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tensors.
The space of all the symmetric covariant tensors on a linear space V can be made into a commutative algebra in the following way. Call Sk(V ) the linear space of covariant symmetric tensors of order k. The total space of symmetric covariant tensors on V will be the direct sum S(V ) = ⊕∞k=0Sk(V ).
the summation taking place over all the permutations P of the indices. Notice that T W ∈ Sk+j(V ). With this symmetrizing operation, the linear space S(V ) becomes a commutative algebra. The same can be made, of course, for contravariant tensors.
§ 6.3.8 An algebra like those above defined (see § 6.3.3 and § 6.3.7), which is a sum of vector spaces, V == ⊕∞k=0Vk, with the binary operation taking
gives a polynomial in the components of the vector v. The definition is actually basis-independent, and p is called a polynomial function of degree k. The space of such functions constitutes a linear space Pk(V ). The sum of these spaces, P (V ) = ⊕∞k=0Pk(V ), is an algebra which is isomorphic to the algebra S(V) of § 6.3.7.
§ 6.3.10 Of special interest are the antisymmetric tensors, which satisfy T (v1, v2, . . . , vk, . . . , vj, . . .) = − T (v1, v2, . . . , vj, . . . , vk, . . .)
for every pair j, k of indices. Let us examine the case of the antisymmetric covariant tensors. At a fixed order, they constitute a vector space by them-selves. The tensor product of two antisymmetric tensors of order p and q is
a (p + q)-tensor which is no more antisymmetric, so that the antisymmetric tensors do not constitute an algebra with the tensor product. We can how-ever introduce another product which redresses this situation. Before that, we need the notion of alternation Alt(T ) of a covariant tensor T of order s, which is a tensor of the same order defined by
Alt(T )(v1, v2, . . . , vs) = s!1 X
(P )
(sign P )T (vp1, vp2, . . . , vps), (6.19)
where the summation takes place on all the permutations P of the num-bers (1, 2, . . . , s). Symbol sign P represents the parity of P . The tensor Alt(T ) is antisymmetric by construction. If n is the number of elementary transpositions (Mathematical Topic 2) necessary to take (1, 2, . . . , s) into (p1, p2, . . . , ps), the parity sign P = (−)n.
§ 6.3.11 Given two antisymmetric tensors, ω of order p and η of order q, their exterior product ω ∧ η is the (p + q)-antisymmetric tensor given by
ω ∧ η = (p+q)!p!q! Alt(ω ⊗ η).
This operation does make the set of antisymmetric tensors into an associative graded algebra, the exterior algebra, or Grassmann algebra. Notice that only tensors of the same order can be added, so that this algebra includes in reality all the vector spaces of antisymmetric tensors. We shall here only list some properties of real tensors which follow from the definition above:
(i) (ω + η) ∧ α = ω ∧ α + η ∧ α (ii) α ∧ (ω + η) = α ∧ ω + α ∧ η
(iii) a (ω ∧ α) = (a ω) ∧ α = ω ∧ (a α), for any a ∈ R; (6.20) (iv) (ω ∧ η) ∧ α = ω ∧ η ∧ α)
(v) ω ∧ η = (−)∂ω∂ηη ∧ ω
In the last property, concerning commutation, ∂ω and ∂η are the orders re-spectively of ω and η.
If {αi} is a basis for the covectors, the space of s-order antisymmetric tensors has a basis
{αi1 ∧ αi2 ∧ αi3 ∧ . . . ∧ αis}, 1 ≤ i1, i2, . . . , is ≤ dim V . An antisymmetric s-tensor can then be written
ω = s!1 ωi1i2i3...isαi1 ∧ αi2 ∧ αi3 ∧ . . . ∧ αis, (6.21)
6.3. TENSORS ON MANIFOLDS 159 the ωi1i2i3...is’s being the components of ω in this basis. The space of anti-symmetric s-tensors reduces automatically to zero for s > dim V .
Notice further that the dimension of the vector space formed by the anti-symmetric covariant s-tensors is dim Vs . The dimension of the whole Grass-mann algebra is 2dim V.
§ 6.3.12 The exterior product is preserved by mappings between manifolds.
Let f : M → N be such a mapping and consider the antisymmetric s-tensor ωf (p) on the vector space Tf (p)N . The function f determines then a tensor on TpM through
(f∗ω)p(v1, v2, . . . , vs) = ωf (p)(f∗v1, f∗v2, . . . , f∗vs). (6.22) Thus, the mapping f induces a mapping f∗ between the tensor spaces,
Figure 6.8: A function f induces a push-foward f∗ and a pull-back f∗.
working however in the inverse sense (see the scheme of Figure 6.8): f∗ is suitably called a pull-back and f∗ is sometimes called, by extension, push-foward . To make [6.22] correct and well-defined, f must be C1. The pull-back has the following properties:
(i) f∗ is linear; (6.23)
(ii) f∗(ω ∧ η) = f∗ω ∧ f∗η; (6.24)
(iii) (f ◦ g)∗ = g∗◦ f∗. (6.25)
The pull-back, consequently, preserves the exterior algebra.
§ 6.3.13 Antisymmetric covariant tensors on differential manifolds are called differential forms. In a natural basis {dxj},
ω = s!1 ωj1j2j3...jsdxj1 ∧ dxj2 ∧ dxj3 ∧ . . . ∧ dxjs.
The well defined behaviour when mapped between different manifolds renders the differential forms the most interesting of all tensors. But of course we shall come to them later on.
§ 6.3.14 Let us now go back to differentiable manifolds. A tensor at a point p ∈ M is a tensor defined on the tangent space TpM . One can choose a chart which gives the transformation of the components under changes of coordi-nates in the charts’ intersection. Changes of basis unrelated to coordinate changes will be examined later on. We find frequently tensors defined by eq.[6.27]: they are those entities whose components transform in that way.
§ 6.3.15 It should be understood that a tensor is always a tensor with re-spect to a given group. In [6.27], the group of coordinate transformations is involved. General basis transformations (section 6.5 below) constitute another group, and the general tensors above defined are related to that group. Usual tensors in E3 are actually tensors with respect to the group of rotations, SO(3). Some confusion may arise because rotations may be rep-resented by coordinate transformations in E3. But not every transformation is representable through coordinates, and it is better to keep this in mind.
6.4. FIELDS & TRANSFORMATIONS 161
6.4
FIELDS & TRANSFORMATIONS6.4.1 Fields
§ 6.4.1 Let us begin with an intuitive view of vector fields. In the preceding sections, vectors and tensors have been defined at a fixed point p of a dif-ferentiable manifold M . Although we have been lazily negligent about this aspect, the natural bases we have used are actually { ∂
∂xi
p}. Suppose now that we extend these vectors throughout the whole chart’s coordinate neigh-bourhood, and that the components are differentiable functions fi : M → R, fi(p) = Xpi. New vectors are then obtained, tangent to M at other points of the coordinate neighbourhood. Through changes of charts, vectors can eventually be got all over the manifold. Now, consider a fixed vector at p, tangent to some smooth curve: it can be continued in the above way along the curve. This set of vectors, continuously and differentiably related along a differentiable curve, is a vector field. At p, Xp : R(M ) → R. At different points, X will map R(M ) into different points of R, that is, a vector field is a mapping X : R(M ) → R(M ). In this way, generalizing that of a vector, one gets the formal definition of a vector field:
a vector field X on a smooth manifold M is a linear mapping X : R(M ) → R(M ) obeying the Leibniz rule:
X(f · g) = f · X(g) + g · X(f ), f, g ∈ R(M ).
§ 6.4.2 The tangent bundle A vector field is so a differentiable choice of a member of TpM for each p of M . It can also be seen as a mapping from M the correctness of this second definition, one should establish a differentiable structure on the 2m-dimensional space TM. Let π be a function
π : T M → M, π(Xp) = p, to be called projection from now on.
§ 6.4.3 As for covering spaces (§ 3.2.15), open sets on T M are defined as those sets which can be obtained as unions of sets of the type π−1(U ), with
U an open set of M (so that π is automatically continuous). Given a chart where Xpi are the components in
Xp = Xpi two charts are, in this way, differentiably related. A complete atlas can in this way be defined on T M , making it into a differentiable manifold. This differentiable manifold T M is the tangent bundle, the simplest example of a differentiable fiber bundle, or bundle space. The tangent space to a point p, TpM , is called, in the bundle language, the fiber on p. The field X itself, defined by eq.[6.28], is a section of the bundle. Notice that the bundle space in reality depends on the projection π for the definition of its topology.
§ 6.4.4 The commutator Take the field X, given in some coordinate neigh-bourhood as X = Xi ∂∂xi. As X(f ) ∈ R(M ), one could consider the action of This expression tells us that the operator Y X, defined by
(Y X)f = Y (Xf ),
does not belong to the tangent space, due to the presence of the last term.
This annoying term is symmetric in XY , and would disappear under anti-symmetrization. Indeed, as easily verified, the commutator of two fields
[X, Y ] := (XY − Y X) = does belong to the tangent space and is another vector field.
6.4. FIELDS & TRANSFORMATIONS 163
§ 6.4.5 ... and its algebra The operation of commutation defines on the space T M a structure of linear algebra. It is also easy to check that
[X, X] = 0,
[[X, Y ], Z] + [[Z, X], Y ] + [[Y, Z], X] = 0,
the latter being the Jacobi identity. An algebra satisfying these two condi-tions is a Lie algebra. Thus, the vector fields on a manifold constitute, with the commutation, a Lie algebra.
§ 6.4.6 Notice that a diffeomorphism f preserves the commutator:
f∗[X, Y ] = [f∗X, f∗Y ].
Furthermore, given two diffeomorphisms f and g, (f ◦ g)∗X = f∗◦ g∗X.
§ 6.4.7 The cotangent bundle Analogous definitions lead to general ten-sor bundles. In particular, consider the union
T∗M = ∪p∈MTpM.
A covariant vector field, cofield or 1-form ω is a mapping ω : M → T∗M
such that ω(p) = ωp ∈ T∗M, p ∈ M .
§ 6.4.8 This corresponds, in just the same way as has been seen for the vectors, to a differentiable choice of a covector on each p ∈ M . In general, the action of a form on a vector field X is denoted
ω(X) = < ω, X >, so that
ω : T M → R(M ). (6.33)
§ 6.4.9 In the dual natural basis (in other words, locally),
ω = ωjdxj. (6.34)
Fields and cofields can be written respectively X = dxi(X) ∂
∂xi = < dxi, X > ∂
∂xi, (6.35)
ω = ω( ∂
∂xi) dxi = < ω, ∂
∂xi > dxi. (6.36)
§ 6.4.10 The cofield bundle above defined is the cotangent bundle, or the bundle of forms.
We shall see later (chapter 7) that not every 1-form is the differential of a function. Those who are differentials of functions are the exact forms.
§ 6.4.11 We have obtained vector and covector fields. An analogous proce-dure leads to tensor fields. We first consider the tensor algebra over a point p ∈ M , consider their union for all p, topologize and smoothen the resultant set, then define a general tensor field as a section of this tensor bundle.
§ 6.4.12 At each p ∈ M , we have two m-dimensional vector spaces, TpM and Tp∗M , of course isomorphic. Nevertheless, their isomorphism is not nat-ural (or canonical ). It depends on the chosen basis. Different basis fix isomorphisms taking the same vector into different covectors. Only the pres-ence of an internal product on TpM (due for instance to the presence of a metric, a case which will be seen later) can turn the isomorphism into a natural one. By now, it is important to keep in mind the total distinction between vectors and covectors.
§ 6.4.13 Think of Enas a vector space: its tangent vector bundle is a Carte-sian product. A tangent vector to En at a point p can be completely specified by a pair (p, V ), where also V is a vector in En. This comes from the iso-morphism between each TpEn and En itself. Forcing a bit upon the trivial, we say that En is parallelizable, the same vector V being defined at all the different points of the manifold En. Given a general manifold M , it is said to be parallelizable if its tangent bundle is trivial, that is, a mere Cartesian product T M = M × Em. In this case, a vector field V can be globally (that is, everywhere on M ) given by (p, V ). Recalling the definition of a vector field as a section on the tangent bundle, this means that there exists a global section. Actually, the existence of a global section implies the triviality of the bundle. This holds for any bundle: if some global section exists, the bundle is a Cartesian product.
All toruses are parallelizable. Of all the spheres Sn, only S1, S3 and S7 are parallelizable. The sphere S2 is not — a result sometimes called the hedgehog theorem: you cannot comb a hairy hedgehog so that all its prickles stay flat. There will be always at least one point like the crown of the head.
The simplest way to find out whether M is parallelizable or not is based on the simple idea that follows: consider a vector V 6= 0. Then, the vector field (, V ) will not vanish at any point of M . Suppose that we are able to show that no vector field on M is everywhere nonvanishing. This would imply that T M is not trivial. A necessary condition for parallelizability is the vanishing
6.4. FIELDS & TRANSFORMATIONS 165 of the Euler-Poincar´e characteristic of M . All Lie groups are parallelizable differentiable manifolds.
§ 6.4.14 Dynamical systems Dynamical systems are described in Classi-cal Physics by vector fields in the “phase” space (q, ˙q). Consider the free fall of a particle of unit mass under the action of gravity: call x the height and y the velocity, y = ˙x. From
˙
y = − g (constant),
one gets the velocity in “phase” space (vx, vy) = (y, − g). A scheme of this vector field is depicted in Figure 6.9.
Figure 6.9: Vector field scheme for ˙y = − g.
A classical system is completely specified by its velocity field in “phase”
space, which fixes its time evolution (Physical Topic 1). Initial conditions simply choose one of the lines in the “flow diagram”. Well, we should per-haps qualify such optimistic statements. In general, this perfect knowledge does not imply complete predictability. Small indeterminations in the initial conditions may be so amplified during the system evolution that after some time they cover the whole configuration space (see Mathematical Topic 3.2).
This happens even with a simple system like the double oscillator with non-commensurate frequencies. The above example is precisely the field vector characterization of the system of differential equations
˙x = y ; ˙y = − g .
The modern approach to systems of differential equations is based on the idea of vector field.4
4 A detailed treatment of the subject, with plenty of examples, is given in the little masterpiece Arnold 1973.
§ 6.4.15 Dynamical systems: maps Dynamical systems are also de-scribed, mainly in model building, by iterating maps like
xn+1 = f (xn) ,
where x is a vector describing the state of some system. To help visualization, we may consider n as a discrete time. The state at the n-th stage is given by a function of the (n − 1)-th stage, and so by its n-th iterate applied on the initial seed state x0. The set of points {xn} by which the system evolves is the orbit of the system. An important concept in both the flow and the map pictures is the following: suppose there is a compact set A to which the sequence xn(or, in the flow case, the state when t becomes larger and larger) converges for a given subset of the set of initial conditions. It may consist of one, many or infinite points and is called an attractor . It may also happen that A is a fractal, in which case it is a strange (or chaotic) attractor.5 This is the case of the simple mapping
f : I → I, I = [0, 1], f (x) = 4λx(1 − x),
popularly known as the “logistic map”, which for certain values of λ ∈ I tends to a strange attractor akin to a Cantor set. Strange attractors are fundamental in the recent developments in the study of chaotic behaviour in non-linear dynamics.6
§ 6.4.16 Let us go back to the beginning of this chapter, where a vector at p ∈ M was defined as the tangent to a curve a(t) onM , with a(0) = p. It is interesting to associate a vector to each point of the curve by liberating the variation of the parameter t in eq.[6.1]:
Xa(t)(f ) = d
dt (f ◦ a)(t). (6.37)
Then, Xa(t) is the tangent field to a(t), and a(t) is the integral curve of X through p. In general, this is possible only locally, in a neighbourhood of p. When X is tangent to a curve globally, the above definition being extendable to the whole M , X is said to be a complete field . Let us for the sake of simplicity take a neighbourhood U of p and suppose a(t) ∈ U , with coordinates (a1(t), a2(t), . . . , am(t)). Then, from [6.37],
Xa(t) = dai dt
∂
∂ai. (6.38)
5 See Farmer, Ott & Yorke 1983, where a good discussion of dimensions is also given.
6 For a short review, see Grebogi, Ott & Yorke 1987.
6.4. FIELDS & TRANSFORMATIONS 167 Thus,
Xa(t)(ai) = dai
dt (6.39)
is the component Xa(t)i . In this sense, the field whose integral curve is a(t) is given by dadt. In particular, Xp =da
dt
t=0. Conversely, if a field is given by its components Xk(x1(t), x2(t), . . . , xm(t)) in some natural basis, its integral curve x(t) is obtained by solving the system of differential equations Xk =
dxk
dt . The existence and unicity of solutions for such systems is in general valid only locally.
6.4.2 Transformations
Let us now address ourselves to what happens to differentiable manifolds under infinitesimal transformations, to which vector fields in a way preside.
More precisely, we examine the behaviour of general tensors under continuous transformations. The basic tool is the Lie derivative, which measures the variation of a tensor when small displacements take place on the manifold.
We start with the study of 1-dimensional displacements along a field (local) integral curve.
§ 6.4.17 The action of the group R of the real numbers on the manifold M is defined as a differentiable mapping
a collective displacement of all the points of M . At fixed p, it is a mapping λp : R → M
λp : t → λ(t, p),
which for each p ∈ M describes a curve γ(t) = λp(t), the “orbit of p generated by the action of the group R”. The mapping λ is a 1-parameter group on M .
§ 6.4.19 The action so defined is a particular example of actions of Lie groups (of which R is a case) on manifolds. We shall see later (section 8.2) the general case. Notice that, being 1-dimensional, group R is abelian.
Mathematicians use to call, by a mechanical analogy, M the phase space, λ the flow , and R × M the enlarged phase space. Due to the group character, it can be shown that only one orbit goes through each point p of M .
§ 6.4.20 Take a classical mechanical system and let its phase space M (see Physical Topic 1) be specified as usual by the points (q, p) = (q1, q2, . . . , qn, p1, p2, . . . , pn). The time evolution of the system, if the hamiltonian function is H(q, p), is governed by the hamiltonian flow , which for a conservative system is
λ(q0,p0)(t) = etH(q0,p0); (q0, p0) → (qt, pt) .
Given a domain U ⊂ M , the Liouville theorem says that the above flow preserves its volume: vol [λ(t)U ] = vol [λ(0)U ]. Suppose now that M itself has a finite volume.
Then, after a large enough time interval, forcibly (λ(t)U ) ∩ U 6= ∅. In words: given any neighbourhood U of a state point (q, p), it contains at least one point which comes back to U for t > some tr. For large enough periods of time, a system comes back as near as one may wish to its initial state. This is Poincar´e’s “th´eor`eme du r´etour”.
§ 6.4.21 Let M = E3, and ¯x = (¯x1, ¯x2, ¯x3) a fixed point different from zero.
Then,
λt(x) = (x1 + ¯x1t, x2 + ¯x2t, x3 + ¯x3t)
defines a C∞action of R on M . For each t ∈ R, λt: E3 → E3 is a translation taking x into x + ¯xt. Each vector ¯x determines a translation. The orbits are the straight lines parallel to ¯x.
§ 6.4.22 To each group λ corresponds a vector field: the infinitesimal oper-ator (or generoper-ator ) of λ is the field X defined by
Xpf = lim
∆t→0
1
∆t [f (λp(∆t)) − f (p)]
, (6.40)
on each p ∈ M and arbitrary f ∈ R(M ). A field X is thus a derivation along the differentiable curve γ(t) = λp(t), which is its integral curve.
6.4. FIELDS & TRANSFORMATIONS 169 The above definition generalizes to manifolds, though only locally, the well known case of matrix transformations engendered by an invertible matrix g(t) = etX, of which the matrix X is the generator,
X =dg
dt
t=0 = XetX|t=0. A matrix Y will transform according to
Y0 = g(t)Y g−1(t) = etXY e−tX ≈ (1 + tX)Y (1 − tX) ≈ Y + t[X, Y ] to first order in t, and we find that the “first derivative” is
[X, Y ] = limt→0 1t {g(t) Y g−1(t) − Y } .
§ 6.4.23 Take M = E2 and λ: R × M → M given by λ(t, (x, y)) = (x + t, y), translations along the x-axis. The infinitesimal operator is then X = dxd.
§ 6.4.24 We have seen that, given the action λ, we can determine the field X which is its infinitesimal generator. The inverse is not true in general but holds locally: every field X generates locally a 1-parameter group. The restriction is related to the fact that to find out the integral curve we have to integrate differential equations (§ 6.4.16), for which the existence and unicity of solutions is in general only locally granted.
§ 6.4.25 Lie derivative In section 6.2 we have introduced the derivative of a differentiable function f along the direction of a vector X: df (Xp) = Xpf . It was a generalization to a manifold M of the directional derivative of a function on Em. Things are a bit more complicated when we try to derive
more general objects. We face, to begin with, the problem of finding the variation rate of a vector field Y at p ∈ M with respect to Xp. This can be done by using the fact that X generates locally a 1-parameter group, which induces an isomorphism λt∗ : TpM → Tλt(p)M , as well as its inverse λ−t∗. It becomes then possible to compare values of vector fields. We shall just state three different definitions, which can be shown to be equivalent. The Lie derivative of a vector field Y on M with respect to the vector field X on M , at a point p ∈ M , is given by any of the three expressions:
Each expression is more convenient for a different purpose. Notice that the vector character of Y is preserved by the Lie derivative: LXY is a vector field.
Let us examine the definition given in the first equality of eqs.[6.42] (see Figure 6.10). The action λp(t) induces an isomorphism between the tangent spaces TpM and TqM , with q = λp(t). By this isomorphism, Yp is taken into λp(t)∗(Yp), which is in general different from Yq, the value of Y at q. By using the inverse isomorphism λp(−t)∗ we bring Yq back to TpM . In this last vector space we compare and take the limit. As it might be expected, the same definition can also be shown to reduce to
LXY = [X, Y ].
One should observe that the concept of Lie derivative does not require any
One should observe that the concept of Lie derivative does not require any