On Samuelson Submanifolds in Four Space
The object of study in this talk is a general class of submanifolds of R^4. The motivation for this work was the derivation of the following equation which answered a question posed by the distinguished economist P.A. Samuelson.
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irk-123456789-63072010-02-24T12:01:07Z On Samuelson Submanifolds in Four Space Cooper, J.B. Russell, T. Геометрія, топологія та їх застосування Праці міжнародної конференції "Геометрія в Одесі - 2008" The object of study in this talk is a general class of submanifolds of R^4. The motivation for this work was the derivation of the following equation which answered a question posed by the distinguished economist P.A. Samuelson. 2009 Article On Samuelson Submanifolds in Four Space / J.B. Cooper, T. Russell // Збірник праць Інституту математики НАН України. — 2009. — Т. 6, № 2. — С. 264-275. — Бібліогр.: 21 назв. — англ. 1815-2910 http://dspace.nbuv.gov.ua/handle/123456789/6307 en Інститут математики НАН України |
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Геометрія, топологія та їх застосування Праці міжнародної конференції "Геометрія в Одесі - 2008" Геометрія, топологія та їх застосування Праці міжнародної конференції "Геометрія в Одесі - 2008" |
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Геометрія, топологія та їх застосування Праці міжнародної конференції "Геометрія в Одесі - 2008" Геометрія, топологія та їх застосування Праці міжнародної конференції "Геометрія в Одесі - 2008" Cooper, J.B. Russell, T. On Samuelson Submanifolds in Four Space |
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The object of study in this talk is a general class of submanifolds of R^4. The motivation for this work was the derivation of the following equation which answered a question posed by the distinguished economist P.A. Samuelson. |
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| author |
Cooper, J.B. Russell, T. |
| author_facet |
Cooper, J.B. Russell, T. |
| author_sort |
Cooper, J.B. |
| title |
On Samuelson Submanifolds in Four Space |
| title_short |
On Samuelson Submanifolds in Four Space |
| title_full |
On Samuelson Submanifolds in Four Space |
| title_fullStr |
On Samuelson Submanifolds in Four Space |
| title_full_unstemmed |
On Samuelson Submanifolds in Four Space |
| title_sort |
on samuelson submanifolds in four space |
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Інститут математики НАН України |
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2009 |
| topic_facet |
Геометрія, топологія та їх застосування Праці міжнародної конференції "Геометрія в Одесі - 2008" |
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| citation_txt |
On Samuelson Submanifolds in Four Space / J.B. Cooper, T. Russell // Збірник праць Інституту математики НАН України. — 2009. — Т. 6, № 2. — С. 264-275. — Бібліогр.: 21 назв. — англ. |
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2025-07-02T09:14:31Z |
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2025-07-02T09:14:31Z |
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1836525994030137344 |
| fulltext |
Çáiðíèê ïðàöüIí-òó ìàòåìàòèêè ÍÀÍ Óêðà¨íè2009, ò.6, �2, 264-275
J. B.Cooper
Johannes Kepler Universität, Linz, Austria
E-mail: cooper@bayou.uni-linz.ac.at
T.Russell
Santa Clara University, USA
E-mail: trussell@scu.edu
On Samuelson Submanifolds in Four
Space
The object of study in this talk is a general class of submanifolds of
R4. The motivation for this work was the derivation of the following
equation which answered a question posed by the distinguished economist
P.A. Samuelson.
1. The Holy Grail Equation
The Holy Grail Equation has the following form:
− b3 ∂2a
∂y2
− a3 ∂2b
∂y2
+ 2
„
∂a
∂x
−
∂b
∂x
«2
− 2b2
„
∂a
∂y
«2
+
∂2a
∂x∂y
!
+ a2
b
∂2b
∂y2
− 2
„
∂b
∂y
«2
+
∂2b
∂x∂y
!!
+ a
„
b2 ∂2a
∂y2
+
∂b
∂y
„
3
∂a
∂y
− 4
∂b
∂x
+
∂a
∂y
∂b
∂x
+ 2b
„
2
∂a
∂y
∂b
∂y
+
∂2a
∂x∂y
+
∂2b
∂x∂y
«««
+
„
∂2a
∂x2
−
∂2b
∂x2
«
+ b
„
∂b
∂y
∂a
∂x
+
∂a
∂y
«„
−4
∂a
∂y
+ 3
∂b
∂x
−
∂2a
∂x2
+
∂2b
∂x2
«
= 0.
We will discuss the origins of this equation below.
We turn to our central definition. For reasons which will soon be
apparent, we write (x, y, u, v) for the coordinates of a typical point
of R4. Consider such a point. If it lies on a given two dimensional
c© J. B.Cooper, T.Russell, 2009
On Samuelson Submanifolds in Four Space 265
submanifold M then by the implicit function theorem, indeed by
the definition of a submanifold, there is a neighbourhood of U of
(x, y, u, v) so that U ∩M is the graph of a smooth function. There
are 6 possibilities:
I. u and v as functions of x and y,
II. x and y as functions of u and v,
III. u and x as functions of v and y,
and the 3 analogous cases: u and y as functions of v and x, v and
x as functions of u and y resp. u and x as functions of v and y.
We say that M is a Samuelson submanifold if, in case I, u =
f(x, y), v = g(x, y), where f and g satisfy the equations
(
g2
J
∂
∂x
+
g1
J
∂
∂y
)(
−f2
J
∂J
∂x
+
f1
J
∂J
∂y
)
=
(
g2
J
∂J
∂x
+
g1
J
∂J
∂y
)(
−f2
J
∂J
∂x
+
f1
J
∂J
∂y
)
.
Here J is the Jacobi-determinant f1g2 − f2g1 – subscripts denote
partial derivatives.
In case II we demand that
J−1J−1
uv = JuJv
where J−1 = f1g2 − f2g1 and x = f(u, v), y = g(u, v).
In case III with u = f(v, x), y = g(v, x), the required equation
is
J
1
f2
∂
∂y
(
∂J
∂v
− f1
f2
∂J
∂y
)
=
1
f2
∂J
∂y
(
∂J
∂v
− f1
f2
∂J
∂y
)
where J = −f1/g2.
Thus definition is motivated by the concept of a Samuelson
configuration which was introduced in [9] following a suggestion
of Professor Samuelson. In view of further applications it seems to
be advantageous to introduce the more symmetric version above.
The relation between the two concepts is clarified below.
266 J.B. Cooper, T.Russell
2. The Connection with Webs
For the general concept of webs, we refer to the classical text
[6]. A more modern version is in [1]. In this paper we will only be
concerned with 1-webs in 2-space.
The relation between the above notion and the subject of webs
lies in the following simple observation.
Suppose we have such a manifold. Then this, generically, de-
scribes a two web in R2 as follows: we project the manifold suc-
cessively onto the (x, y, u) and (x, y, v) spaces and then consider
the intersections with planes parallel to the (x, y) plane.
At this point, we describe briefly various ways to specify webs
in the plane:
(1) As the level curves u(x, y) = c of a smooth real valued
function. (Example: if u(x, y) = x2 + y2 we get the family
of concentric circles centered at the origin.)
(2) As indexed families x = c1(λ, t), y = c2(λ, t) of
parametrized curves. (Example: x = λ cos t, y = λ sin t
describes the same family.)
(3) As the flow lines of a vector field F = (F1, F2) i.e. the
solution curves of the system
dx
dt
= F1(x(t), y(t))
dy
dt
= F2(x(t), y(t)).
(Example: F = (−y, x) again produces the family of con-
centric circles.)
(4) As the family of solutions of the differential equation dy
dx =
a(x, y) for a given smooth functions on the plane (a(x, y)
is the gradient of the curve through (x, y)). This can
be regarded as a special case of 3 where F is the direc-
tional field (1, a(x, y)), (the concentric circles correspond
to a(x, y) = −x
y ).
On Samuelson Submanifolds in Four Space 267
A key point in our discussion is the fact that these associations
are not uniquely determined. Thus if we are given two functions u
and v, then their level curves form a 2-web in R2. However these
webs do not uniquely determine u and v. This is the motivation
for the following definition.
Let u be a smooth real-valued function on the plane. Then a
recalibration of u is a function of the form U = φ ◦ V where φ is a
diffeomorphism between open intervals of R. More formally, U is
a member of the orbit of u under the group of diffeomorphisms of
the line (more accurately, the pseudo-group of diffeomorphisms of
the above type).
We now turn to our main result which gives various characteri-
zations of Samuelson submanifolds and which we will formulate in
the spirit of classical differential geometry (i.e. without detailing
what may occur in non-generic cases). Hence the quotation marks
around the word equivalent in our formulation.
Let be M a 2-manifold in R4, W the 2-web it induces on R2
(i.e. the configuration of two webs).
Then following concepts are “equivalent”:
(1) M is a Samuelson submanifold.
(2) W is an S-configuration [9].
(3) The Jacobi-determinant J = uxvy−uyvx splits multiplica-
tively as a function of u and v.
(4) The gradient fields a(x, y) and b(x, y) of the webs satisfy
the Holy Grail equation.
(5) The derivative of the differential form
divF · ω1 − ω1([F,G])ω2
vanishes. Here ω1 and ω2 are the differential forms dual
to the bases generated by the fields F,G. In coordinates,
ω1 =
1
F1G2 −G2F1
(G2,−G1)
268 J.B. Cooper, T.Russell
and
ω2 =
1
F1G2 −G2F1
(−F2, F1).
(6) There are two vector fields F and G with
divF = divG = [F,G] = 0
so that the webs consist of the flow lines of F and G resp.
(7) There are recalibrations U and V of u and v for which the
Jacobi-determinant is 1.
In the formulation of the next conditions we remark that
Diff (R)2 acts on R4(R5) and so on the family of submani-
folds of R4(R5) via the formula
(x, y, u, v) 7→ (x, y, φ(u), ψ(v))
(x, y, u, v,E) 7→ (x, y, φ(u), φ(ψ), E).
(8) M is equivalent to a Lagrangian submanifold of R4 with
the canonical symplectic structure x∧ y−u∧ v, under this
group action.
(9) The manifold is equivalent to the projection of a Legen-
drian submanifold of R5 onto R4, when we regard R5 as a
contact manifold with the differential form E±x dv±u dy.
(The four possible choices of sign depend on which of the
four cases in III above holds) [4].
We remark that in [17] Tabachnikov has introduced a connection
on 2-webs in the symplectic space (R2, x∧ y) where he states that
any of the above conditions is equivalent to the vanishing of this
connection.
We mention briefly connections with themes in economics, car-
tography, thermodynamics, elementary geometry.
3. Economics
The first use of the mathematical theory of webs in economics
was by Debreu, [11, 12]. Debreu had earlier obtained conditions
for a preference ordering to be representable by a numerical func-
tion, Debreu [10]. Once such a function exists, it was natural for
On Samuelson Submanifolds in Four Space 269
economists to ask when it would be additively separable. Debreu
showed that this question was equivalent to requiring that a 3-
web in the plane given by the level curves of the function, the
verticals, and the horizontals be equivalent to the trivial 3-web.
This in turn required the satisfaction of the Blasche/Thomsen/Bol
hexagon condition, Thomsen [18], Blaschke [5], see also Vind [19]
and Wakker [20].
Here we will use web theory in a quite different way. As noted
in the introduction, the original motivation for the derivation of
the Holy Grail equation was to provide a test for the presence of
maximization processes in economics. Such processes, Samuelson
[16], are mathematically equivalent to the processes of classical
thermodynamics. In this section we will show how web theory can
also be used to provide a test for maximizing behavior. We focus
on the economic situation, but, suitably reinterpreted, the under-
lying ideas are equally applicable to classical thermodynamics.
We use the same framework as Samuelson [16]. We consider a
firm which hires 2 inputs, say L1 and L2 , at competitive prices
W1 and W2 respectively. The firm sells its output as a monopolist
into a downward sloping demand function. In principle we can
experimentally construct two families of demand functions. The
first, steeper, family, L1(W1,L2) gives the demand for L1 at input
price W1 when L2, the quantity of the other input, is held fixed.
The second, flatter family L1(W1,W2), gives the demand when
the price of the second input is held fixed, see Samuelson [16].
As discussed earlier, view this data as a 2-web given by the
projection of a manifold in R4 onto 3-dimensional W1L1W2 and
W1L1L2 space respectively, the resulting projected manifold being
then intersected by planes parallel to the W1L1 plane.
A key question in economics is the identification of the condi-
tions on this 2-web necessary for the firm to be acting so as to
maximize profits. Samuelson proposed the following simple test.
Samuelson Maximization Test: A firm whose input demand
data is given by a 2-web cannot be a profit maximizer unless ab =
270 J.B. Cooper, T.Russell
cd, where a, b, c, d are areas of quadrilaterals bounded by “threads”
of the web.
As Tabachnikov has shown, Tabachnikov [17], when the Samuel-
son area condition is satisfied, the 2-web is trivial under an area
preserving transformation. Thus we can calibrate the leaves of
the web in such a way that there is unit area between the threads
labeled x and x+ 1 and y and y + 1, for each respective member
of the family. In classical thermodynamics this calibration corre-
sponds to the passage from empirical temperature and entropy to
absolute temperature and entropy. In economics this recalibration
is not possible, so the relevant test for maximization is whether or
not the already calibrated 2-web, when trivialized to the horizon-
tal/vertical web already satisfies the equal area condition.
Tabachnikov [17] gives a further characterization of the profit
maximizing condition. If we place the standard area (symplec-
tic) form dL1 ∧ dW1 on R2, the 2-web of factor demands is a
Lagrangian 2-web since all curves on the symplectic plane are La-
grangian sub-manifolds. Tabachnikov’s theorem 0.1 now applies
directly and a symplectic, torsion free, connection (the Hess con-
nection [15], can be associated with the web. As Tabachnikov
shows, when the Samuelson test is satisfied with equality, this
connection is flat. This provides an alternative characterization of
profit maximizing behavior.
Finally we note that if we impose the standard symplectic form
dL1 ∧ dW1 + dL2 ∧ dW2
on R4, the 2-dimensional sub-manifold is a Lagrangian sub-
manifold of R4. This means that the mapping from L1,W1 to
L2,W2 is area preserving and orientation reversing as we have
just seen. Since maximizing economic processes take place on La-
grangian sub-manifolds, economics, too, succumbs to Weinstein’s
Lagrangian creed. That everything is a Lagrangian submanifold
[21].
On Samuelson Submanifolds in Four Space 271
4. Cartography
(The Central Problem of Mathematical Cartography.)
Our starting points are the two basic map projections:
• the Lambert projection which is an equal area projection,
• the Mercator projection which is a conformal projection
(see, for example, Brown [7]).
All equal area (conformal) projections arise from the compo-
sition of these projections with area preserving maps (conformal
maps) of the plane. (We remark that it follows from our main
result that a 2 web is the family of meridians resp. parallels of an
equal-area projection iff they form an S-configuration.)
The central problems of cartography were to characterize all
equal area (conformal) projections whose meridians and parallels
are (generalized) circles (cf. [7]).
The conformal case was solved by Lagrange, the equal area case
by Gravé, a mathematician at the University of Kiev. In order to
get the flavour of Gravé’s result, consider the equation
x = δ cos
u+ ǫ
k
− sin
u+ ǫ
k
√
2kv + e,
y = δ sin
u+ ǫ
k
+ cos
u+ ǫ
k
√
2kv + e
(δ, ǫ, k, l are parameters), which is taken from article [14] with his
solution to the second problem. Then the mapping from (u, v)
to (x, y) described by this formula is an equal area mapping of
the plane with the required property. Gravé’s result consists of
a list of analogous formulae which includes all possible equal-area
projections with the above property.
We note that it follows easily from the main result that certain
special S-configurations can be generated by groups in either of
the following ways:
A) if we are given 2 one-parameter groups of area-preserving
transformations A(t), B(t) so that A and B commute, then
their orbits form an S-configuration;
272 J.B. Cooper, T.Russell
B) if A(t) is as above and Γ is a generic curve, then the webs
formed by the orbits of A resp. the images of Γ under A
form an S-configuration.
It is natural to express the result of Gravé’s in terms of such
groups. Thus we require one-parametric groups whose orbits are
generalized circles. There are three natural candidates:
• the translation group.
• the rotation group.
• the dilatation group.
We can then use these three groups together with an additional
generalized circle to generate S-configurations as in B) and hence
equal area projections. It turns out that all of Gravé’s configura-
tions arise in this manner and this provides a much more accessible
version of his characterization.
5. Thermodynamics
Suppose we are given two functions u and v (empirical temper-
ature and empirical entropy). Then the basic facts of thermody-
namics are contained in the statement: u and v can be recalibrated
to satisfy the Maxwell relations, if and only if their level curves (the
“isotherms” and “adiabatics”) form an S-configuration. These re-
calibrations, which are essentially unique, are absolute tempera-
ture and absolute entropy.
We remark that this application was our motivation for seek-
ing a more symmetric version of Samuelson’s area condition. The
equations of state for a thermodynamic substance involve two re-
lationships between the four quantities p, V , T , S (pressure, vol-
ume, (empirical) temperature, (empirical) entropy) but there is
no necessity (empirical or theoretical) which dictates the choice of
dependent and independent variables.
Examples:
• Ideal gas: u = xy, v = xyγ ,
On Samuelson Submanifolds in Four Space 273
• Van der Waals gas: u = (x+ 1
y2
)(y−1), v = (x+ 1
y2
)(y−1)γ ,
• Feynman gas: u = xy, v = xyγ(xy),
• u = (x+ 1
y2
)(y − 1), v = (x+ 1
y2
)(y − 1)
γ((x+ 1
y2 )(y−1))
.
All of these satisfy the Holy Grail theorem and so have unique
recalibrations for which they satisfy the Maxwell relationships.
The resulting computations provide insights into the nature of
real gases.
6. Geometry
We close with the remark that a number of elegant geometrical
configurations which we call Apostol-Tabachnikov configurations
are examples of S-configurations and this gives a simple and ele-
gant approach to some of their properties. These are generated by
a single curve and a web of tangential curves, one at each point.
The second web is obtained by moving the initial curve along the
latter in a suitable manner.
(1) Apostol configurations. In this case we move a curve along
its tangents. Analytically, thus is given by the equations
x(u, v) = c1(u) + v
(ċ1(u), ċ2(u))√
ċ21(u) + ċ22(u)
,
y(u, v) = c2(u) + v
(ċ1(u), ċ2(u))√
ċ21(u) + ċ22(u)
.
(2) Tait-Kneser configurations. Here the second foliation con-
sists of the family of osculating circles:
(x(u, v), y(u, v)) =
(c1(u), c2(u)) +
[
cos v − sin v
sin v cos v
] [
c1(u) − C1(u)
c2(u) − C2(u)
]
.
(C(u) is the center of curvature).
274 J.B. Cooper, T.Russell
(3) Tait-Kneser-Tabachnikov configurations generated by the
curve y = f(x).
x(u, v) = u+ v,
y(u, v) =
n∑
r=0
vr
r!
f (r)(u).
(For these configurations, see references [2], [3] and [13].) In
each case it is a routine matter to compute the Jacobi-determinant
and to verify that it splits multiplicatively in the required manner.
References
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