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

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-

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

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

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

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

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

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

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.

− (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

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

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

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.


1/2
.
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

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


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