . . . . . . . . . . . . . 593
7.1.4 Hamiltonian approach . . . . . . . . . . . . . . . . . . 594
7.1.5 Exterior differential formulation . . . . . . . . . . . . . 595
7.2 B Functional differential approach . . . . . . . . . . . . . . . . 596
7.2.1 Functional Forms . . . . . . . . . . . . . . . . . . . . . 596
7.2.2 The space of gauge potentials . . . . . . . . . . . . . . 598
7.2.3 Gauge conditions . . . . . . . . . . . . . . . . . . . . . 601
7.2.4 Gauge anomalies . . . . . . . . . . . . . . . . . . . . . 602
7.2.5 BRST symmetry . . . . . . . . . . . . . . . . . . . . . 603
7.3 C Chiral fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 603
8 GENERAL RELATIVITY 605
8.1 Einstein’s equation . . . . . . . . . . . . . . . . . . . . . . . . 605
8.2 The equivalence principle . . . . . . . . . . . . . . . . . . . . . 608
8.3 Spinors and torsion . . . . . . . . . . . . . . . . . . . . . . . . 612

9 DE SITTER SPACES 615
9.1 General characteristics . . . . . . . . . . . . . . . . . . . . . . 615
9.2 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619
9.3 Geodesics and Jacobi equations . . . . . . . . . . . . . . . . . 620
9.4 Some qualitative aspects . . . . . . . . . . . . . . . . . . . . . 621
9.5 Wigner-In¨on¨u contraction . . . . . . . . . . . . . . . . . . . . 621
10 SYMMETRIES ON PHASE SPACE 625
10.1 Symmetries and anomalies . . . . . . . . . . . . . . . . . . . . 625
10.2 The Souriau momentum . . . . . . . . . . . . . . . . . . . . . 628
10.3 The Kirillov form . . . . . . . . . . . . . . . . . . . . . . . . . 629
10.4 Integrability revisited . . . . . . . . . . . . . . . . . . . . . . . 630
10.5 Classical Yang-Baxter equation . . . . . . . . . . . . . . . . . 631
VI Glossary and Bibliography 635
Part I
MANIFOLDS
1

Chapter 1
GENERAL TOPOLOGY
Or, the purely qualitative properties of spaces.
1.0 INTRODUCTORY COMMENTS
§ 1.0.1 Let us again consider our ambient 3-dimensional euclidean space
E
3
. In order to introduce ideas like proximity between points, boundedness
of subsets, convergence of point sequences and the dominating notion —
continuity of mappings between E
3
and other point sets, elementary real
analysis starts by defining open r-balls around a point p:

1
B
r
(p) =

q ∈ E
3
such that d(q, p) < r

.
The same is done for n-dimensional euclidean spaces E
n
, with open r-balls
of dimension n. The question worth raising here is whether or not the real
analysis so obtained depends on the chosen distance function. Or, putting
it in more precise words: of all the usual results of analysis, how much is
dependent on the metric and how much is not? As said in the Preamble,
Physics should use experience to decide which one (if any) is the convenient
metric in each concrete situation, and this would involve the whole body
of properties consequent to this choice. On the other hand, some spaces of
physical relevance, such as the space of thermodynamical variables, are not
explicitly endowed with any metric. Are we always using properties coming
from some implicit underlying notion of distance ?
1
Defining balls requires the notion of distance function
†
, which is a function d taking
pairs (p, q) of points of a set into the real positive line R
+
and obeying certain conditions.

A complete definition is found in the Glossary. Recall that entries in the Glossary are
indicated by an upper dagger
†
.
3
4 CHAPTER 1. GENERAL TOPOLOGY
§ 1.0.2 There is more: physicists are used to “metrics” which in reality do
not lead to good distance functions. Think of Minkowski space, which is R
4
with the Lorentz metric η:
η(p, q) =

(p
0
− q
0
)
2
− (p
1
− q
1
)
2
− (p
2
− q
2
)
2

− (p
3
− q
3
)
2

1/2
.
It is not possible to define open balls with this pseudo-metric, which allows
vanishing “distances” between distinct points on the light cone, and even
purely imaginary “distances”. If continuity, for example, depends upon the
previous introduction of balls, then when would a function be continuous on
Minkowski space?
§ 1.0.3 Actually, most of the properties of space are quite independent of
any notion of distance. In particular, the above mentioned ideas of proximity,
convergence, boundedness and continuity can be given precise meanings in
spaces on which the definition of a metric is difficult, or even forbidden.
Metric spaces are in reality very particular cases of more abstract objects,
the topological spaces, on which only the minimal structure necessary to
introduce those ideas is present. That minimal structure is a topology, and
answers for the general qualitative properties of space.
§ 1.0.4 Consider the usual 2-dimensional surfaces immersed in E
3
. To be-
gin with, there is something shared by all spheres, of whatever size. And
also something which is common to all toruses, large or small; and so on.
Something makes a sphere deeply different from a torus and both different
from a plane, and that independently of any measure, scale or proportion. A
hyperboloid sheet is quite distinct from the sphere and the torus, and also

from the plane E
2
, but less so for the latter: we feel that it can be somehow
unfolded without violence into a plane. A sphere can be stretched so as to be-
come an ellipsoid but cannot be made into a plane without losing something
of its “spherical character”. Topology is that primitive structure which will
be the same for spheres and ellipsoids; which will be another one for planes
and hyperboloid sheets; and still another, quite different, for toruses. It will
be that set of qualities of a space which is preserved under suave stretching,
bending, twisting. The study of this primitive structure makes use of very
simple concepts: points, sets of points, mappings between sets of points. But
the structure itself may be very involved and may leave an important (even-
tually dominant) imprint on the physical objects present in the space under
consideration.
§ 1.0.5 The word “topology” is – like “algebra” – used in two different
senses. One more general, naming the mathematical discipline concerned
1.1. TOPOLOGICAL SPACES 5
with spacial qualitative relationships, and another, more particular, naming
that structure allowing for such relationships to be well defined. We shall
be using it almost exclusively with the latter, more technical, meaning. Let
us proceed to make the basic ideas a little more definite. In order to avoid
leaving too many unstated assumptions behind, we shall feel justified in
adopting a rather formal approach,
2
starting modestly with point sets.
1.1 TOPOLOGICAL SPACES
§ 1.1.1 Experimental measurements being inevitably of limited accuracy,
the constants of Nature (such as Planck’s constant , the light velocity c,
the electron charge e, etc.) appe aring in the fundamental equations are not
known with exactitude. The pro cess of building up Physics presupposes this

kind of “stability”: it assumes that, if some value for a physical quantity is
admissible, there must be always a range of values around it which is also
acceptable. A wavefunction, for example, will depend on Planck’s constant.
Small variations of this constant, within experimental errors, would give other
wavefunctions, by necessity equally acceptable as possible. It follow s that,
in the modeling of nature, each value of a mathematical quantity must be
surrounded by other admissible values. Such neighbouring values must also,
by the same reason, be contained in a set of acceptable values. We come thus
to the conclusion that values of quantities of physical interest belong to sets
enjoying the following property: every acceptable point has a neighbourhood
of points equally acceptable, each one belonging to another neighbourhood
of acceptable points, etc, etc. Sets endowed with this property, that around
each one of its points there exists another set of the same kind, are called
“open sets”. This is actually the old notion of open set, abstracted from
euclidean balls: a subset U of an “ambient” set S is open if around each
one of its points there is another set of points of S entirely contained in U.
All physically admissible values are, therefore, necessarily members of open
sets. Physics needs open sets. Furthermore, we talk frequently about “good
behaviour” of functions, or that they “tend to” some value, thereby loosely
conveying ideas of continuity and limit. Through a succession of abstractions,
the mathematicians have formalized the idea of open set while inserting it in
a larger, more comprehensive context. Open sets appear then as members
of certain families of sets, the topologies, and the focus is concentrated on
the properties of the families, not on those of its members. This enlarged
2
A commendable text for beginners, proceeding constructively from unstructured sets
up to metric spaces, is Christie 1976. Another readable account is the classic Sierpi´nski
1956.
6 CHAPTER 1. GENERAL TOPOLOGY
context provides a general and abstract concept of open sets and gives a clear

meaning to the above rather elusive word “neighbourhood”, while providing
the general background against which the fundamental notions of continuity
and convergence acquire well defined contours.
§ 1.1.2 A space will be, to begin with, a set endowed with some decompo-
sition allowing us to talk about its parts. Although the elements belonging
to a space may be vectors, matrices, functions, other sets, etc, they will be
called, to simplify the language, “points”. Thus, a space will be a set S of
points plus a structure leading to some kind of organization, such that we
may speak of its relative parts and introduce “spatial relationships”. This
structure is introduced as a well-performed division of S, as a convenient fam-
ily of subsets. There are various ways of dividing a set, each one designed to
accomplish a definite objective.
We shall be interested in getting appropriate notions of neighbourhood,
distinguishability of points, continuity and, later, differentiability. How is a
fitting decomposition obtained? A first possibility might be to consider S
with all its subsets. This conception, though acceptable in principle, is too
particular: it leads to a quite disconnected space, every two points belonging
to too many unshared neighbourhoods. It turns out (see section 1.3) that
any function would be continuous on such a “pulverized” space and in con-
sequence the notion of continuity would be void. T he family of subsets is
too large, the decomposition would be too “fine-grained”. In the extreme
opposite, if we consider only the improper subsets, that is, the whole point
set S and the empty set ∅, there would be no real decomposition and again
no useful definition of continuity (subsets distinct from ∅ and S are called
proper subsets). Between the two extreme choices of taking a family with
all the subsets or a family with no subsets at all, a compromise has been
found: good families are defined as those respecting a few well chosen, suit-
able conditions. Each one of such well-bred families of subsets is called a
topology.
Given a point set S, a topology is a family of subsets of S (which are

called, by definition, its open sets) respecting the 3 following conditions:
(a) the whole set S and the empty set ∅ belong to the family;
(b) given a finite number of members of the family, say U
1
, U
2
, U
3
, . . . , U
n
,
their intersection

n
i=1
U
i
is also a member;
(c) given any number (finite or infinite) of open sets, their union belongs to
the family.
1.1. TOPOLOGICAL SPACES 7
Thus, a topology on S is a collection of subsets of S to which belong the
union of any subcollection and the intersection of any finite subcollection, as
well as ∅ and the set S proper. The paradigmatic open balls of E
n
satisfy, of
course, the above conditions. Both the families suggested above, the family
including all subsets and the family including no proper subsets, respect
the above conditions and are consequently accepted in the club: they are
topologies indeed (called respectively the discrete topology and the indiscrete

topology of S), but very peculiar ones. We shall have more to say about them
later (see below, §’s 1.1.18 and 1.3.5). Now:
a topological space is a point set S
on which a topology is defined.
Given a point set S, there are in general many different families of subsets
with the above properties, i.e., many different possible topologies. Each such
family will make of S a different topological space. Rigour would require that
a name or symbol be attributed to the family (say, T ) and the topological
space be given name and surname, being denoted by the pair (S, T ).
Some well known topological spaces have historical names. When we say
“euclidean space”, the set R
n
with the usual topology of open balls is meant.
The members of a topology are called “open sets” precisely by analogy with
the euclidean case, but notice that they are determined by the specification
of the family: an open set of (S, T ) is not necessarily an open set of (S, T

)
when T = T

. Think of the point set of E
n
, which is R
n
, but with the discrete
topology including all subsets: the set {p} containing only the point p of R
n
is an open set of the topological space (R
n
, discrete topology), but not of the

euclidean space E
n
= (R
n
, topology of n-dimensional balls).
§ 1.1.3 Finite Space: a very simple topological space is given by the set
of four letters S = {a, b, c, d} with the family of subsets
T = {{a}, {a, b}, {a, b, d}, S, ∅}.
The choice is not arbitrary: the family of subsets
{{a}, {a, b}, {b, c, d}, S, ∅},
for example, does not define a topology, because the intersection
{a, b} ∩ {b, c, d} = {b}
is not an open set.
8 CHAPTER 1. GENERAL TOPOLOGY
§ 1.1.4 Given a point p ∈ S, any set U containing an open set belonging
to T which includes p is a neighbourhood of p. Notice that U itself is not
necessarily an open set of T: it simply includes
3
some open set(s) of T . Of
course any p oint will have at least one neighbourhood, S itself.
§ 1.1.5 Metric spaces
†
are the archetypal topological spaces. The notion of
topological space has evolved conceptually from metric spaces by abstraction:
properties unnecessary to the definition of continuity were progressively for-
saken. Topologies generated from a notion of distance (metric topologies) are
the most usual in Physics. As an experimental science, Physics plays with
countings and measurements, the latter in general involving some (at least
implicit) notion of distance. Amongst metric spaces, a fundamental role will
be played by the first example we have met, the euclidean space.

§ 1.1.6 The euclidean space E
n
The point set is the set R
n
of n-uple s
p = (p
1
, p
2
, . . . , p
n
), q = (q
1
, q
2
, . . . , q
n
), etc, of real numbers; the distance
function is given by
d(p, q) =

n

i=1
(p
i
− q
i
)
2


1/2
.
The topology is formed by the set of the open balls. It is a standard practice
to designate a topological space by its point set when there is no doubt as
to which topology is meant. That is why the euclidean space is frequently
denoted simply by R
n
. We shall, however, insist on the notational differ-
ence: E
n
will be R
n
plus the ball topology. E
n
is the basic, starting space,
as even differential manifolds will be presently de fined so as to generalize it.
We shall see later that the introduction of coordinates on a general space S
requires that S resemble some E
n
around each one of its points. It is impor-
tant to notice, however, that many of the most remarkable properties of the
euclidean space come from its being, besides a topological space, something
else. Indeed, one must be careful to distinguish properties of purely topolog-
ical nature from those coming from additional structures usually attributed
to E
n
, the main one being that of a vector space.
§ 1.1.7 In metric spaces, any point p has a countable set of open neighb our-
hoods {N

i
} such that for any set U containing p there exists at least one N
j
included in U. Thus, any set U containing p is a neighbourhood. This is not
a general property of topological spaces. Those for which this happens are
said to be first-countable spaces (Figure 1.1).
3
Some authors (Kolmogorov & Fomin 1977, for example) do define a neighbourhood
of p as an open set of T to which p belongs. In our language, a neighbourhood which is
also an open set of T will be an “open neighbourhood”.
1.1. TOPOLOGICAL SPACES 9
Figure 1.1: In first-countable spaces, every point p has a countable set of open neigh-
bourhoods {N
k
}, of which at least one is included in a given U  p. We say that “all
points have a local countable basis”. All metric spaces are of this kind.
§ 1.1.8 Topology basis In order to specify a topological space, one has to
fix the point set and tell which amongst all its subsets are to be taken as
open sets. Instead of giving each member of the family T (which is frequently
infinite to a very high degree), it is in general much simpler to give a subfamily
from which the whole family can be retraced. A basis for a topology T is a
collection B of its open sets such that any member of T can be obtained as
the union of elements of B. A general criterium for B = {U
α
} to be a basis
is stated in the following theorem:
B = {U
α
} is a basis for T iff, for any open set V ∈ T and all p ∈ V , there
exists some U

α
∈ B such that p ∈ U
α
⊂ V .
The open balls of E
n
constitute a prototype basis, but one might think of open
cubes, open tetrahedra, etc. It is useful, to get some insight, to think about
open disks, open triangles and open rectangles on the euclidean plane E
2
. No
two distinct topologies may have a common basis, but a fixed topology may
have many different basis. On E
2
, for instance, we could take the open disks,
or the open squares or yet rectangles, or still the open ellipses. We would
say intuitively that all these different basis lead to the same topology and
we would be strictly correct. As a topology is most frequently intro duced
via a basis, it is useful to have a criterium to check whether or not two basis
correspond to the same topology. This is provided by another theorem:
10 CHAPTER 1. GENERAL TOPOLOGY
B and B

are basis defining the same topology iff, for every U
α
∈ B and
every p ∈ U
α
, there exists some U


β
∈ B

such that p ∈ B

β
⊂ U
α
and
vice-versa.
Again, it is instructive to give some thought to disks and rectangles in E
2
. A
basis for the real euclidean line E
1
is provided by all the open intervals of the
type (r −1/n, r + 1/n), where r runs over the se t of rational numbers and n
over the set of the integer numbers. This is an example of countable basis.
When a topology has at least one countable basis, it is said to be second-
countable. Second countable topologies are always first-countable (§ 7) but
the inverse is not true. We have said above that all metric spaces are first-
countable. There are, however, metric spaces which are not second countable
(Figure 1.2).
Figure 1.2: A partial hierarchy: not all metric spaces are second-countable, but all of
them are first-countable.
We see here a first trial to classify topological spaces. Topology frequently
resorts to this kind of practice, trying to place the space in some hierarchy.
In the study of the anatomy of a topological space, some variations are sometimes helpful.
An example is a small change in the concept of a basis, leading to the idea of a ’network’.
A network is a collection N of subsets such that any member of T can be obtained as the

union of elements of N. Similar to a basis, but accepting as members also sets which are
not open sets of T .