the role of differential parameters in beltrami's work

HISTORIA MATHEMATICA 24 (1997), 25–45ARTICLE NO. HM972121

The Role of Differential Parameters in Beltrami’s Work*

ROSSANA TAZZIOLI

Dipartimento di Matematica ed Applicazioni, Universita di Palermo, Via Archirafi 34,90123, Palermo, Italy

Differential parameters play a relevant role in Beltrami’s mathematical work. They areemployed in different contexts, in order to express some well-known results in a new wayand to extend potential theory and the theory of elasticity to a Riemannian manifold. Theauthor aims to show that differential parameters enabled Beltrami to solve many mathematicalquestions and that they constitute the first step toward the conception of tensor calculus. 1997

Academic Press

Les parametres differentiels jouent un role important dans l’oeuvre mathematique deBeltrami. Ils sont employes en contextes differents, pour exprimer dans une maniere nouvellequelques resultats bien-connus et pour generaliser la theorie du potentiel et la theorie del’elasticite aux varietes riemanniennes. Le but de l’auteur est de montrer que les parametresdifferentiels permettent a Beltrami de resoudre beaucoup de questions mathematiques etqu’ils constituent le premier pas vers la conception du calcul tenseuriel. 1997 Academic Press

I parametri differenziali giocano un ruolo importante nell’opera matematica di Beltrami.Essi vengono impiegati in diversi contesti, per esprimere alcuni noti risultati in modo nuovoe per estendere le teorie del potenziale e dell’elasticita alle varieta riemanniane. Scopo dell’au-trice e di mostrare che i parametri differenziali permettono a Beltrami di risolvere moltequestioni matematiche e che essi costituiscono il primo passo verso la concezione del calcolotensoriale. 1997 Academic Press

MSC 1991 subject classification: 01A55, 53B50.KEY WORDS: Beltrami, differential parameters, Riemannian manifold, tensor calculus.

INTRODUCTION

Beltrami wrote about a hundred papers on geometry, analysis, and mathematicalphysics. In many of them, he expressed key definitions and theorems in terms ofdifferential parameters. In addition, these quantities were the basic instruments forformulating new geometric and analytical concepts and also for describing theseconcepts in a mechanical language.

The first and second differential parameters are, indeed, invariant expressions forchanges of coordinates, and therefore they enabled Beltrami to describe invariantquantities and equations in a simpler manner. In particular, he also used differentialparameters in order to extend some basic results of potential theory and of thetheory of elasticity to Riemannian manifolds with constant curvature.

Differential parameters played a prominent role in the development of differentialgeometry, and they strongly influenced the conception of tensor calculus. Even

* This paper has been supported by C.N.R.

25

0315-0860/97 $25.00Copyright 1997 by Academic Press

All rights of reproduction in any form reserved.

26 ROSSANA TAZZIOLI HM 24

though they were not absolutely necessary for deducing results, they representedvery effective notations. But ‘‘in mathematical sciences a good notation has thesame philosophical importance as a good classification in natural sciences,’’ as RicciCurbastro and his student, Levi Civita, noticed [61, 188]. In fact, they admitted theinfluence of Beltrami’s differential parameters in their formulation of tensor cal-culus.

1. CURVILINEAR COORDINATES AND DIFFERENTIAL PARAMETERSBEFORE BELTRAMI

Curvilinear coordinates and differential parameters were both introduced byLame, who also attached a geometric and physical meaning to these expressions.Lame’s studies in the theory of elasticity inspired him to consider three families oforthogonal surfaces whose intersection constituted a system of curvilinear coordi-nates. In this regard, he wrote: ‘‘In fact, it results from this theory [of elasticity]that at every point of a solid body in elastic equilibrium it is always possible to findthree orthogonal plane-elements, to which the elastic forces are normal, while allthe other elements, at the same point, can only undergo oblique tractions andpressures’’ [60, vii–viii]. The totality of the plane-elements of the body then leadsto the consideration of three families of orthogonal surfaces.

Lame applied his theory of curvilinear coordinates to different cases includingheat propagation in triangular prisms and polyhedra [53] and the equilibrium oftemperature within a cylindrical body [55]. He also successfully examined the equi-librium of a spherical elastic envelope subjected to a given distribution of load witha system of polar coordinates—the so-called Lame problem [59].

In two papers published in 1839 [56; 57], Lame considered the Laplace equationin ellipsoidal coordinates in order to deduce the laws of stationary heat in a solid,homogeneous body with an ellipsoidal boundary. He found the differential for-mula—referred to as Lame’s equation—which described this phenomenon, and healso showed some solutions of Lame’s equation (Lame’s functions). This new classof functions plays the same role for the ellipsoid as the spherical functions play forthe sphere.

Lame adopted formalism and applied results, obtained from the theories ofelasticity and heat, to study light propagation. The theory of partial differentialequations could also be employed in the analysis of Fresnel’s theory. In 1821,Fresnel’s experiment seemed to suggest the existence of a medium, the ether, whichfilled space and propagated physical actions, in particular light and electromagneticforces. Lame understood the ether as an elastic, isotropic and homogeneous me-dium, whose deformations could be studied by means of the mathematical theoryof elasticity. In a long, five-part memoir printed in 1834 [54], Lame established thelaws of equilibrium for the ethereal fluid and devoted many pages to the analysisof light propagation.

In order to deduce the equations of elastic equilibrium for the ether, Lameintroduced two differential invariant expressions of a function F, which he calledthe first and second differential parameters,

HM 24 DIFFERENTIAL PARAMETERS IN BELTRAMI’S WORK 27

D1F 5 !SFx1D2

1 SFx2D2

1 SFx3D2

(1.1)

D2F 52Fx2

11

2Fx2

21

2Fx2

3, (1.2)

where x1, x2, x3 are Cartesian orthogonal coordinates. He then expressed (1.1) and(1.2) in curvilinear coordinates and stressed their important role in mathematicalphysics. Lame observed that ‘‘the expansion of a solid body in elastic equilibrium,the temperature of the same body in equilibrium of heat, and the potential ofattraction between spheroids are all functions having the second differential parame-ter equal to zero’’ [58, 69]. In fact, (1.2) equated to zero coincides with the Laplaceequation. The first differential parameter was also relevant in physics, accordingto Lame. As he put it, ‘‘[t]his [differential] parameter is not so essential as thesecond differential parameter in the equations of mathematical physics, but itsnatural role is very important’’ [60, 28]. The intensity of the force H due to thepotential F is described by the first differential parameter of F (H 5 D1F).

Lame expressed the equations of elastic equilibrium and the Laplace equationin curvilinear orthogonal coordinates by means of differential parameters. As theyare invariant for changes of coordinates, their application to the theory of elasticity,heat theory, and potential theory is fundamental for obtaining new invariant quanti-ties and equations.

In 1847, Jacobi [47] also analyzed the transformation properties of the Laplaceequation D2F 5 0. He considered a system of Cartesian coordinates (x,y,z) and theline element

ds 5 Ïdx2 1 dy2 1 dz2. (1.3)

He then deduced the new line element expressed in curvilinear orthogonal coordi-nates from the old one (1.3). From this formula he obtained the first and the seconddifferential parameter in the new system of coordinates. They coincided with theformulas already established by Lame (namely, (1.1) and (1.2)).

Jacobi applied the same procedure in order to express differential parameters incurvilinear coordinates u1, u2, u3 which were not necessarily orthogonal. He con-sidered a function K depending on the coordinates and studied the equationeKdxdydz 5 eKDdu1du2du3, where D is the Jacobian of the transformation ofcoordinates. By varying these integrals, Jacobi found a differential equation whichwas generally very difficult to integrate. He could deduce from it, and for particularexpressions K only, the second differential parameter expressed in the new coordi-nates u1, u2, u3. Nevertheless, Jacobi assumed that the line element was eitherÏdx2 1 dy2 1 dz2 or reducible to this form. Only after 1868 and the printing ofRiemann’s Habilitationsvortrag [79] was it conceivable to consider the line elementin a more general manner.

Chelini [36] and Codazzi [38] also found expressions for differential parametersin any coordinate system. They adopted two different procedures: while Chelini


derived them from the analogous formulas in Cartesian coordinates by using geo-metric methods, Codazzi obtained them by means of purely analytical tools.

Another alternative approach to the theory of differential parameters in threecurvilinear coordinates was carried on by Somoff in 1865 [80]. His method wassimilar to Jacobi’s in the way that he tried to deduce the first and second differentialparameters from the line element. However, Somoff attributed a particular dynamicmeaning to this procedure by considering the variables as coordinates of a movingpoint and the quantity ds2/dt2 (the kinetic energy of the system) instead of thesquare of the line element, namely, ds2. This point of view was developed mostlyby Lipschitz, who pointed out that configuration space could be interpreted as aRiemannian manifold (see [85]).

Although Lame, Jacobi, Chelini, Codazzi, and Somoff proved interesting invariantproperties of differential parameters as illustrated, the most relevant results wereobtained by Brioschi in his Teorica dei determinanti [32, Section 10]. He did notadvance any restrictive hypotheses for the change of coordinates yi 5 yi (x1, . . . , xn),for i 5 1, 2, . . . , n and deduced the most general transformation law for the Laplacianof a function F(x1, . . . , xn). This transformation includes the results given by Lameas particular cases.

2. INTRINSIC QUANTITIES ON A SURFACE

In his Ricerche di analisi applicata alla geometria printed in 1864–1865 [2], whichwas inspired by Bertrand’s Traite de calcul differentiel [22, 1], Beltrami analyzedsome aspects of the theory of curves and surfaces and of systems of lines in space.He also developed a systematic analysis of intrinsic properties (‘‘proprieta assolute’’)of a surface, that is, of the properties independent of the form of the surface.In this context, he defined two invariant expressions, which he recognized to begeneralizations of the first and second differential parameters introduced by Lame.

Beltrami considered a surface defined by the square of the line element:

ds2 5 E du2 1 2 F du dv 1 G dv2, (2.1)

where u, v are curvilinear coordinates and E, F, G are functions of u, v dependingon the particular surface. These functions had been introduced by Gauss [42] and,if u, v are Cartesian orthogonal coordinates, then (2.1) becomes ds2 5 du2 1 dv2

(see (1.3)). If f is a function on the surface, Beltrami deduced the formula

Sdf

dsD2

5

E Sf

vD2

2 2Ff

vf

u1 G Sf

uD2

EG 2 F 2 . (2.2)

The quantity df represents the increase of f by passing from one curve f 5 const.to an infinitely close one, and ds is the normal distance between these curves.Beltrami pointed out that the right side of (2.2), which he denoted (D1f)2, is ‘‘verysimilar’’ to the square of the first differential parameter of f defined by Lame inhis theory of surfaces.


Since the quantity D1f is invariant for changes of coordinates, it is independentof the coordinate lines: u 5 const., v 5 const. In the case where the normal distancebetween two consecutive curves is constant at each point, the first differentialparameter varies from one curve to another but is constant along every curve;therefore it is a function of f only, that is,

D1f 5 f(f). (2.3)

As Beltrami pointed out, (2.3) can be interpreted in mechanical terms; Weingarten[89] had observed that (2.3) is linked to the equation of motion of a point notsubjected to external forces on a surface. In addition, Beltrami wrote, ‘‘the previousequation [(2.3)] characterizes a kind of parallelism on a surface which can be definedas geodesic’’ [2, 117]. Since the first differential parameter is an invariant quantity,the ‘‘geodesic parallelism’’ is an intrinsic property of the surface. ‘‘Geodeticallyparallel curves,’’ as Betti [23] called the curves with the property of geodesic parallel-ism, were employed also by Liouville, Minding, and Roberts (see [73, 311–312]).Darboux investigated the connections between (2.3) and Hamilton–Jacobi theoryin chapitre VI, livre V of his Lecons sur la theorie generale des surfaces [39, 2].

In order to determine other intrinsic quantities on a surface, Beltrami developeda general method by remarking that all surfaces having the same line element mustbe considered identical. An intrinsic function (‘‘funzione assoluta’’), describing anintrinsic property, was defined by Beltrami as depending only on the quantities E,F, G and on their partial derivatives. Casorati [34; 35] had succeeded in determininga great number of intrinsic functions and then intrinsic properties of the surface.However, the method used by Casorati, which evidently does not involve differentialparameters, is very complicated and involves very long calculations.

In addition to intrinsic functions, Beltrami introduced invariable functions (‘‘fun-zioni invariabili’’) which depend on E, F, G and on other functions f, c, h . . . ofu, v and are invariant for changes of coordinates. If f, c, h . . . are intrinsic functions,then an invariable function made with these quantites will be an intrinsic function.‘‘In my opinion,’’ Beltrami wrote, ‘‘this is the usefulness of considering invariablefunctions, the search for which is less difficult than that for intrinsic functions’’ [2,142]. In the context of invariable functions, Beltrami reconsidered the first differen-tial parameter and defined the mixed differential parameter D(fc) of two givenfunctions f, c,

D(fc) 5

Ef

vc

v2 F Sf

vc

u1

f

uc

vD1 Gf

uc

u

EG 2 F 2 . (2.4)

From formulas (2.4) and (2.2), the mixed differential parameter becomes D(f2) 5(D1f)2, if f 5 c. Beltrami then proved that, if f, c are intrinsic functions,D(fc), D1f, and the second differential parameter D2f will be intrinsic functionsas well.

In order to find other intrinsic elements on a surface, Beltrami introduced particu-


lar systems of coordinates, referred to as isothermal systems, by means of differentialparameters. He proved that a new orthogonal system (f, c) could be deduced fromthe curvilinear coordinates (u, v) if the condition D(fc) 5 0 was satisfied. If f wasa solution of

D2f 5 0, (2.5)

then the analogous equation for c : D2c 5 0 was valid, and, moreover, D1f 5 D1c.By setting 1/l 5 1/(D1f) 5 1/(D1c), the new line element becomes

ds2 51l

(df2 1 dc 2). (2.6)

In the case where the original coordinates (u, v) are isothermal as well, one candeduce that f 1 ic is an analytic function of the complex variable u 1 iv.

In a manner similar to that used by Lame in his Lecons sur les coordonneescurvilignes [60, 30–36], Beltrami characterized an isothermal system (f, c) byproving that the lines f 5 const. and their orthogonal trajectories constitute anisothermal system if and only if the expression D2f/(D1f)2 depends only on thevariable f.

By introducing isothermal coordinates, Beltrami could easily examine the ‘‘geode-sic curvature’’ of a curve. This quantity had already been considered in 1830 byMinding [70], who had studied the expression cosu/R, where R is the radius ofcurvature of the curve and u is the angle between its osculating plane and thetangent plane of the surface. Bonnet [27] used the name ‘‘courbure geodesique’’for the quantity cosu/R, but this name had already been introduced by Liouville[62], as Bonnet himself remarked [28] (see also [66, 749–750]).

Beltrami [2, 160] considered the geodesic curvature of a curve as the projectionof the absolute curvature on the tangent plane to the surface. According to him,it is the simplest definition given by Bertrand [22, 1: 736–741], who had introduceda great number of different and equivalent definitons for the geodesic curvature.After pages of long calculations, Beltrami proved that the geodesic curvature in-volved the first and second differential parameters of intrinsic functions and there-fore ‘‘it does not change after bending the surface’’ [2, 162]. Minding [70] had statedthe same theorem, which Beltrami proved once again in a paper printed in 1868[9], without using differential parameters. He indeed developed an elementarymethod by starting from some simple formulas deduced by Chelini [37].

Beltrami applied intrinsic properties of surfaces in order to deduce an important,classic theorem, Gauss’s theorema egregium. Since the line element of the surfaceis invariant for changes of coordinates, Beltrami, by considering the two isothermalsystems (f, c) and (f9, c9), found that (1/l2)(df2 1 dc2) 5 (1/l92)(df92 1 dc92)(see (2.6)). He could then prove the relation D2 log l9 5 D2 log l from previousresults on isothermal coordinates. This formula depends solely on the line elementof the surface, and so Beltrami concluded that the expression D2 log l is an intrinsicfunction: ‘‘From this result and from the invariability of the parameter D2 . . . . it iseasy to find the general expression of this intrinsic function, which I denote by Q’’


[2, 189]. He found that Q coincides with the product of the two principal radii ofcurvature—the Gaussian curvature—and therefore it does not change for isometrictransformations. As a result, Beltrami obtained the ‘‘memorable formula’’ of Gausswith ‘‘an analytical and very simple proof’’ [2, 189].

As a consequence of his research on intrinsic properties of a surface, Beltrami[3] elaborated a method in order to determine the ruled surfaces which wereisometrically transformable into another given ruled surface. This problem hadalready been studied by Minding [71], who had obtained all the analytical formulasof a ruled surface which was subjected to this transformation. These formulasdepended on a single arbitrary function which was sometimes very difficult todeduce (see [25, 1: 260–262]). Beltrami formulated a new procedure for findingthe expressions of a surface’s directrix after bendings. Each of these expressionscorresponded to a particular form of the surface which could be analytically deducedby assuming that the geodesic curvature and the angle between directrix and genera-trix were intrinsic properties, that is, they were invariant for isometric transforma-tions.

Therefore the whole class of intrinsic and invariable functions on a surface isobtained from results and theorems involving the first and second differential param-eters. These quantities lead to characterizations of isothermal systems, which arestrictly connected with the theory of functions of a complex variable. Intrinsicfunctions are very important also in differential geometry because of their indepen-dence of the choice of coordinates. Their prominent role in mathematics was com-pletely understood by Ricci and Levi Civita, who, stimulated by Beltrami’s differen-tial parameters, formulated the theory of tensor calculus.

3. DIFFERENTIAL PARAMETERS AND ANALYTIC PROBLEMS

The intrinsic and invariable functions—including differential parameters—allowed Beltrami to find many intrinsic elements of a surface and to prove someclassic theorems in a new manner. In two subsequent papers published in 1867, heapplied properties of differential parameters in order to solve questions in complexanalysis [4] and to elaborate a new method for deducing minimal surfaces [5]. Hisresults in differential parameters led him to interpret some well-known questionsin the context of intrinsic geometry and also to deduce relevant theorems in geome-try and analysis.

Beltrami [4] considered a surface S with line element expressed by

ds2 5 E du2 1 2 F du dv 1 G dv2 (3.1)

and a finite portion V of S, where the functions E, F, G, H 2 5 EG 2 F 2 and theirsquare roots are real, finite, and continuous. From his hypotheses, it follows that‘‘every small region of the surface surrounding a point in V can be covered by anetwork of coordinate curves. This network, small deflections excepted, is verysimilar to the one made by two systems of parallel straight lines on the plane. Sucha region can be called an ordinary region’’ [4, 319–320]. Beltrami did not mentionthe intersections between ordinary regions; moreover, he supposed that the coeffi-


cients in (3.1) are continuous only, but, in what follows, he also considered their de-rivatives.

After his preliminary assumptions, he noticed that it did not seem effective tochoose the most natural variable u 1 iv on S in order to apply the theory of complexvariables and their functions to the study of surfaces ‘‘advantageously’’ [4, 323].From his results, he was led to consider the variable w such that

dw 5 x(Udu 1 Vdv), (3.2)

where x is an integrating factor of Udu 1 Vdv and

U 5 ÏE, V 5F 1 iH

U, and H 5 ÏEG 2 F 2. (3.3)

Thus, Beltrami noticed, the variable w depends on an integration which is possiblein some particular cases only. However, functions f of w can be defined a prioriwith respect to the variables u, v. Indeed, Beltrami proved that the condition

D1 f(u, v) 5 0 (3.4)

completely characterizes f ; that is, as he [4, 324] stated, a function f(u, v) can beexpressed as a function of the variable w if and only if its first differential parametervanishes. By writing f 5 f 1 ic as the sum of its real and complex parts, Beltramifound from (3.4) the two equations

D1f 5 D1c and D(fc) 5 0. (3.5)

Formulas (3.5) are the necessary and sufficient conditions since a functionf(u, v) 5 f(u, v) 1 ic(u, v) can be considered as a function of w. As consequencesof (3.5), Beltrami proved that the second differential parameters of the functionsf, f, and c vanish. Moreover, in analogy with the usual theory of functions of acomplex variable, if one of the two functions f, c is known, then the other one isdetermined as well.

From some results established in his Ricerche [2] it follows that the system ofcoordinates f, c is isothermal. Then isothermal coordinates were not only relevantfor the theory of surfaces, as Beltrami himself had shown, but also useful in orderto examine some aspects of complex analysis.

Beltrami carried on his analysis by assuming that the ordinary region V can beconsidered as a collection of domains Wi such that each boundary of Wi intersectsthe coordinate lines u 5 const., v 5 const. at two points only. That is to say, theregions Wi are convex and their boundaries are regular. If W is one of these regions,Beltrami examined the integral

P 5 EW

D(fc) dW, (3.6)

with dW 5 H du dv. In order to integrate (3.6), he set

Mc 5

Gc

u2 F

c

vH

and Nc 5

Ec

v2 F

c

uH

. (3.7)


FIGURE 1

Then, from the definition of the mixed differential parameter (2.4), he obtained

P 5 E E Mc

f

udu dv 1 E E Nc

f

vdu dv. (3.8)

In the domain W, he considered a particular coordinate curve v 5 const. and thetwo values of u, u 5 a and u 5 b, which corresponded to the intersection pointsbetween the boundary of W and the coordinate curve v 5 const. (see Fig. 1).

The two integrals in (3.8) were calculated at first with respect to u (v 5 const.)and then by varying v between u 5 a and u 5 b. Beltrami integrated (3.8) by partsand obtained

2P 5 ES

cf

nds 1 E

Wc D2f dW, (3.9)


where s is the integration line on the boundary of W and n is the normal to Wdrawn inwards. By considering the function c instead of f in the definition (3.7)and by developing the integral (3.8) as before, Beltrami deduced the expression

2P 5 ES

fc

nds 1 E

Wf D2c dW. (3.10)

By subtracting (3.9) from (3.10), he could write

EW

(f D2c 2 cD2f) dW 1 ESSf

c

n2 c

f

nD ds 5 0. (3.11)

This equation expresses, according to Beltrami, ‘‘a theorem valid on a surface withany curvilinear coordinates; that theorem is analogous to another very famous onevalid in a three dimensional space with usual coordinates’’ [4, 339]. In fact, (3.11)is a generalization of Green’s formula. It is valid for a particular W but can alsobe extended to the ordinary domain V by summing up the contributions of all theregions dividing V.

In addition, if the function c on W becomes infinite as log 1/r at an interior pointO (r is the geodesic distance between O and a boundary point of W), the value f0

of f at O is given by the equation

2ff0 5 EW

(f D2c 2 cD2f) dW 1 ESSf

c

n2 c

f

nD ds

(s is the boundary of W). This extends Green’s integral formula to a system ofcurvilinear coordinates. In order to prove it, Beltrami followed a common procedureby examining the region W 2 Q, where Q is a circle contained in W with centerO and radius R, and then estimated the limit of the integral as R R 0.

These formulas allowed Beltrami to determine the complex function w, whichwas the most suitable one for studying functions of a complex variable f 5 f(w)on a surface. The original problem of the explicit determination of w was solvedif the integrating factor x was found; in that case, the variable w could, indeed, bededuced from (3.2) and (3.3). From some previous results and from Green’s theorem(3.11), Beltrami obtained

EV

D2 log uxu dV 5 2Es

d log uxudn

, (3.12)

where V is an ordinary region with boundary s and dV 5 H du dv. The integratingfactor x must then satisfy (3.12). But what is the geometrical meaning of thiscondition? Beltrami gave such an intepretation by remarking that the left side of(3.12) represents the integral (or total) curvature G of the surface, while the rightside can be transformed by introducing the angle t (‘‘angolo di contingenza geode-tica’’), which is an integral of the geodesic curvature. So he deduced

G 5 2f 2 T, (3.13)


where T is the sum of all the angles t valued with respect to the boundary s of V.This expression had already been obtained by Bonnet [30, 217]. If V is a geodesicpolygon on the surface, (3.13) will coincide with Gauss’s formula, G 5 2f 2 o ci,ci being the internal angles of the polygon [42, Section XXVI].

Therefore, Beltrami established the relations required to define the ‘‘right’’ com-plex variables w on a surface by using differential parameters and their properties.From these invariant expressions, he also proved theorems which are analogous tothe relative results in geometry and in the usual theory of functions of a complexvariable. Beltrami [5] also employed differential parameters in order to determinea system of equations which analytically describe minimal surfaces. After a lucidand interesting historical introduction to the theory of minimal surfaces from La-grange to Weierstrass,1 he proved some formulas involving differential parameterson a surface defined by the fundamental form (3.1). These equations are actuallyreferred to as Beltrami’s formulas and express the mixed, the first, and the seconddifferential parameters of the coordinates x, y, z at a point P as functions of theprincipal radii of curvature R1, R2 at P and of the cosines X, Y, Z of the anglesbetween the coordinate axes and the normal to the surface at P. The idea thatcoordinates x, y, z and cosines X, Y, Z can be more easily studied as functions ofu, v had been formulated by Weingarten [88; 89] and Brioschi [33].

Beltrami applied these results to minimal surfaces which were characterized bythe condition2 1/R1 1 1/R2 5 0 and deduced that

D2x 5 0, D2 y 5 0, and D2z 5 0. (3.14)

Beltrami interpreted x, y as independent variables and introduced two new orthogo-nal coordinates u, v such that the coefficients of the line element (3.1) satisfyE 5 G, F 5 0, and H 5 E. This is the characterization of an isothermal systemwhich Beltrami had deduced in his Ricerche [2, 150–162] (see also Section 2). Inthese assumptions, (3.14) becomes

2xu2 1

2xv2 5 0;

2yu2 1

2yv2 5 0;

2zu2 1

2zv2 5 0,

whose solution is expressed as x 1 ix1 5 f(u 1 iv); y 1 iy1 5 c(u 1 iv); z 1 iz1

5 x(u 1 iv), where x, y, z, x1, y1, z1, are real functions of u, v and where

df2 1 dc 2 1 dx 2 5 0. (3.15)

By integrating (3.15), Beltrami deduced the formulas for f, c, x, namely,

1 Darboux wrote in the chapter devoted to minimal surfaces, ‘‘[t]his beautiful work [namely [5]] . . .contains a very extensive historical Note which, hopefully, did not let us forget any important worksabout our subject published before 1860’’ [39, 1: 280].

2 This condition was rigorously proved by Meusnier in 1776 and published in 1785 [69].


f(w) 5dfdw

sin w 1d 2fdw2 cos w,

(3.16)

c(w) 5dfdw

cos w 2d 2fdw2 sin w,

and

x(w) 5 if(w) 1 id 2fdw2 ,

where f(w) is an arbitrary function of the complex variable w 5 u 1 iv.Beltrami concluded that, in this way, ‘‘the most general forms of the three func-

tions f, c, x will be obtained. After that, by setting x, y, z equal to the real parts—orto the coefficients of i—of the right sides of the previous equations [(3.16)], twosystems of formulas will be deduced. These systems are both able to represent allminimal surfaces (the plane excepted)’’ [5, 28]. Minimal surfaces are thus introducedtwo by two (‘‘per coppie’’ [5, 28]). Two surfaces of the same couple, which aredescribed by the same function f(w), have similar and linked geometric properties:the tangent to u 5 const. on one surface is parallel to the tangent to the curvev 5 const. on the other surface, and then the normals at the corresponding pointsare parallel as well. Beltrami further deduced that two minimal surfaces of thesame couple can be isometrically transformed one onto the other. This result hadalready been noticed by Bonnet in 1853 [29].

Consequently, due to his new approach, Beltrami proved some well-known theo-rems on minimal surfaces in a completely different way and formulated generalconsiderations. He observed that his analytical representation of minimal surfaces(3.16) was equivalent to the analogous result obtained by Weierstrass [87] usingalgebraic techniques. The procedure adopted by Beltrami pointed out how differen-tial parameters are also prominent in studying the theory of minimal surfaces.Results published in Ricerche enabled him to single out significant connectionsbetween differential parameters, functions of a complex variable, minimal surfaces,geodesic lines, and isothermal coordinates.

4. APPLICATIONS TO POTENTIAL THEORY

In 1868 Dedekind published Riemann’s Habilitationsvortrag [79] on the principlesof geometry. There the concept of manifold is introduced by means of the fundamen-tal form

ds2 5 Oni,j51

gij dxi dx j, (4.1)

where x1, . . . , xn are coordinates and gij is a symmetric, positive-definite matrix.Riemann’s lecture induced Beltrami to publish his Saggio di interpretazione dellageometria non euclidea [7], where he tried to build a Euclidean model of theLobatschevski plane. Writing to Houel on 18 November, 1868 Beltrami explainedthat ‘‘[t]he paper (which is entitled ‘‘Interpretation. . . [Saggio]) was written during


the autumn of last year [1867], in consequence of some ideas which I conceivedafter reading your translation of Lobatschewsky’s work.’’3 However, after readingRiemann’s lecture, Beltrami decided to publish his work because it was substantiallyin agreement with Riemann’s ideas.4

In many subsequent papers, Beltrami applied his deep knowledge of non-Euclid-ean geometry in order to extend definitions and concepts in analysis, geometry,and mathematical physics to spaces with constant curvature. Since differential pa-rameters were relevant quantities in several fields of mathematics, Beltrami [10]generalized them to a Riemannian manifold. He intended to establish the theoryof differential parameters on a purely analytical footing, without assumptions aboutthe number and the meaning of the variables. He then defined the first and seconddifferential parameters in a space whose line element is described by (4.1) bygeneralizing the procedure developed in his Ricerche (see Section 2). Beltramiset g 5 det(gij) and introduced the first differential parameter of a function F 5F(x1, . . . , xn):

D1F 5 Oni,j51

gij Fxi

Fxj

. (4.2)

Since this denomination had already been used by Lame and had then been acceptedby mathematicians and physicians, Beltrami showed that his formula (4.2) consti-tuted a real generalization of Lame’s. In order to reach this goal, Beltrami showedthat D1F 5 (F/s)2, where ds is normal to the surface F 5 const., by using resultsobtained in a previous work [6] about quadratic forms. This formula had alreadybeen expressed by Lame [60, 11 and 22] in order to characterize the first differen-tial parameter.5

By adopting an analytical method, similar to that used by Jacobi [47], Beltramiestimated the variation of the integral

E D1FÏg dx1 . . . dxn, (4.3)

which led to the definition of the second differential parameter:

D2F 51

ÏgOni,j51

xiSgij Ïg

Fx1D. (4.4)

Differential parameters on Riemannian manifolds are invariant for changes ofcoordinates. Moreover, in order to describe them in a new system of coordinates,

3 This letter is in the ‘‘Dossier Beltrami’’ in the Archives de l’Academie des Sciences in Paris.4 Beltrami wrote to Angelo Genocchi in a letter dated 23 July, 1868 that ‘‘[The Saggio] will appear

in the Giornale of Naples [Giornale di matematiche], in its original form, except for some additionswhich I can now hazard, because they substantially agree with some of Riemann’s ideas.’’ This letteris published in [65, 415–416].

5 It is interesting to remark that if the coordinates are Cartesian orthogonal, from (4.2), D1F 5

(F/x1)2 1 (F/x2)2 1 (F/x3)2. So, this expression is the square of the first differential parameter(1.1) defined by Lame.


it is sufficient to know the expression of the line element in the new variables. Thismethod enables the line element to be interpreted as an essential quantity fromwhich not only the kind of the manifold but also the expressions of differentialparameters can be derived. As Jacobi [47] had remarked, this approach is veryfruitful; Beltrami himself appreciated Jacobi’s method in his formulation of thetheory of differential parameters on a surface developed in Ricerche.

Beltrami considered Gauss’s theorem, according to which, if a set of equal lengthgeodesics is orthogonal to a given curve, their final points form a curve orthogonalto the geodesics themselves and, thanks to his new formalism, extended the theoremto the case of manifolds. In 1869, Lipschitz also generalized this theorem in a spacewhose line element f(dx) was homogeneous in its differentials dx1, . . . , dxn of degreep $ 2, positive-definite, and with determinant 2f(dx)/dxadxb ? 0. Lipschitz gavea mechanical interpretation of Gauss’s theorem in his Lipschitzian manifolds bymeans of Hamilton–Jacobi theory (see [85]).

Differential parameters were also useful for examining problems in mechanicsand in potential theory on a surface. In particular, Beltrami rigorously provedsome classic theorems stated by Thomson, Clausius, Kirchhoff, Dirichlet, and CarlNeumann. His main results on potential theory dealt with Newtonian and logarith-mic potential, symmetric functions, potential functions due to the attraction of anellipsoid, and superficial potential [12–16; 18; 21]. However, his most original andinteresting papers on potential theory were devoted to formulating well-knownconcepts and theorems in a Riemannian manifold by means of differential param-eters.

The idea that the Newtonian attractive law could be extended to a curved spacewas not new at all. In 1835, Green [46] had observed that classic theorems onpotential theory could be proved in a space with n dimensions in a natural way,that is, without substantial changes. The same opinion was expressed by Schlafli insome unpublished notes dating from 1850–1852, which were gathered and publishedby Graf [44]. Dirichlet tried to express the Newtonian law in a space with constantnegative curvature about 1850 (see [84, 5]).

Before all of them, Janos Bolyai had expressed the possibility that potentialtheory—and generally mechanics—could be founded on a geometry independentof Euclid’s postulate in two manuscripts most probably dated from 1835 and 1850–1851.6 This was a consequence of his research on absolute geometry, printed in1832 in the Appendix to the Tentamen of his father [26] and based on some axioms,which did not include the parallel postulate. In fact, according to Bolyai, it wasimpossible to verify Euclid’s axiom on the basis of experimental data. He noticedthat ‘‘[i]t is not possible to distinguish between Euclidean geometry and absolutegeometry by employing terrestrial measurements; we need a distinction based oncalculations’’ [81, 64]. These calculations concerned the motion of planets by assum-ing that ‘‘the force of attraction between two bodies is always inversely proportionalto the spherical surface whose radius is the distance between the bodies’’ [52, 277].

6 The first was published by Stackel [81] and the latter by Kurshak and Stackel [52].


Then, if space is Euclidean, planets will interact according to the Newtonian law,but if the curvature of space is constant, the law of attraction will be proportional to

1

sin2 rk

, (4.5)

where r is the distance between the planets and k is the curvature of space. In 1885,Killing [49] would also find the same expression (4.5) (see also [85]).

Beltrami also considered problems of potential theory in curved spaces andgenerally in Riemannian manifolds and solved them with methods involving thefirst and second differential parameters. By setting (4.4) equal to zero, he obtainedthe Laplace equation valid for a system of n coordinates; it is actually commonlyknown as the Laplace–Beltrami equation.

As a remarkable application of his theory, Beltrami proved the relation:

ESn

(F D2G 2 G D2F) dSn 5 ESn21

SFGn

2 GFnD dSn21, (4.6)

where F, G are C1 in the domain Sn, Sn21 is the boundary of Sn, and n is normalto Sn. As Beltrami explained: ‘‘This latter equation [(4.6)] contains the generalizationof a well-known and useful theorem of integral calculus, generalization which—itseems to me—is as great as possible’’ [10, 108]. The well-known theorem is Green’sformula, which Beltrami had already established for n 5 2 in a previous paper [4](see (3.11)).

In a paper printed in 1884 [19], Beltrami studied some questions on potentialtheory in a three-dimensional space in order to examine the stress system of elasticfluid propagating phenomena.7 He considered the line element (4.1) of ‘‘a certaindetermined and invariable space’’ [19, 154] S where he considered a function Up

which was C 1 on S, became infinite of first order at a point p, and satisfied theequation D2U 5 0 on S. He then introduced

V 5 O Up mp or V 5 E Up dmp, (4.7)

if points with masses mp or solid bodies having infinitesimal mass dmp were consid-ered. The function V is similar to the usual potential function; by employing proce-dures analogous to those in the ordinary case when space is Euclidean, Beltramiproved the generalization of the Gaussian flux theorem [41, 9], namely

Es

Vn

ds 5 4fMs , (4.8)

where s is a closed surface in S with no masses on the boundary, n is its normal,and Ms is the sum of masses in s.

7 This problem was deeply investigated by Beltrami (see [84]).


From this ‘‘general property [(4.8)]’’ [19, 155], Beltrami deduced theorems ‘‘allvalid in S and identical to those theorems which, in analogous circumstances, arevalid for the ordinary potential functions’’ [19, 158]. In order to obtain a potentialtheory in a curved space for functions (4.7), which was similar to the other onevalid for ordinary potential functions, Beltrami had to investigate what happenedat infinity. Thus, he supposed space S as ‘‘unlimited in all directions’’ [19, 158], butit was ‘‘difficult’’ to establish a general potential theory, that is, a theory whichcould be applied to an n-dimensional space with constant or, more generally, nonconstant curvature. ‘‘In order to justify the foregoing considerations’’ [19, 158],Beltrami showed two cases where the potential function Up is defined in all space—also at infinity: the ordinary Euclidean space, where Up 5 1/r, and pseudosphericspace with radius R, where Up 5 R(coth r/R 2 1). As Beltrami had proved in hisSaggio [7], the quantity coth r/R is proportional to the straight distance Ïx2 1 y2

from P 5 (x, y) to the origin of coordinates O, and r represents the geodesicdistance between P and O.

In Germany, similar attempts to solve the Laplace equation expressed in anycoordinate system were made by Lipschitz, Schering, and Killing, who made im-portant contributions to mechanics and potential theory in curved spaces (see [84;85]). They all were aware of Beltrami’s works and, in particular, of his paper ondifferential parameters, which was very closely connected to their research.

5. CONCLUSIONS

By means of differential parameters, Beltrami expressed many geometric, ana-lytic, and mechanical quantities in order to interpret these quantities as intrinsicelements of a surface and, therefore, to point out their independence of coordinates.Differential parameters played a prominent role not only in differential geometrybut also in potential theory and in the theory of elasticity. Main definitions andresults in both theories were described by the first and second differential parame-ters; thus, for formulating these theories in curved spaces, it was sufficient to expressdifferential parameters in Riemannian manifolds.

Beltrami influenced Lipschitz, Schering, and Killing in Germany, and many Italianmathematicians, such as Bianchi, Alberto Tonelli, Cesaro, Padova, and Somigliana.While Bianchi and Tonelli made interesting contributions to potential theory innon-Euclidean spaces, Cesaro, Padova, and Somigliana tried to extend some resultsin the theory of elasticity to spaces with constant curvature (see [84]). These attemptsaimed at explaining electromagnetic phenomena by assuming that the deformationsof an elastic medium, filling a curved space, transmitted physical forces. All thesemathematicians often followed Beltrami’s methods in their research; in particular,they applied the properties of differential parameters in order to show that someexpressions or physical quantities were independent of the choice of coordinates.This point of view would be shown to be very fruitful in the conception of ten-sor calculus.

In 1900, Ricci and his student Levi Civita published their famous paper wherethey advanced Beltrami’s work by introducing a new ‘‘algorithm’’ [61, 189], the


‘‘differential absolute calculus’’ [61, 189] or tensor calculus as it is now commonlyknown. That theory led as the first result to ‘‘the discovery of a chain of invariantdifferential expressions containing one or several arbitrary functions, and D2U isthe first and the most important ring in the chain’’ [61, 188]. These ‘‘differentialinvariant expressions’’ were fundamental in deducing theorems and equations ingeometry, analysis, theory of elasticity, heat theory, and potential theory, as Ricciand Levi Civita showed in the last part of their memoir [61, 266–271].

Several applications of tensor calculus to the theory of elasticity were deducedby Ricci in his lectures on the mathematical theory of elasticity [78],8 where hedescribed the most relevant results in the theory of elasticity in tensorial formalism.Therefore, his formalism suggested the ‘‘character of independence of the choiceof variables, which in general is not pointed out by commonly used notations’’ [76,245]. This ‘‘character,’’ which is evident in Ricci’s calculus, appears in nuce inBeltrami’s theory of differential parameters.

In my opinion, it was not accidental that Ricci Curbastro was a student of Padova,who was a follower of Beltrami. However, Ricci’s ideas about the connectionsbetween mathematics and physics were very different from those expressed byBeltrami and Padova themselves. Ricci explained that theoretical physics just fur-nished a mathematical representation of the physical world. He thought that nohypothesis about the nature of space—such as the existence of ether—and howphenomena propagated through space could be assumed. Indeed, Ricci wrote inhis commemoration of Padova:

The equations of dynamics are another subject favored by Padova, and he tried to found themon solid bases. These bases also enabled him to explain not only the theory of motion of systemsbut also the mechanico-physical theories which aimed at describing optical and electromagneticphenomena by means of the ether’s vibrations. Nevertheless there are many opponents tothese theories. [77, 68–69]

And Ricci was one of them. In particular, Ricci observed that the connectionsbetween the physical world and mathematical description are very complicated.Theoretic physics must thus limit itself to giving a mathematical and symbolicrepresentation of the physical world, with no assumption on the nature of space.From this point of view, ether becomes a useless physical structure and only geomet-ric space has to be studied. These ideas revealed their relevance in Einstein’s theoryof relativity.

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8 These Lectures were published in the second volume of Ricci’s Opere [75] posthumously.


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the role of differential parameters in beltrami's work

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