Gliklikh Y.E. Global and Stochastic Analysis with Applications to Mathematical Physics

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26 1 Manifolds and Related Objects

or polyvector (polyvector ﬁeld). Often one also omits the words “exterior” or

“diﬀerential”, saying “s-form”.

Real-valued functions are included in this terminology as 0-forms, covec-

tors as 1-forms and vectors as 1-vectors. Note that the term “1-form” is far

more commonly used than the word “covector”. In what follows we shall gen-

erally refer to 1-forms, only rarely using the term “covector”. Similarly, the

term “1-vector” is rarely used, so we shall generally refer to vectors.

Clearly there are two parallel theories for s-forms and r-vectors. Our main

focus shall be on s-forms with the understanding that analogous results hold

for r-vectors. Only when we need to use both will polyvectors appear.

Exterior product

As for symmetric tensors, as described above, there exists a certain special

reduction of the tensor product which is very useful when dealing with s-

forms. It is called the exterior product.

Let ˜a

, ˜a

...˜a

be 1-forms. We introduce the following formal matrix with

1-form coeﬃcients

A =

⎛

⎜

⎝

˜a

...˜a

˜a

...˜a

............

˜a

...˜a

⎞

⎟

⎠

. (1.27)

Just as for a matrix with real-valued coeﬃcients, for (1.27) one can con-

sider its determinant det A where the ordinary product of real numbers is

replaced by the tensor product of 1-forms. Since the tensor product is not

commutative (unlike the ordinary numerical product) this yields a non-trivial

expression. For a permutation σ of {1,...,n}, the coeﬃcient of the summand

˜a

σ(1)

˜a

σ(2)

···˜a

σ(n)

in this determinant is equal to the sign of σ.

Deﬁnition 1.61.

˜a

∧ ˜a

∧···∧˜a

=detA.

Immediately from Deﬁnition 1.61 we see that ˜a

∧ ˜a

∧···∧˜a

is a skew-

symmetric (0,l)-tensor; changing the order of any two forms in the product

results in a change of the sign. It follows that if there are at least two equal

factors in the product, the product is equal to zero (since the product remains

unchanged if we swap two such identical factors).

For two 1-forms a and b, by Deﬁnition 1.61, we obtain a∧b =(a⊗b−b⊗a).

Remark 1.62. Note that there are l! summands in the determinant of (1.27).

Thus it is tempting to insert the factor

before det A in the deﬁnition of the

exterior product, imitating the deﬁnition of the symmetric tensor product in

Section 1.5. We emphasize that we do not do so since it is rather convenient

to omit it in the formulae below. It should be pointed out that that the factor

does appear in some text-books. This leads to some changes in consequent

formulae. So, if you use formulae from diﬀerent books, you should check what

1.6 Diﬀerential Forms and Polyvectors 27

deﬁnitions of exterior product the authors use in order to avoid running into

absurdities of the sort 1 = 0.

From the deﬁnition of the exterior product we can easily see that any s-

form can be presented as a linear combination of exterior products of basis

1-forms (covectors) dq

. The problem is that these products are not linearly

independent: for example dq

∧dq

,dq

∧dq

and dq

∧dq

diﬀer from one another only by sign. Moreover, a product having at least two

common factors is equal to zero (see the previous section). In order to avoid

this problem we shall choose the order of factors in the exterior products of

basis 1-forms according to increasing order of indices and exclude the prod-

ucts with equal factors. The resulting products become linearly independent

and so they form the basis of the space of forms of speciﬁed degree at a point

of M.

Thus any s-form at a given point of M has the expansion

ω = ω

...i

∧···∧dq

(1.28)

where i

< ···<i

The space of all s-forms at a point m ∈ M is denoted by ∧

∗

M and

the bundle of all s-forms by ∧

∗

M. We can now calculate the dimension of

∧

∗

M for the n-dimensional manifold M . By the construction it is equal to

the binomial coeﬃcient C

.NotethatC

= C

n−s

. In particular this means

that ∧

∗

M is isomorphic to ∧

n−s

∗

M since the dimensions of these ﬁnite-

dimensional linear spaces coincide. Unfortunately, as usual, there are many

isomorphisms between these spaces but none can be considered as canonical

– the best one for all problems. Later we shall ﬁnd a candidate for the ‘best’

isomorphism on a Riemannian manifold with some additional property.

Theorem 1.63 dim ∧

∗

M =1, and hence ∧

∗

M is isomorphic to the

space of 0-forms at m, i.e., to R

Indeed, C

= C

= 1. The unique basis form in ∧

∗

M is dq

∧ dq

∧

···∧dq

We say that an n-form ω at some point m is identically zero if its value is

zero on any multiplicity of n vectors from T

Corollary 1.64

(i) Let ω

,ω

∈∧

∗

M and ω

be not identically zero. Then ω

= λω

for some real number λ.

(ii) Let ω

,ω

be two diﬀerential n-forms on M and ω

be nowhere iden-

tically zero on M.Thenω

= λ(m)ω

where λ(m) is a real-valued

function on M.

Theorem 1.65 The forms of degree greater than n on a manifold with di-

mension n are identically zero.

28 1 Manifolds and Related Objects

Indeed, if the degree is greater than n and there are only n basis 1-forms

, there will be equal factors in the exterior products of expansion (1.28).

By the properties of the exterior product, all such products are equal to zero.

The exterior diﬀerential d

Recall that in Section 1.1 we deﬁned the notion of the diﬀerential dF of a

map F sending a certain manifold M,dimM = n, to a linear space R

This operation d was very useful for dealing with a real-valued function f on

M (here k = 1). It transforms f into the diﬀerential 1-form df on M (see

Proposition 1.14).

Here we extend the operation d to diﬀerential forms of higher degree so

that it transforms an s-form α into an (s+1)-form dα. This extension is called

the exterior diﬀerential. For the sake of simplicity we introduce the exterior

diﬀerential in terms of coordinates. This is probably the only exception to

our general idea to ﬁrst introduce a new object in invariant form and then to

describe it in terms of coordinates. The invariant construction of the exterior

diﬀerential is more complicated than the coordinate construction.

We deﬁne the exterior diﬀerential d on 0-forms as the operation coinciding

with the above-mentioned d. For an s-form α, s>0, with the expansion

(1.28), i.e.,

α = α

...i

∧···∧dq

we deﬁne the action of d by the formula

dα =dα

...i

∧ dq

∧···∧dq

(1.29)

where dα

...i

is the diﬀerential of the function α

...i

, i.e., a 1-form that can

be expanded by general formula (1.14): dα

...i

∂α

...i

∂q

so that (1.29)

is transformed into

dα =

∂α

...i

∂q

∧ dq

∧···∧dq

. (1.30)

Note that (1.30) does not satisfy the above convention that the factors in

the exterior products dq

∧ dq

∧ ··· ∧dq

should be ordered according

to increasing size of indices, so for each form α it needs to be rewritten

accordingly. Note also that dq

∧ dq

∧···∧dq

may have equal factors so

that some summands in (1.30)mayvanish.

Deﬁnition 1.66. If dα =0theformα is called closed.Ifα =dβ for some

form β, α is called exact.

Theorem 1.67 d

=d◦ d=0, i.e., all exact forms are closed.

Remark 1.68. Note that in R

all closed forms are exact. The same is true of

any contractible manifold. In the general case the diﬀerence between closed

1.6 Diﬀerential Forms and Polyvectors 29

and exact forms indicates the topological structure of the manifold. This

diﬀerence is described by means of the so-called de Rham cohomologies.

Physically equivalent r-forms and r-vectors

Let M be a Riemannian manifold, i.e. a manifold M on which a Riemannian

metric ·, · = g

⊗ dq

is given. As for ordinary tensors, in this case the

notion of physical equivalence is well-deﬁned for skew-symmetric tensors of

types (s, 0) and (0,s). For the sake of convenience we shall denote physically

equivalent forms and polyvectors by the same letter putting a tilde over the

form and a bar over the polyvector. For instance,

X and

X denote a form

and polyvector, respectively, each physically equivalent to the other.

Theorem 1.69 Let ˜α =˜a

∧···∧ ˜a

be a k-form where a

are 1-forms.

Then ¯α =¯a

∧···∧¯a

where ¯a

are the vectors physically equivalent to ˜a

i =1,...,k.

This theorem follows directly from the construction.

Recall that dq

and

∂

∂q

are generally not physically equivalent to each

other. This means that the transition to the physically equivalent object

should be done by means of the general formulae (1.20)and(1.21)inthe

same manner as in Section 1.5 for general tensors.

The interior product

This is a version of a contraction (or trace, see Section 1.5) adapted to the

language of diﬀerential forms.

Consider an r-vector X and an s-form α, r<s.Theirinterior product is

denoted by X  α and is deﬁned as follows. Create the tensor product of X

and α, both expanded as in (1.25). This yields an (r, s)-tensor. Now contract

it with respect to all r contravariant factors and the ﬁrst r covariant factors in

each summand of the expansion (1.25) so that the k-th contravariant factor

is contracted with the k-th covariant factor. X  α is the resulting tensor of

type (0,s−r). Since X and α are skew-symmetric, so too is X  α, i.e., it is a

diﬀerential (exterior) form. Its expansion should be found in the form (1.28).

Volume forms. Orientable manifolds

Recall (see Corollary 1.64)thatonann-dimensional manifold M for two

n-forms α and β at some point m ∈ M ,whereβ is not identically zero, we

have α = λβ where λ is a real number. Thus for two diﬀerential forms α and

β,

such that β is identically zero nowhere on M ,wehaveα = λ(m)β where

λ is a real-valued function on M.

The problem is whether there exist a diﬀerential n-form on M which is

nowhere identically zero. There are manifolds on which any n-form is iden-

30 1 Manifolds and Related Objects

tically zero at at least one of its points. Such manifolds are called non-

orientable.

If there exist an n-form that is nowhere identically zero on M, M is called

orientable. Obviously if a single such form exists, there must be inﬁnitely

many such forms: for example one can multiply the form by any real number

or by a non-zero real-valued function on M. The set of all such n-forms on an

orientable manifold M is naturally divided into two classes as follows: specify

a certain set of n linearly independent vectors at a given point m ∈ M,then

two n-forms belong to the same subclass if their values on the above set

have the same sign. We summarize this as follows: there exist two possible

orientations on an orientable manifold.

On an orientable manifold M it is convenient to specify a certain nowhere

identically zero n-form. In this case we say that an orientation has been

chosen on M and call M an oriented manifold. This form is called the volume

form on the oriented manifold M.

The latter term originates from the following. Let M be an oriented Rie-

mannian (or semi-Riemannian) manifold (as is commonly found in many

physical problems). In this case there is a canonical volume form Ω con-

structed as follows. Choose an orientation on M , i.e., specify a certain n-form

ω that is nowhere identically zero. Note that any tangent space T

M, m ∈ M,

is a Euclidean (or semi-Euclidean, respectively) space. For any set of vectors

,...,X

∈ T

M deﬁne the value Ω(X

,...,X

) to be the volume of the

parallelepiped spanning X

,...,X

, with the sign + if ω(X

,...,X

) > 0

and with the sign − if ω(X

,...,X

) < 0. Note that if X

,...,X

are not lin-

early independent, the volume of the spanning parallelepiped is equal to zero,

i.e. the above construction is well-deﬁned. Obviously Ω is skew-symmetric,

i.e., it is an n-form at m. Doing this for all m ∈ M we obtain a diﬀerential

n-form that is clearly nowhere identical zero.

Deﬁnition 1.70. The above form Ω is called the Riemannian volume form.

In a Euclidean space with orthonormal basis

∂

∂q

,...,

∂

∂q

(i.e., dq

(

∂

∂q

here) we have Ω =dq

∧,...,∧dq

. In a chart of a Riemannian manifold

the Riemannian volume form is described by the formula (see [202])

Ω =



det(g

)dq

∧,...,∧dq

. (1.31)

On a manifold the integrals of real-valued functions are well-deﬁned if the

integrator is a volume form. We refer the reader, say, to [212] for details.

The operations ∗, δ and Δ

Let M be a Riemannian (or semi-Riemannian) oriented manifold with Rie-

mannian volume form Ω.

The operation ∗ is an isomorphism of the space of k-forms onto the space

of (n−k)-forms at any point m ∈ M. For a certain k-form ˜α the (n−k)-form

∗˜α is deﬁned by the formula:

1.6 Diﬀerential Forms and Polyvectors 31

∗ ˜α =

¯α  Ω (1.32)

where ¯α is the k-vector physically equivalent to ˜α. Note that the coeﬃcient

is involved in order to naturally coordinate the formulas with each other. For

instance, one can easily see that in the Euclidean space R

, by formula (1.32),

∗(dq

∧ dq

)=dq

while it would be equal to 2dq

without the coeﬃcient.

Since for all k =0,...,n, ∗ : ∧

∗

M →∧

n−k

∗

M is an isomorphism,

there exists an inverse ∗

−1

: ∧

∗

M →∧

n−s

∗

M for all s =0,...,n.(Note:

if s = n − k then k = n − s.)

Deﬁnition 1.71. The operator δ = ∗

−1

d ∗ is called the codiﬀerential.

Let α be a k-form. Then ∗α is an (n−k)-form, d ∗α is an (n −k +1)-form

and δα = ∗

−1

d ∗α is a (k − 1)-form.

Theorem 1.72 δ

=0.

Indeed, δ

=(∗

−1

d∗)(∗

−1

d∗)=∗

−1

∗ =0sinced

= 0 by Theorem 1.67.

Deﬁnition 1.73. Δ =(d+δ)

is called the Laplace-de Rham operator (or

Kodaira-Hodge Laplacian).

Since d

=0andδ

= 0, one can easily calculate that

Δ =dδ + δd (1.33)

(note that d and δ do not commute and so the summands on the right-hand

side of (1.33) are not equal to each other).

If α is a k-form, then since d increases the degree by one and δ decreases

thedegreebyone,Δα is also a k-form, i.e., Δ preserves the degree.

Relations to operators of classical vector analysis

Making use of the operations with diﬀerential forms introduced above, one

can generalize the operators gradf,divX and rotX of vector analysis in

three-dimensional Euclidean space to spaces of higher dimension and to Rie-

mannian (semi-Riemannian) manifolds.

In the remainder of this section we consider a Riemannian (or semi-

Riemannian) manifold M, a smooth vector ﬁeld

X and a smooth real-valued

function f on M. As introduced above, we use the symbols bar and tilde over

a certain expression to denote physically equivalent polyvectors and forms,

respectively. In particular, by

X we denote the 1-form physically equivalent

X. Recall that the coordinates of

X are denoted by X

and those of

X by

Recall also the deﬁnition of the gradient of a function given in Deﬁnition

1.55. In our new notation gradf =

df. One can easily see that in the Eu-

clidean space R

, i.e., where dq

∂

∂q

, this formula yields the usual deﬁnition

of gradient from vector analysis.

32 1 Manifolds and Related Objects

Deﬁnition 1.74. The divergence of the vector ﬁeld

X is the function div

X =

By direct calculation one can easily show that in the Euclidean space R

the above deﬁnition gives the ordinary divergence.

Remark 1.75. Below (see Deﬁnition 1.86) we give an alternative deﬁnition

of divergence in terms of the so-called Lie derivative. It will be applicable in

even broader settings (for instance, the manifold M may not be Riemannian).

Deﬁnition 1.76. The rotation of the vector ﬁeld X is the (n−2)-vector ﬁeld

rot

X =

∗d

Notice that only for n = 3 do we have n − 2 = 1, i.e., only in this case is

rot

X a 1-vector ﬁeld. As in the case of divergence, by direct calculations one

can easily show that in the Euclidean space R

the above deﬁnition gives the

ordinary rotation.

1.7 The Lie Derivative

Here we present a method of diﬀerentiation of various objects along a vector

ﬁeld on a manifold. Note that even in Euclidean spaces this method, generally

speaking, does not coincide with the ordinary derivative.

First we should describe the general notion of diﬀerentiating something

along a smooth curve. Let Q be a vector space. Associate a copy of Q to each

point m of a manifold M and denote it by Q

.Letq(m) ∈ Q

be chosen so

that the “ﬁeld” q(m) is given on M. Consider a smooth curve m(t), t ∈ [0,l].

Suppose that there exists a set of linear operators A(t):Q

m(t)

→ Q

m(0)

Consider the curve q(t)=A(t)q(m(t)) in the vector space Q

m(0)

. Calculate

q(t)att = 0. This derivative (if it exists) is a vector of Q

m(0)

, i.e., an object

of the same sort as the ﬁeld q(m). Note that this manner of diﬀerentiating

depends on the operator A(t); having determined A(t) we obtain a speciﬁc

type of diﬀerentiation.

For example, if q(m) is a smooth vector ﬁeld on M and A(t) is the opera-

tor of parallel translation, the above derivative coincides with the covariant

derivative

(see Section 2.2).

The Lie derivative is deﬁned according to the above scheme where the

operator A(t) is determined by a particular vector ﬁeld on M , as we describe

below.

Let a smooth vector ﬁeld X be given on M. Consider its ﬂow g

(m)(see

Deﬁnition 1.6). Let for a point m

∈ M the integral curve g

) exist for

t ∈ [0,ε

]andsoforanyt ∈ (0,ε

) the neighboring integral curves also

exist. Thus we can consider the map g

on a neighborhood of m

and since it

is jointly smooth in t and m, we can consider its tangent Tg

and its cotangent

1.7 The Lie Derivative 33

∗

mappings. These mappings will be used as operators A(t) for translating

vectors and forms along g

) in the construction of Lie derivatives. Note

that since g

is a diﬀeomorphism, Tg

is also one-to-one and there exists an

inverse map Tg

−1

We begin the construction of the Lie derivative with that for real-valued

functions. Consider such a function f : M → R.Letm

∈ M and consider

the integral curve g

). Note that here the general construction is reduced

since by deﬁnition all values f(g

)) belong to the same real line R and so

we needn’t move them along. Thus, according to the general idea, we should

create the function f(g

)) and diﬀerentiate it at t = 0. The resulting

number L

f(m

) is called the Lie derivative of f along X at m

.Having

done this at all points of M we obtain the function L

f called the Lie

derivative of f along X.

Proposition 1.77 L

f coincides with the ordinary derivative Xf of f

along X.

Indeed, the statement is obvious since the constructions of L

f and Xf

coincide (cf. Deﬁnition 1.7).

Now we generalize the construction to diﬀerential forms of higher degree.

Let α be a k-form, 0 <k≤ n. Consider the forms α

)

along g

). In

ordertomovethemtom

along g

) we should generalize the construction

of the cotangent mapping (see Deﬁnition 1.16)sothatitappliestok-forms.

We do this for the map g

(the general construction is quite analogous).

Denote by T

∗

(α

)

)thek-form at the point m

whose value on the k

vectors

,...,

∈ T

M is given by:

∗

(α

)

)(

,...,

)=α

)

(Tg

,...,Tg

). (1.34)

Deﬁnition 1.78. The k-form

∗

(α

)

|t=0

at m

is called the Lie

derivative of α along X at m

and is denoted by L

α(m

). Having carried

out this procedure at all points of M we obtain the k-form L

α on M which

is called the Lie derivative of α along X.

Theorem 1.79 L

α = X  dα +d(X  α).

Note that both summands on the right-hand side of the last formula have

degree k (this follows from the properties of d and the interior product). We

leave the proof to the reader as a (not so simple) exercise (for a proof, see

[212]).

The last modiﬁcation of the construction deals with vector ﬁelds. Let Y be

a smooth vector ﬁeld on M. Consider its restriction Y (g

)) to the curve

). Now we translate those vectors into T

M by Tg

−1

which by its

construction sends T

)

M into T

Deﬁnition 1.80. The vector

−1

(Y (g

))

|t=0

∈ T

M is called the

Lie derivative of Y along X at m

and is denoted by L

Y (m

). Having

34 1 Manifolds and Related Objects

carried out this procedure at all points of M we obtain the vector ﬁeld L

Y ,

called the Lie derivative of Y along X.

Theorem 1.81 L

Y =[X, Y ], the Lie bracket of X and Y .

It is important to understand what is meant if the Lie derivative of an

object along X is equal to zero. Directly from the construction of the Lie

derivative we obtain:

Theorem 1.82

(i) If L

f =0, f is constant along integral curves of X.

(ii) If L

α =0, T

∗

(α

(m)

)=α

for all m ∈ M.

(iii) If L

Y =0, Tg

= Y

(m)

for all m ∈ M.

Deﬁnition 1.83. If T

∗

(α

(m)

)=α

(Tg

= Y

) for all m ∈ M we

say that α (Y , respectively) is constant along the ﬂow of X.

So, a zero value of a Lie derivative means that the corresponding object

is constant along the ﬂow of X. Hence, in particular, if we apply g

to an

integral curve of Y such that [X, Y ] = 0, the image is also an integral curve

of Y . Since [X, Y ]=−[Y,X] (see above), the same is true of the image of an

integral curve of X under the action of the ﬂow of Y .

Deﬁnition 1.84. If [X, Y ]=0wesaythatX and Y commute.

An obvious example of commuting vector ﬁelds is the pair of coordinate

ﬁelds

∂

∂q

and

∂

∂q

Theorem 1.85 (see, e.g., [26]) If [X, Y ]=0in some chart, in this chart

there exists a coordinate system (q

,...,q

) such that X =

∂

∂q

and Y =

∂

∂q

Using the Lie derivative one can extend the notion of divergence of a vector

ﬁeld. Let Ω be a volume form on an orientable manifold M and let X be a

smooth vector ﬁeld. The Lie derivative L

Ω is also an n-form and so it is

presented as the product of a real-valued function and Ω.

Deﬁnition 1.86. The function divX in the equality L

Ω =divX · Ω is

called the divergence of the vector ﬁeld X on the manifold M with speciﬁed

volume form Ω.

Let M be a Riemannian manifold.

Proposition 1.87 If the form Ω in Deﬁnition 1.86 is the Riemannian vol-

ume form, divX from Deﬁnition 1.86 coincides with the divergence in Deﬁ-

nition 1.74.

Proof. By Theorem 1.79 L

Ω = X  dΩ +d(X  Ω). Since Ω is an n-form,

dΩ =0andsoL

Ω =d(X  Ω). But d(X  Ω)=d(∗

X)(see(1.32)). On

the other hand, by construction divX from Deﬁnition 1.86 equals ∗

−1

Ω.

Hence divX = ∗

−1

d ∗

X = δ

X (see Deﬁnition 1.71). 

Chapter 2

Connections

2.1 The Structure of a Tangent Bundle to a Vector

Bundle

Let π : Θ → M be a vector bundle with standard ﬁber R

,dimM = n.

Denote by Θ

the ﬁber at m ∈ M and by (m, ϑ)=ϑ

the points of this

ﬁber. Consider a chart U

on a manifold M with local coordinates (q

,...,q

)

and a trivialization F

of the bundle over that chart. Let e

,...,e

be the

standard basis in R

.SinceF

(π

−1

)=U

× R

, this basis generates a

basis in every ﬁber Θ

, m ∈ U

. We obtain a smooth ﬁeld of bases that

will also be denoted by e

,...,e

. Thus every cross-section ϑ of the bundle

Θ can be represented in terms of coordinates with respect to these bases in

the form ϑ = ϑ

, i =1,...,d.InU

× R

the set of vectors U

×{X

} for

some X

∈ R

corresponds to the vectors ϑ from Θ

, m ∈ U

that have the

same coordinates with respect to e

,...,e

as X

. Another trivialization of

the bundle over U

would generate another set of vectors equivalent to X

that is diﬀerent from the former.

In the vector bundle the set F

(π

−1

)=U

× R

can be considered

as a chart on the total space Θ. Denote by ϑ

the coordinates in the ﬁbers

, m ∈ U

, whose coordinate axes are spanned by the basis vectors e

the ﬁbers. We obtain the coordinate system (q

, ..., q

,ϑ

, ..., ϑ

)inthechart

× R

on Θ. By a general scheme (see Section 1.1) this system generates

a basis

∂

∂q

,...,

∂

∂q

∂

∂ϑ

,...,

∂

∂ϑ

in the tangent space T

(m,ϑ)

Θ to the total

space Θ of the bundle at every point (m, ϑ), m ∈ U

The symbols

∂

∂q

,...,

∂

∂q

for the ﬁrst “half” of the basis vectors coincide

with the symbols for the basis vectors

∂

∂q

,...,

∂

∂q

in the tangent space T

m ∈ U

,toM generated by coordinates (q

,...,q

). This is natural since

the former vectors are tangent to a submanifold U

×{V } in F

(π

−1

)

where the coordinates (q

,...,q

) are generated by the projection of the

same coordinates from U

, and that projection is an isomorphism. On the

Y.E. Gliklikh, Global and Stochastic Analysis with Applications

to Mathematical Physics, Theoretical and Mathematical Physics,

DOI 10.1007/978-0-85729-163-9