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There is no general agreement as to what one should understand by “Cartan lemmas” or “Cartan theorems” when it comes to the development of the differential geometric concepts. As a matter of fact, as Robert Hermann says in his rich Addendum to the English version of Cartan’s Géométrie des Espaces de Riemann (Cartan, 1983), these are algebraic results scattered over all the places in Cartan’s work, and used as momentarily needed. We will extract here, from among these result, only the ones that have a direct connection with what we would like to call the ‘theoretical physics’ of surfaces, to be applied in a theoretical physics of the tire. These were first presented systematically in some of the works of old pupils of Cartan, who took his course on Riemannian geometry in the late 1920’s (Finikov, 1948). The spirit of those lectures is also available in the recent English translation of the Russian version Finikov’s Riemannian Spaces (Cartan, 2001). The general mathematics of differential forms is available in many modern developments which can be found quite easily, if needed. In regards to applications of theoretical mechanics’ nature, the work of Vladimir Arnold, by now a classical treatise, seems to be worth recommending (Arnold, 1976).

We will limit here our considerations to the space of three dimensions. The vectors are conceived as entities written either by components, or in the Dirac form as matrices. For instance the position vector can be written in one of the two forms:

(1)

The first of these is the regular geometrical writing, in terms of the base unit vectors  . Here, and everywhere else, we adopt the summation convention over repeated indices, and x^k are the (contravariant) components of our position vector. The second writing – the matrix notation – disregards the reference frame of the three base unit vectors. It is therefore worth considering for calculations in the same reference frame, or in situations when the reference frame is fixed once and for all. This is, for instance the case of the Euclidean space, where the reference unit vectors do not vary. In general, however, the reference frame is local: it can vary from a point to another, and may not be always orthogonal. Using for the reference frame a matrix notation of the kind above, one can thus write in general

(2)

Here the matrix g is the metric tensor – a matrix having 1 as diagonal elements. In case this matrix is the identity matrix, we have to do with the Euclidean case in Cartesian coordinates.

      The grounds of Cartan’s considerations is the observation that an elementary displacement means both a variation in position per se, and a variation in the reference frame:

(3)

According to general geometrical rules, the variations of the unit vectors can be expressed with respect to the unit vectors themselves by linear relations:

(4)

Obviously, the matrix ω has zeros along the main diagonal, but it is by no means symmetrical. In view of the definition of the metric tensor, we have

(5)

So, only if the reference frame is orthonormal, the matrix ω is skew symmetric, otherwise it has no symmetry. It is thus sometimes very convenient to discuss the geometry in an orthonormal reference frame.

      Now, in view of (4), the equations (3) can be written as

(6)

Obviously, both the components of the vector  and those of the vectors  are exact differentials. In the language of the differential forms this fact can be stated by the equations:

(7)

It is the simple consequences of these two equations that are the ground of the whole Cartan geometrical construction. Just following the rules of working with the exterior differentiation and exterior multiplication, one can find from (7) the following equations relating the components of the vector  to the matrix ω:

(8)

Here we maintain the rule of summation on dummy indices, only the monomials are not defined by the usual multiplication but by exterior multiplication. For obvious reasons the first equation is called the compatibility equation: it gives the compatibility conditions between the variation of the reference frame and the displacements in space.

      It can be easily proved, just following the same rules of exterior multiplication and exterior differentiation, that the first equation (8) is a direct consequence of the second one and the definition of s^k from equation (6). However, the equations (8) are very general: they are valid regardless of definition (6). So, the two equations from (8) may not be always equivalent in an obvious way. This means that there are situations, mostly corresponding to physical reasons, when one has to define the variations of coordinates directly in terms of the reference frame, in which case s^k don’t have that neat structure given by equation (6). In such cases we don’t know precisely how much from the displacement vector is pure displacement and how much is contribution of the variation of reference frame. All we can assume is that the components of displacement vector are differential 1-forms, and the equations (8) still stand.

      The physical reasons we were mentioning above have always geometrical expressions. In order to catch the idea, let’s take for instance an arbitrary motion. It is done along a certain path, whose image is a curve in space. This curve can be represented, at least in limited local extensions, as intersection of two surfaces. Even more precisely, if we consider the local situation in a certain point, the curve is a straight line represented by the intersection of the tangent planes of those surfaces in that point. And the tangent planes in a point can be represented always by 1-forms. Thus we can come up with the idea that, in spite of the fact that the space is three-dimensional, in order to describe the motion locally we only need two 1-forms, not three. True, these 1-forms are linear combinations of the three basic components of the elementary differential of position vector, but this is an entirely different story. The main point of the exercise is that the local description of the motion only needs two differential forms, and those are linearly independent. In a way this means that they are sufficient for the purpose of description of motion.

      Thus the immediate point is to define the linear independence of two differential 1-forms. As the above example suggests in terms of the tangent planes in a point, these have to be necessarily different, otherwise they cannot properly determine a line. This can be expressed simply by the statement: two 1-forms ω and f are linearly independent if, and only if, their exterior product is non-null. The same way three 1-forms, ω, f, y are linearly independent if their exterior product is non-null. This simply expresses the fact that the volume of the parallelepiped defined by the three 1-forms as edges is nonzero: the parallelepiped is nontrivial. Such three 1-forms define a basis of 1-forms in space, which can be characterized as usual: any linear combination of them, with numerical coefficients, uniquely represents a 1-form.

      Here an interesting fact occurs. We can say that a 1-form is null if all the coefficients of its linear expansion in a base of 1-form are zero. This fact can however be ascertained otherwise. Namely, if {y^k; k = 1, 2, 3} is such a base, then the 1-form W is zero if, and only if, the following relations are satisfied:

(9)

In a way, this relation is equivalent to the known fact that in a vectorial space a vector is characterized by its projections on the base vectors. If all these projections are zero, then the vector is zero. Only, here the dot multiplication of vector is replaced by the exterior multiplication of the differential forms.

      The exterior product of two 1-forms is a 2-form. The 2-forms in space are usually associated with fluxes. Well known examples are: the flux of magnetic lines through a surface, whose magnitude is the magnetic induction, the flux of electric lines whose magnitude is the electric induction, the flux of particles, whose magnitude is known as current, an so on. The magnitude is usually a skew symmetric tensor, because the base of two forms in space is given by the three 2-forms representing the projections of a surface element on the three coordinate planes. Thus a 2-form, ‘b’ say, can be written in this basis as

(10)

where the summation convention is respected. The magnitude here is given by the skew symmetric tensor b, formed with the coefficients of the 2-form.

      By the same reasoning, a differential 3-form represents densities. These have magnitudes represented by the coefficients of the 3-form in terms of the elementary oriented volumes:

(11)

where the summation is done over the six permutations of the indices.

      Examples of differential forms of all degrees will be provided here all along the development of the physical theory of tire. But in order to properly develop that theory we need a few results of Cartan, helping in draw particular conclusions from the equations (8). First there is the theorem that Finikov terms as Cartan’s lemma (Finikov, 1948). We give it in a little modernized formulation of Guggenheimer (Guggenheimer, 1977): if ω^r, r = 1, 2, 3 are r linearly independent 1-forms, and there are r 1-forms π^r such that

(12)

then there is a r´r symmetric matrix c such that:

(13)

The proof is as follows: if r = 3, then ω^r can be taken as a basis of the 1-forms and thus ω^r can be expressed linearly with respect to them. Therefore equation (13) is valid, without any qualification on the matrix c. However, the equation (12) shows that this matrix is symmetrical. Indeed, that condition becomes

(14)

whence the symmetry of the matrix, in view of the skew symmetry of the exterior product. If r < 3, then the system (ω) can be completed to a basis, and the proof follows the same lines.

      Another important theorem which, in our opinion, is instrumental in physical applications, is the following: the 2-form

(15)

is zero as a consequence of vanishing of the r (£ 3) linearly independent 1-forms f¹, f²,…,f^r, if, and only if, it can be written in the form:

(16)

where f₁, f₂,…, f_r are r appropriately chosen 1-forms, and the summation convention is respected. The proof goes as before: in case r = 3 the system of 1-forms {f^k} can be taken as a basis, in which case F can be written as

(17)

with b the skew symmetric matrix derived from a by changing the basis of 1-forms. For r < 3, the system {f^a} can be completed up to a basis, and the proof goes exactly like in the case r = 3.

References

Arnold, V. (1976): Les Méthodes Mathématiques de la Mécanique Classique, Editions MIR, Moscou

Cartan, E. (1983): Geometry of Riemannian Spaces, Mathematical Science Press, Brookline, Massachusetts, USA

Cartan, E. (2001): Riemannian Geometry in an Orthogonal Frame, World Scientific Publishing, Singapore

Finikov, S. P. (1948): Cartan’s Method of Exterior Forms in Differential Geometry, OGIZ, Moscow (Метод Внешних Форм Картана в Дифференциальной Геометрии, ОГИЗ, Москва)