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V. Barbilian’s Differential Geometry

The advantage of this last approach to characterizing the cubic equation is mostly physical: as mentioned before, more often than not, in Physics and Engineering problems we have to do with quantities qualifying as coefficients of the Hessian of a cubic function. However, having now concentrated on the pure mathematical side of the problem, we ought to consider a pure algebraical advantage: the values of variables h, h^* and k can be scanned by a simply transitive continuous group with real parameters. This group has been exhibited for the first time by Dan Barbilian (Barbilian, 1938) with the occasion of a study of the Riemannian space associated with the previous family of cubics. We will briefly review Barbilian’s theory insisting on some particular technical points necessary for further reference. The basis of approach is the fact that the simply transitive group with real parameters (Baker, 1901)

(45)

where x_k are the cubic roots previously discussed, induces a simply transitive group for the quantities h, h^* and k:

(46)

which will be called Barbilian group. The structure of this group is that of SL (2, R), which we take in the standard form

(47)

where B_k are the infinitesimal generators of the group. Because the group is simply transitive these generators can be easily found as the components of the Cartan frame (Fels, Olver, 1998) from the formula

(48)

where w^k are the components of the Cartan coframe to be found from the system

(49)

Thus we have immediately both the infinitesimal generators and the coframe by identifying the right hand side of equation (48) with the standard dot product of SL (2, R):

(50)

so that we have

(51)

and

(52)

In real terms h = u + iv, k = e^iΦ, these last equations can be written as

(53)

Mention should be made that in his original paper (Barbilian, 1938) Barbilian does not work with the above differential forms but with the absolute invariant differentials

(54)

or, in real terms, exhibiting a 3D Lorentz structure of this space

(55)

The advantage of this representation is that it makes explicitly obvious the connection with the Poincaré representation of the Lobatchewsky plane. Indeed, the metric here is

(56)

This metric reduces to that of Poincaré in case where W¹ = 0 which, as Barbilian observes, defines the variable f as the ‘angle of parallelism’ of the hyperbolic plane (the connection). In fact, recalling that in modern terms (du/v) represents the connection form of the hyperbolic plane (Flanders, 1989), the equations (55) then represent a general Bäcklund transformation of that plane (Sasaki, 1979). It is interesting to expound a little on this association of the above theory related to cubic equation and the theory of conservation laws related to the evolution formally described by pseudospherical surfaces. In a way, this association was to be expected, at least for the simple cases where the cubic equation is the characteristic equation of stress (or strain) tensor for instance. Indeed, the stresses are equilibrium quantities: a statistical equilibrium of inter-particle forces in a solid. So it is legitimate to expect some conservation laws characterizing this kind of equilibrium. It turns out that there are conservation laws regarding the evolution of connection in the above-accepted sense (Sasaki, 1979), as well as conservation laws characterizing the type of statistics we have to use in such problems. These issues are to be discussed, at different levels, on this very page.

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