FRAMES 177 and their permutations in the lower indices (in which the coefficients are

clearly antisymmetric). Notice: the coefficients are not necessarily constant and depend on the chosen basis. Clearly, a necessary condition for the basis to be holonomic is that C_ij^k = 0 for all commutators of the basis members.

This condition, C_ij^k = 0 for all basis members, may be shown to be also sufficient for holonomy. The Jacobi identity, required by the Lie algebra, implies

Cⁿ_klCⁱ_jn+ Cⁿ_jkCⁱ_ln+ Cⁿ_ljCⁱ_kn= 0. (6.54)

§ 6.5.5 Let us re-examine the question of frame transformations. Given two natural basis on the intersection of two charts, a field X will be written

X = X^{i ∂}_∂xi = X^{i0 ∂}

∂xⁱ⁰. The action of X on the function X^j0 leads to

X^j0 = ∂x^j0

∂xⁱ Xⁱ. (6.55)

This expression gives the way in which field components in natural bases change when these bases are themselves changed. Here, basis transformations are intimately related to coordinate transformations. However, other basis transformations are possible: for example, going from the holonomic basis (∂_r, ∂_θ, ∂_ϕ) to the basis (X_r, X_θ, X_ϕ) of eq.[6.52] in the spherical case above is a basis transformation unrelated to a change of coordinates.

§ 6.5.6 Given an anholonomous basis {X_i}, it will always be possible to write locally each one of its members in some coordinate basis as

X_i = X_i^j ∂

∂x^j .

By using the differentiable atlas, the components can be in principle obtained all over the manifold. Each change of natural basis will give new components according to

X_i^k0 = X_i^j ∂x^k0

∂x^j . (6.56)

Notice that basis {X_i} would be holonomous only if X_i^j = ^∂x_∂y^ji, where {yⁱ} is some other coordinate system. In that case, {Xi = ^∂x_∂y^ji}. General matrices (X_i^j) are not of this form, and an holonomous basis is more of an exception than a rule. More generally, a basis transformation will be given by

X_i^k0 = X_i^jA^k_i⁰, (6.57)

where A is some matrix. Notice that each basis is characterized by the matrix (X_i^k) of its components in some previously chosen basis. Just above, a natural basis was chosen. The tangent spaces, being isomorphic to E^m, possess each one a “canonical basis” of the type

v1 = (1, 0, 0, . . . , 0), v2 = (0, 1, 0, 0, . . . , 0), . . . , vm = (0, 0, 0, . . . , 1).

The important point is that we can choose some starting basis from which all the other basis are determined by the matrices of their components. Such m × m matrices belong to the general linear space of m × m real matrices. As they are forcibly non-singular (otherwise the linear independence would fail and we would have no basis) and consequently invertible, they constitute the linear group GL(m, R). Starting from one basis we obtain each other basis in this way, one basis for each transformation, one basis for each element of the group. The set of all basis at each point p ∈ M is thus isomorphic to the linear group. But the transformation matrices A of eq.[6.57] also belong to the group, so that we have a case of a group acting on itself. Due to the peculiar form of the action shown in [6.57], we say that the transformations act on the right on the field basis, or that we have a right-action of the group.

The frequent use of natural basis (in general more convenient for calculations) is responsible for some confusion between coordinate transformations and basis transformations, which are actually quite distinct.

§ 6.5.7 The case of covector field basis is analogous. Two natural basis are related by so that the group of transformations acts on the left on the 1-form basis. Dual basis transform inversely to each other, so that, under the action, the value

< ω, X > is invariant. That is to say that < ω, X > is basis-independent.

§ 6.5.8 The bundle of linear frames Let B_pM be the set of all linear basis for TpM . As we have said, it is a vector space and a group, just GL(m, R).

In a way similar to that used to build up T M as a manifold, the set

6.5. FRAMES 179 BM = ∪_p∈MB_pM

of all the basis on the manifold M can be viewed as a manifold. To begin with, we define a projection π : BM → M , with π({X_(p)i} ∈ BpM ) = p.

A topology is defined on BM by taking as open sets the sets π⁻¹(U ), for U open set of M . Given a chart (U, x) of M , a basis at p ∈ U is given by (x¹, x², . . . , x^m, X₁¹, X₁². . . , X₁^m, X₂¹. . . X₂^m. . . X_m^m), where X_i^j is the j-th component of the i-th basis member in the natural basis. This gives the (m + m²) coordinates of a “point” on BM . It is possible to show that the mapping U × GL(m, R) → E^m+m2 is a diffeomorphism. Consequently, BM becomes a smooth manifold, the bundle of linear frames on M. We arrive thus to another fundamental fiber bundle. Let us list some of its character-istics:

(i) the group GL(m, R) acts on each BpM on the right (see eq.[6.57]);

B_pM is here the fiber on p; this group of transformations is called the struc-ture group of the bundle;

(ii) BM is locally trivial in the sense that every point p ∈ M has a neigh-bourhood U such that π⁻¹(U ) is diffeomorphic to U × GL(m, R).

(iii) concerning dimension: dim BM = dim M + dim GL(m, R) = m + m².

§ 6.5.9 The fiber itself is GL(m, R). A fiber bundle whose fiber coincides with the structure group is a principal fiber bundle. A more detailed study of bundles will be presented later on. Let us here only advance another concept.

The tangent bundle has the spaces T_pM as fibers. The action of GL(m, R) on the basis can be thought of as an action on TpM itself: it is the group of linear transformations, taking a vector into some other. A bundle of this kind, on whose fibers (as vector spaces) the same group acts, is said to be an associated bundle to the principal bundle. Most common bundles are vector bundles on which some group acts. The main interest of principal bundles comes from the fact that properties of associated bundles are deducible from those of the principal bundle.

Coordinates, which are in general local characterizations of points on a manifold, are usually related to a local frame. One first chooses a frame at a certain point, consider the euclidean tangent space supposing it as “glued”

to the manifold at the point, make its origin as a vector space (that is, the zero vector) to coincide with the point, then introduce cartesian coordinates, and finally move to any other coordinate system one may wish. By a change

of frame, the set of coordinates will transform according to x⁰ = Ax, or x^j0

= A^j_i⁰xⁱ, as any contravariant vector. This leads to dx^j0 = dA^j_i⁰xⁱ + A^j_i⁰dxⁱ.

Many physical problems involve comparison of rates of change of vector quan-tities in two different frames (recall for example the case of the “body” and the “space” frames in the rigid body motion, Physics Topic 2, from section 2.3.5 on). Consider a general vector u, with

du^j0 = dA^j_i⁰uⁱ + A^j0duⁱ.

The rate of change with a parameter t (usually time) will be

du^j0 dt = ^dA

j0 i

dt uⁱ+ A^j0du_dtⁱ.

A velocity, for example, as seen from two frames, will have its components related by

v^j0 = A^j_i⁰vⁱ+dA^j_i⁰ dt vⁱ.

Of course, we are here supposing that also the frames are in relative motion.

We shall come back to such “moving frames” later (§ 7.2.17 , § 7.3.12 and

§ 9.3.6).

6.6

METRIC & RIEMANNIAN MANIFOLDS

The usual 3-dimensional euclidean space E³ consists of the set R³ of ordered triples plus the topology defined by the 3-dimensional balls. Such balls are defined through the use of the euclidean metric, a tensor whose components are, in the global cartesian coordinates, constant and given by g_ij = δ_ij. We may thus say that E³ is R³ plus the euclidean metric. We use precisely this metric to measure lengths in our everyday life. It happens frequently that an-other metric is simultaneously at work on the same R³. Suppose, for example, that the space is permeated by a medium endowed with a point-dependent refractive index (that is, a point-dependent electric and/or magnetic perme-ability) n(p). Light rays (see Physical Topic 5) will in this case “feel” another metric, which will be g⁰_ij = n²(p)δ_ij if n(p) is isotropic. To “feel” means that they will bend, acquire a “curved” aspect if looked at by euclidean eyes (like ours). Light rays will become geodesics of the new metric, the “straightest”

possible curve if measurements are made using g_ij⁰ instead of g_ij. As long as we proceed to measurements using only light rays, distances will be different from those given by the euclidean metric. Suppose further that the medium

6.6. METRIC & RIEMANNIAN MANIFOLDS 181 is some compressible fluid, with temperature gradients and all which is nec-essary to render point-dependent the derivative of the pressure with respect to the fluid density at fixed entropy. The sound velocity will be given by c²_s =

∂p

∂ρ



S

and the sound propagation will be governed by geodesics of still another metric, g_ij⁰⁰ = _c¹2

S

δ_ij. Nevertheless, in both cases we use also the eu-clidean metric to make measurements, and much of geometrical optics and acoustics comes from comparing the results in both metrics involved. This is only to call attention to the fact that there is no such a thing like the metric of a space. It happens frequently that more than one is important in a given situation (for an example in elasticity, see Physical Topic 3, section 3.3.2).

Let us approach the subject a little more formally.

§ 6.6.1 In the space of differential forms, a basis dual to the basis {X_i} for fields in T M is given by those ω^j such that

ω^j(Xi) = < ω^j, Xi > = δ^ji, (6.61) so that ω = < ω, X_j > ω^j. Given a field Y = YⁱX_i and a form z = z_jω^j,

< z, Y > = z_jYⁱ. (6.62)

§ 6.6.2 Bilinear forms are covariant tensors of second order, taking T M × T M into R(M ). Recall that the tensor product of two linear forms w and z is defined by

(w ⊗ z)(X, Y ) = w(X) · z(Y ). (6.63) Given a basis {ω^j} for the space of 1-forms, the products ωⁱ ⊗ ω^j, with i, j = 1, 2, . . . , m, form a basis for the space of covariant 2-tensors, in terms of which a bilinear form g is written

g = gijωⁱ⊗ ω^j. (6.64)

Of course, in a natural basis,

g = gijdxⁱ⊗ dx^j. (6.65)

The most fundamental bilinear form appearing in Physics is the Lorentz metric on R⁴, which defines Minkowski space and whose main role is to endow it with a partial ordering, that is, causality.⁸

§ 6.6.3 A metric on a smooth manifold is a bilinear form, denoted g(X, Y ), X · Y or < X, Y >, satisfying the following conditions:

(i) of course, it is bilinear :

8 See Zeeman 1964.

X · (Y + Z) = X · Y + X · Z (X + Y ) · Z = X · Z + Y · Z;

(ii) it is symmetric:

X · Y = Y · X;

(iii) it is non-singular :

if X · Y = 0 for every field Y , then X = 0.

§ 6.6.4 In the basis introduced in § 6.6.2, we have

g(Xi, Xj) = Xi· Xj = gmnω^m(Xi) ωⁿ(Xj), so that

g_ij = g(X_i, X_j) = X_i· X_j. (6.66) The relationship between metrics and general frames (in particular, tetrads) will be seen in § 9.3.6. As g_ij = g_ji and we commonly write simply ωⁱω^j for the symmetric part of the bilinear basis, then

ωⁱω^j = ω⁽ⁱ⊗ ω^j) = ¹₂ (ωⁱ⊗ ω^j + ω^j⊗ ωⁱ) , we have

g = g_ijωⁱω^j (6.67)

or, in a natural basis,

g = g_ijdxⁱdx^j. (6.68)

§ 6.6.5 This is the usual notation for a metric. Notice also the useful sym-metrizing notation (ij) for indices. All indices (ijk . . .) inside the parenthesis are to be symmetrized. For antisymmetrization the usual notation is [ijk . . .], meaning that all the indices inside the brackets are to be antisymmetrized.

Knowledge of the diagonal terms is enough: the off-diagonal may be obtained by polarization, that is, by using the identity

g(X, Y ) = ¹₂[g(X + Y, X + Y ) − g(X, X) − g(Y, Y )].

§ 6.6.6 A metric establishes a relation between vector and covector fields:

Y is said to be the contravariant image of a form z if, for every X, g(X, Y ) = z(X).

6.6. METRIC & RIEMANNIAN MANIFOLDS 183 If, in the dual bases {X_i} and {ω^j}, Y = YⁱX_i and z = z_jω^j, then g_ijY^j = z_i. In this case, we write simply z_j = Y_j. That is the usual role of the covariant metric, to lower indices, taking a vector into the corresponding covector. If the mapping Y → z so defined is onto, the metric is non-degenerate. This is equivalent to saying that the matrix (g_ij) is invertible. A contravariant metric ˆg can then be introduced whose components are the elements of the matrix inverse to (gij). If w and z are the covariant images of X and Y , defined in a way inverse to the image given above, then

ˆ

g(w, z) = g(X, Y ). (6.69)

§ 6.6.7 All this defines on each T_pM and T_p^∗M an internal product

(X, Y ) := (w, z) := g(X, Y ) = ˆg(w, z). (6.70) A beautiful case of the field-form duality created by a metric is found in hamiltonian optics, in which the momentum (eikonal gradient) is related to the velocity by the refractive index metric (see Physical Topic 5.2). There are many other in Physics. Let us illustrate by a howlingly simple example not only the relation of 1-forms to fields, but also that of both to linear partial differential equations. Consider on the plane the function (x, y are cartesian coordinates, a and b real constants)

f (x, y) = ^x_a²2 +^y_b2².

Each case f (x, y) = C (constant) represents an ellipse. The complete family of ellipses is represented by the gradient form df ; that family is just the set of solutions of the differential equation df = 0. But f is also solution of the set of differential equations X(f ) = 0, where X is the field

X = ^a_x² ∂x−^b_y² ∂y.

Thus, a differential equation is given either by df = 0 or by a vector field.

In the first case the form is the gradient of the solution, which vanishes at each value C. In the second case the solution must be tangent to the given field. The form is “orthogonal” to the solution curve, that is, it vanishes when applied to any tangent vector: df (X) = X(f ) = 0. Thus, a curve is the integral of a field through tangency, and of a cofield through “gradiency”.

The word “orthogonal” was given quotation marks because no metric con-notation is given to df (X) = 0. Of course, multiplying f by a constant will change nothing. The same idea is trivially extended to higher dimensions.

In the example, we have started from a solution. We may start at a region around a point (x, y) and eventually obtain from the form

df = ^2x_a2 dx + ^2y_b2 dy

some local solution f = T df (see § 7.2.12 for a systematic method to get it);

this solution can be extended to the whole space, giving the whole ellipse.

This is a special case, as of course not every field or cofield is integrable. In most cases they are only locally integrable, or nonintegrable at all.

Suppose now that a metric g_ij is present, which relates fields and cofields.

In the case above gij = diag(1/a², 1/b²) is of evident interest, as f (v) = g(v, v), with v the position vector (x, y). To the vector v of components (x^j) will correspond the covector of components (p_k = g_kjx^j) and the action of this covector on v will give simply p(v) = pkx^k = gijxⁱx^j. As we are also in a euclidean space, the euclidean metric m_ij = δ_ij may be used to help intuition.

We may consider p and v as two euclidean vectors of components (p_k) and (x^k). Comparison of the two metrics is made by using g(v, v) = m(p, v).

Consider the curve p(v) = g(v, v) = m(p, v) = C, which is an ellipse. The vector v gives a point on the ellipse and the covector p, now assimilated to an euclidean vector, is orthogonal to the curve at each point, or to its tangent at the point. This construction, allowing one to relate a 1-form to a field in the presence of a non-trivial metric, is very much used in Physics. For rigid bodies, the metric m is the inertia tensor,, the vector v is the angular velocity and its covector is the angular momentum. The ellipsoid is the inertia ellipsoid, the whole construction going under the name of Poinsot (more details can be found in Physical Topic 2, section 2.3.10). In crystal optics, the Fresnel ellipsoid ε_ijxⁱx^j = C regulates the relationship between the electric field E and the electric displacement D = ε E, where the metric is the electric permeability (or dielectric) tensor. In this case, another ellipsoid is important, given by the inverse metric ε⁻¹: it is the index, or Fletcher’s ellipsoid (Physical Topic 5.6). In all the cases, the ellipsoid is defined by equating some hamiltonian to a constant.

§ 6.6.8 An important property of a space V endowed with an internal prod-uct is the following: given any linear function f ∈ R(V ), there is a unique v_f ∈ V such that, for every u ∈ V , f (u) = (u, v_f). So, the forms include all the real linear functions on T_pM (which is expected, they constituting its dual space), and the vectors include all the real linear functions on T_p^∗M (equally not unexpected, the dual of the dual being the space itself). The presence of a metric establishes a natural (or canonical) isomorphism between a vector space (here, TpM ) and its dual.

§ 6.6.9 The above definition has used fixed bases. As in general no base cov-ers the whole manifold, convenient transformations are to be performed in the intersections of the definition domains of every pair of bases. If some of the

6.6. METRIC & RIEMANNIAN MANIFOLDS 185 above metric-defining conditions are violated at a point p, it can eventually come from something wrong with the basis: for instance, it may happen that two of the X_i are degenerate at p. A real singularity in the metric should be basis-independent. Non-degenerate metrics are called semi-Riemannian. Al-though physicists usually call them just Riemannian, mathematicians more frequently reserve this denomination to non-degenerate positive-definite met-rics, g : T M × T M → R⁺. As it is not definite positive, the Lorentz metric does not define balls and is consequently unable to provide for a topology on Minkowski spacetime.

§ 6.6.10 A Riemannian manifold is a smooth manifold on which a Rie-mannian metric is defined. A theorem (see Mathematical Topic 3.6) due to Whitney states that

it is always possible to define at least one

Riemannian metric on an arbitrary differentiable manifold .

§ 6.6.11 A metric is presupposed in any measurement: lengths, angles, vol-umes, etc. We may begin by introducing the length of a vector field X through

§ 6.6.12 Given two points p, q ∈ M , a Riemannian manifold, we consider all the piecewise differentiable curves γ with γ(a) = p and γ(b) = q. The distance between p and q is the infimum of the lengths of all such curves between them:

In this way a metric tensor defines a distance function on M .

§ 6.6.13 A metric is indefinite when ||X|| = 0 does not imply X = 0. It is the case of Lorentz metric for vectors on the light cone.

§ 6.6.14 Motions are transformations of a manifold into itself which pre-serve a metric given a priori. They are also called isometries in modern texts, but this term in general includes also transformations between differ-ent spaces. When represdiffer-ented by field vectors on the manifold, eq.[6.51] will give the components of the Lie derivative:

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