Eschrig H. Topology and Geometry for Physics

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Seminorm in a vector space: a function p : X ! R

with the properties

• p(x + x

) B p(x)+p(x

) (subadditivity),

• pðkxÞ¼jkjpðxÞ

• If moreover p(x) = 0, iff x = 0, then pðxÞ¼kxk is called a norm.

A family of seminorms {p

ja 2 A} in the vector space X is separating, if for

every x 2 X, x = 0 there is some a with p

(x) [ 0.

Locally convex space Vector space topologized with a separating family of

seminorms and with the neighborhood base at 0 of open sets B

fða

Þg;n

ð0Þ¼

fx 2 X jp

ði ¼ 1; ...; nÞ; e

[ 0g; 0\n\1:

A locally convex space with a countable neighborhood base at 0 can be metrized

with the distance function dð x; yÞ¼

i¼1

i

ðx yÞð1 þ p

ðx yÞÞ

1

Normed vector space A neighborhood base at 0 and a metric are deﬁned by a

single norm: dðx; yÞ¼kx  yk:

Fréchet space Complete metrizable locally convex space.

Banach space Complete normed vector space.

Banach algebra Banach space X with a norm continuous multiplication X  X !

X : ðx; yÞ7!xy which makes X into an algebra with unity e, and

• kek¼1,

• kxykkxkkyk:

Linear function (linear operator) L: X ? Y, X, Y normed vector spaces:

L(kx + k

) = kL(x)+k

L(x

L is bounded if kLk¼sup

06¼x2X

kLðxÞk

=kxk

\1: A bounded linear function

is continuous.

LðX; YÞ¼fL : X ! Y jkLk\1g is a normed vector space by ðkL þ

ÞðxÞ¼kLðxÞþk

ðxÞ; k; k

scalars in Y. It is Banach, if Y is Banach.

Topological dual X



¼LðX; KÞ of a topological vector space X Set of all

continuous linear functionals f : X ! K : x 7!hf ; xi2K; hf ; kx þ k

i¼

khf ; xiþk

hf ; x

i, made into a vector space by hkf þ k

; xi¼khf ; xiþ

; xi (observe the difference to an inner product). X

is always Banach

(since K is Banach).

There is an embedding J : X ! X



¼ðX



: x 7!

x, were

x is deﬁned by

x; f i¼hf ; xi for all f 2 X

.IfJ is surjective, X = X

is called reﬂexive.

Weak topology on X Given by the neighborhood base at 0 of open sets

2jhf ; xij\1=ngn ¼ 1; 2... for all f 2 E

, a base of the vector space X

.In

general, X is only locally convex in the weak topology; in a ﬁnite dimensional

Compendium 357

X weak and norm topologies are equal.

Weak

topology on X

Given by the neighborhood base at 0 of open sets {f 2 X

|hf, xi| \ 1/n}, n = 1, 2… for all x 2 E, a base of X , X

Inner product space A sesquilinear form (inner product) X  X ! C ðor RÞ :

ðx; yÞ7!ðxjyÞ is deﬁned with the properties

• ðxjyÞ¼

ðyjxÞ; z : complex conjugate of z,(z ¼ z for z 2 R),

• ðxjy

þ y

Þ¼ðxjy

Þþðxjy

Þ;

• ðxjkyÞ¼kðxjyÞðphysics conventionÞ;

• ðxjxÞ[ 0 for x 6¼ 0:

kxk¼ðxjxÞ

1=2

is a norm for which the Schwarz inequality jðxjyÞjkxkkyk holds.

Two vectors x, y 2 X are called orthogonal to each other, if (x|y) = 0.

Hilbert space Complete inner product space. In a direct sum X  Y of Hilbert

spaces, X and Y = X

are orthogonal by deﬁnition. The tensor product

X  Y of Hilbert spaces is the product space with the inner product

(ðx  yjx

 y

Þ¼ðxjx

Þðyjy

Þ).

Unitary space Finite dimensional complex Hilbert space.

Euclidean space Finite dimensional real Hilbert space. Angles are deﬁned by

cos ]ðx; yÞ¼ðxjyÞ=ðkxkkykÞ:

Directional derivative of a function F: X ? Y from an open set X of a normed

vector space X into a topological vector space Y:

Fðx

Þ¼

Fðx

þ txÞ



t¼0

¼ lim

t6¼0; t!0

þtx2X

Fðx

þ txÞFðx

; k xk¼1:

Gâteaux derivative, if D

F(x

) is a continuous linear function of x 2 X.

Total derivative (Fréchet derivative) of a function F: X ? Y from an open set X

of a normed vector space X into a normed vector space Y : DFðx

Þ2LðX; YÞ

so that

Fðx

þ xÞFðx

Þ¼DFðx

Þx þ RðxÞkxk; lim

x!0

RðxÞ¼0;

If D

F(x

) is a continuous function of x

2 X, then D

F(x

) = DF(x

)x.

Functional derivative of a functional F: X ? K from an open set X of a normed

vector space X into its scalar ﬁeld K.

If X ¼ L

ðK

; d

zÞ3f ðzÞthen DFðf

Þ¼gðzÞ2L

ðK

; d

zÞ; 1=p þ1=q ¼ 1:

358 Compendium

Taylor expansion

Fðx

þ xÞ¼Fðx

ÞþDFðx

Þx þ

Fðx

Þxx þþ

Fðx

Þxx x

|ﬄﬄﬄ{zﬄﬄﬄ}

ðk factorsÞ

þ;

provided x

and x

+ x belong to a convex domain X , X on which F is deﬁned

and has total derivatives to all orders, which are continuous functions of x

in X

and provided this Taylor series converges in the norm topology of Y. Here,

Fðx

Þ2LðX; LðX; ...; LðX; YÞ...ÞÞ is a k-linear function from X 9 X 9 _

9 X (k factors) into Y.

LðX; LðX; ...; LðX; YÞ...ÞÞ

|ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}

depth k

¼ sup

ð1Þ

x

ðkÞ

ð1Þ

x

ðkÞ

ð1Þ

kx

ðkÞ

Chain rule F: X . X ? Y, F(X) , X

, G: Y . X

? Z and H = G  F:

X . X ? Z. Then,

DHðx

Þ¼DGðFðx

ÞÞ  DFðx

if the right hand side derivatives exist. In this case, DFðx

Þ2LðX; YÞ and

DGðFðx

ÞÞ 2 LðY; ZÞ and hence DHðx

Þ2LðX; ZÞ: Moreover, if DF : X !

LðX; YÞ is continuous at x

2 X and DG : X

!LðY; ZÞ is continuous at

F(x

) 2 X

, then DH : X !LðX; ZÞ is continuous at x

2 X.

Class C

(X, Y) function n = 0, 1, …, ?, x: F:(X , X) ? Y which for n = 0is

continuous, has continuous derivatives up to order n, n = 1, 2,…,?, and has a

converging Taylor expansion for n = x.Forn = ? it is called smooth, for

n = x it is called analytic.

diffeomorphism a bijective mapping from X , X onto X

, Y which, along

with its inverse, is C

, n [ 0, (C

, C

Derivative of a product in a C

-algebra, n C 1:

DðFGÞ¼ðDFÞG þ FðDGÞ:

Implicit function theorem Let X, Z be normed vector spaces and let Y be a

Banach space. Let F 2 C

((X , X 9 Y), Z) and consider the equation

Fðx; yÞ¼c; c 2 Z fixed:

Assume that F(a, b) = c and that Q = D

F(a, b) is a linear bijection from

Y onto Z, so that Q

1

2LðZ; YÞ: Then, there are open sets A 3 a and B 3 b in

X and Y, so that for every x 2 A the above equation has a unique solution y 2 B

which deﬁnes a continuous function G : A ! Y : x 7!y ¼ GðxÞ implicitly by

the above equation. The function G has a continuous total derivative at x = a,

and

Compendium 359

DGðaÞ¼ðD

Fða; bÞÞ

1

 D

Fða; bÞ; b ¼ GðaÞ:

Open cover of a set X in a topological space X: Family of open sets U

a 2 A so

that [

 X: A subfamily which is also a cover is called a subcover.

Compact set C Every open cover of C contains a ﬁnite subcover, [

i¼1

 C: A

Hausdorff compact set is a compactum. A set is relatively compact, if its

closure is compact.

A compactum is closed. A closed subset of a compact set is compact.

Every inﬁnite subset of a compact set has a cluster point.

Every sequence in a compact set has a convergent subsequence.

Locally compact topological space X Every point x 2 X has a relatively compact

neighborhood.

Compact function (operator) It is continuous and maps bounded sets into

relatively compact sets.

Lower (upper) semicontinuous function F : X !

R : For every x 2 X and every

e [ 0 there is a neighborhood U

ðxÞ in which F [ Fð xÞeðF\FðxÞþeÞ:

F is ﬁnite from below,ifF(x) [ -? for all x.

Extremum problems A continuous real-valued function on a compact set takes on

its maximum and minimum values.

If F is a ﬁnite from below and lower semicontinuous function from a non-empty

compactum A into R; then F is even bounded below and the minimum problem

min

x2A

FðxÞ¼a has a solution x

2 A, F(x

) = a.

Fixed point theorems Banach: A strict contraction F : X ! X; dðFx; Fx

Þ

kdðx; x

Þ; k\1, on a complete metric space X has a unique ﬁxed point.

Tychonoff: A continuous mapping F: C ? C in a compact convex set C of a

locally convex space has a ﬁxed point.

Schauder: A compact mapping F: C ? C in a closed bounded convex set C of a

Banach space has a ﬁxed point.

Banach-Alaoglu theorem The unit ball of the dual X

of a Banach space X is

compact in the weak

topology.

The unit ball of a reﬂexive Banach space is compact in the weak topology.

A Banach space is in general not ﬁrst countable in the weak and weak

topologies; this is why instead of sequences nets are needed.

360 Compendium

Net Set of points x

2 X indexed with a directed set I 3 a (every pair

(a, b) 2 I 9 I has an upper bound c 2 I, a B c, b B c).

Every net in a compact set has a convergent subnet.

Function of compact support F: X ? Y, X locally compact space, Y normed

vector space, supp F is contained in some compactum; supp F is the smallest

closed set outside of which F(x) = 0. A class C

(X, Y) function is a class

(X, Y) function with compact support (n = 0, …, ?).

Partition of unity A family fu

ja 2 Ag of C

ðX; RÞ-functions such that

• there is a locally ﬁnite open cover, X [

b2B

(every x 2 X has a

neighborhood W

which intersects only with ﬁnitely many U

• the support of each u

is in some U

• 0 u

ðxÞ1onX for every a,

•

a2A

ðxÞ¼1onX.

The partition of unity is called subordinate to the cover [

b2B

Paracompact space A Hausdorff topological space for which every open cover,

X [

a2A

, has a locally ﬁnite reﬁnement, that is a locally ﬁnite open cover

[

b2B

for which every V

is a subset of some U

It is a space which permits a partition of unity.

Connectedness A topological space is connected,if

• it is not a union of two disjoint non-empty open sets, or equivalently,

• it is not a union of two disjoint non-empty closed sets, or equivalently,

• the only open-closed sets are the empty set and the space itself.

Otherwise it is disconnected.

The connected component of a point x of a topological space X is the largest

connected set in X containing x.

X is totally disconnected, if its connected components are all its one point sets

{x}. (A discrete space is totally disconnected; the rational line Q in the relative

topology as a subset of R is totally disconnected, but not discrete.)

X is locally connected, if every point has a neighborhood base of connected

neighborhoods.

The image F(A) of a connected set A in a continuous mapping is a connected

set.

Homotopy Continuous function H: [0, 1] 9 X ? Y translating the continuous

function F

: X ? Y into the continuous function F

: X ? Y: H(0, ) = F

and

H(1, ) = F

. F

and F

are called homotopic, F

% F

. By concatenating two

Compendium 361

homotopies, H

translating F

into F

and H

translating F

into F

, their

product H

is deﬁned as a homotopy translating F

into F

. Homotopic, %,is

an equivalence relation dividing C

(X, Y) into homotopy classes [F]. The

homotopy class of a constant function mapping X into a single point of Y is

called the null-homotopy class.

Homotopy equivalent Two topological spaces X and Y are homotopy equivalent,

if their exists continuous functions F: X ? Y and G: Y ? X so that

F  G % Id

and G  F % Id

. X is called contractible, if it is homotopy

equivalent to a one point space.

Pathwise connected (also called arcwise connected) A topological space X is

pathwise connected, if for every pair (x, x

) of points of X there is a continuous

function H: [0, 1] ? X, H(0) = x, H(1) = x

. A general space X consists of the

set p

(X) of its pathwise connected components.IfX is a topological group,

then p

(X) is its zeroth homotopy group.

Locally pathwise connected A space X is locally pathwise connected, if every

point has a neighborhood base of pathwise connected sets.

nth homotopy group p

(X) of a pathwise connected topological space X: The

homotopy classes of functions from the n-dimensional sphere S

into X,

mapping the north pole of the sphere into a ﬁxed point of X. By an intermediate

homeomorphism from the n-sphere to the one-point compactiﬁed n-cube, two

mappings may be concatenated along the x

-axis of the cube. Concatenation as

group operation yields a group structure in p

(X). If X is a (not necessarily

pathwise connected) topological group, then the group multiplication yields an

isomorphic group structure on p

) for the pathwise connected component X

of X, and p

(X) = p

(X) 9 p

). p

(X) is called the fundamental group of X.

n-connected A topological space is called n-connected (also n-simple), if every

continuous image in X of the n-dimensional sphere S

is contractible. A

topological group X is n-connected, if p

ðXÞp

ðXÞ: A 0-connected space is

pathwise connected, a 1-connected space is called simply connected.

C.3 Smooth Manifolds

See for instance [4].

Finite-dimensional smooth manifold M

• M is a paracompact topological space, or slightly more special, M is locally

compact, Hausdorff and second countable,

• every point x 2 M has a neighborhood U

which is homeomorphic to an open

subset U

of the Euclidean space R

by a homeomorphism u

: U

! U

m is the dimension of M,

362 Compendium

• the charts ðU

; u

Þ form an atlas of M, that is, the U

form a cover of M, and

there is a complete family of C

-diffeomorphisms w

from U

to U

compatible with the u

with respect to composite mappings.

• There exists always a complete atlas A

which is not a proper subset of any

other atlas of M; it is also called the differentiable structure of M.

Coordinate neighborhood of x

2 M:

• A base {e

, …, e

} with u

ðx

Þ as origin in R

so that x ¼

2 U

and

ðxÞ¼ðu

ðxÞ; ...; u

ðxÞÞ with u

¼ p

 u

and p

(x) = x

• transition functions w

¼ðw

ðxÞ; ...; w

ðxÞ with a regular Jacobian.

• The set fu

g is called a local coordinate system on U

2 M; fu

ðxÞg are

local coordinates of x 2 M.

Orientable manifold A smooth manifold M which permits a complete atlas the

transition functions of which all have only positive Jacobians.

Product manifold of two manifolds ðM; A

Þ and ðN; A

Þ is the manifold ðM 

N; A

A

Þ with the product topology and the complete atlas containing the

natural product of the complete atlases A

and A

; its dimension is dim

M + dim N.

Smooth parametrized curve in M : R a; b½3t 7!xðtÞ2M; so that every

ðtÞ¼u

 xðtÞ for a restriction of x(t) to the corresponding open interval of

t is a smooth vector function with values in U

Smooth real function on M : M 3 x 7! FðxÞ2R: It deﬁnes in every coordinate

neighborhood U

a smooth real function F

ðx

; ...; x

Þ¼ðF  u

1

Þðx

Þ:

Tangent vector X

on M at x

: For every smooth real function F; X

F 7!X

F 2 R : in local coordinates

F ¼

; n

ðw

; ðw

¼ p

 w

 p

;

where p

ðx

Þ¼ð0; ...; 0; x

; 0; ...; 0Þ2R

; the tangent vectors at x

span an m-

dimensional real vector space, the tangent space T

ðMÞ:

Cotangent vector x

on M at x

: For every X

; x

: X

7!hx

; X

i2R : in

local coordinates

; X

i¼

; x

ðw

1

; ðw

1

¼ðw

;

the cotangent vectors span the m-dimensional real cotangent space T



ðMÞ;in

particular

Compendium 363



; hdF

; X

i¼X

Local bases

o=ox

; x

;

hdx

; o=ox

i¼d

; dx

ðw

; o = o x

ðw

1

o=ox

Tangent and cotangent vector ﬁelds Functions X : M 3 x 7!X

2 T

ðMÞ and

x : M 3 x 7!x

2 T



ðMÞ, so that XF : x 7!X

F and xðXÞ : x 7!hx

; X

i are

smooth real functions on M for every smooth F and every smooth X,

respectively. In any local coordinate system the components n

of X and x

of x

are smooth real functions of the local coordinates.

Lie product of two tangent vector ﬁelds: It is a tangent vector ﬁeld

½X; Y¼XY  YX; ½X; YF ¼

 g



the tangent vector ﬁelds on M form a Lie algebra.

Smooth mapping of manifolds F: M ? N induces at every point x 2 M as a

push forward a linear mapping F

: T

(M) ? T

F(x)

(N) and at every point

F(x) 2 N as a pull back a linear mapping F

F(x)

: T

F(x)

(N) ? T

(M) with



FðxÞ

ðx

FðxÞ

ÞÞ; X

i¼hx

FðxÞ

; F



ðX

Þi;

the composite of F: M ? N and G: N ? P yields (G  F)

= G

F(x)

 F

and

(G  F)

G(F(x))

= F

F(x)

 G

G(F(x))

, that is, F

composes covariantly and F

composes contravariantly; if F

is injective at every point x 2 M, then the

mapping F is an immersion of M into N; if in addition F itself is injective, then

it is an embedding of M into N and M is an embedded submanifold of N.

Tensor ﬁelds on M: Functions t mapping x 2 M into a tensor product of tangent

and cotangent spaces on M at x, in local coordinates

tðxÞ¼t

...i

...j

ðxÞ



 dx

dx

;

with Einstein summation over pairs of equal upper and lower indices

understood, so that the component functions t

...i

...j

ðxÞ are smooth in every

coordinate neighborhood and transform for each index like tangent and

cotangent vector components, respectively; t is of type (r, s).

Tensor product t  t

, in coordinate neighborhoods

ðt  t

...i

rþr

...j

sþs

¼ t

...i

...j

rþ1

...i

rþr

sþ1

...j

sþs

364 Compendium

Tensor contraction C

p,q

(t), in coordinate neighborhoods

p;q

ðtÞ

...i

r1

...j

s1

¼ t

...i

p1

...i

r1

...j

q1

...j

s1

Lie derivative L

with respect to a tangent vector ﬁeld X, in coordinate

neighborhoods

ðL

uÞ

...i

...j

¼ n

...i

...j



p¼1

...i

p1

pþ1

...i

...j

qþ1

...i

...j

q1

qþ1

...j

;

it maps tensors of type (r, s) to tensors of the same type and is the derivative of

u along integral curves (‘ﬁeld lines’) of X; L

[X,Y]

= [L

, L

Exterior r-form is an alternating tensor of type (0, r).

Exterior product For r-, s-, and t-forms x, r, and s and functions F and G,

• x ^ r = (- 1)

r ^ s,

• x ^ (Fr + Gs) = Fx ^ r + Gx ^ s,

• (x ^ r) ^ s = x ^ (r ^ s);

the general r-form in a coordinate neighborhood is

x ¼

\...\i

...i

ðxÞdx

^^dx

; x ¼ 0ifr [ m;

x ¼ t

...i

dx

; x

...i

¼ r! t

...i

; t alternating:

½x; r¼

\\i

rþs

...i

rþ1

...i

rþs

 r

rþ1

...i

rþs

...i



^^dx

rþs

;

1-forms: [x, r] = x ^ r (in general = -r ^ x for non-commutative

quantities x

and r

Exterior derivative d in coordinate neighborhoods

dx ¼

rþ1

s¼1

ð1Þ

sþ1

\\i

rþ1

...i

s1

sþ1

...i

rþ1

^^dx

rþ1

Interior multiplication For a tangent vector ﬁeld X and an r-form x,

ðxÞ¼C

1;1

ðX  xÞ¼r

i;i

\\i

...i

r1

^^dx

r1

Exterior calculus Basic relations are

¼ d  i

þ i

 d; ½d; L

¼0; ½i

; L

¼i

½Y;X

; d

¼ 0; ði

¼ 0:

Compendium 365

Stokes’ theorem

dx ¼

where X is a domain in M with boundary oX and x is an m-form, m = dim M.

C.4 Topological Groups

See for instance [5, 4].

Topological group G

• G is a group (the abstract group of G),

• G is a topological space,

• G  G ! G : ðg; hÞ7!gh

1

is continuous.

If B

¼fB

g is a neighborhood base of the unity e 2 G, then B

¼fgB

g is a

neighborhood base of g for every g 2 G.

Subgroup H

• H is an abstract subgroup of G,

• H is a closed subset of G.

If H is an abstract subgroup of G, then

H is a subgroup of the topological group G.

Homomorphism f : G ? G

• f is an algebraic homomorphism,

• f is continuous.

An isomorphism is an algebraic isomorphism and a homeomorphism.

The kernel N ¼ ker f of a homomorphism f : G ? G

is an invariant subgroup

of G, and the quotient group G=N with the quotient topology is a topological

group.

The direct product of groups of two topological groups is a topological group in

the product topology.

If U is any neighborhood of the unity e 2 G, then [

n=1

= G

is the

connected component of e of G. G

is an invariant subgroup of G, and G



G=G

is totally disconnected. (If G

is pathwise connected, then G

= p

(G),

the zeroth homotopy group of G.)

Lie group G:

• G is a topological group,

366 Compendium