Download Frölicher versus differential spaces: A Prelude to Cosmology

Frölicher versus differential spaces:
A Prelude to Cosmology
Paul Cherenack∗
Department of Mathematics and Applied Mathematics, University of
Cape Town, 7701 Rondebosch, South Africa.
ABSTRACT: Differential spaces due to R. Sikorski and Frölicher spaces
(called earlier smooth spaces and studied by A. Frölicher) are compared. The
category of Frölicher spaces is known to be topological over sets and Cartesian
closed under the usual set adjunction. We see that the category of differential
spaces is topological over sets but not Cartesian closed under the usual set
adjunction. The tangent and cotangent bundles of an arbitrary Frölicher space
X are defined and these notions are seen to coincide with the usual ones if X
is a smooth manifold. We show that one can define, in a fairly natural way,
the notions metric, connection, Ricci tensor, Riemannian curvature tensor and
finally Einstein tensor for an arbitrary Frölicher space.
Keywords: Frölicher space, differential space, topological functor,
Cartesian closed, tangent bundle, connection, Einstein tensor.
AMS Classifications: 18F15, 57R55, 58D15.
Introduction
In physics an object is often known by various scalar fields such as temperature and pressure defined on it. Local estimates of temperature can
be put together to obtain a global temperature function. Various scalar
functions can be combined via a smooth function of several variables
to determine other scalar fields. Thus, if I denotes the current in an
electric line and R is resistance, then the function V = f (I, R) = IR
determines the scalar field V denoting potential. Such ideas lead to the
notion of differential space and its generalizations which have played
some role in mathematical physics. See [8], [12] among the wide variety
of papers here. Penrose and Rindler [10] use related concepts. To be
precise we now define the notion of differential space and map between
differential spaces. Let X be a set and F be a set of real valued functions
on X called scalar fields or just scalars which define an initial topology
T on X. Thus, as one can show, U ∈ T if and only if U is the union of
∗
This paper is dedicated with thanks to Bernhard Banaschewski on his seventieth
birthday
c 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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sets of the form f −1 (V ) where f is a finite (pointwise defined) product
of elements in F and V is an open subset of the reals IR.
DEFINITION 0.1. The pair (X, F) is a differential space (assumed to
possess the topology T ) if
− for each open covering {Ui }i∈K of X and function g : X → IR,
if g|Ui = hi |Ui for some hi : X → IR ∈ F and each i ∈ K, then
g ∈ F . Thus, if locally a function comes from F, it must be in F.
This is a sheaf theoretic property.
− if f1 , · · · , fn is a collection of functions in F and g(x1 , · · · , xn ) is
a smooth real-valued function (i.e., infinitely differentiable), then
g ◦ (f1 , · · · , fn ) is again in F.
A map f : (X, FX ) → (Y, FY ) of differential spaces is a map f : X → Y
such that f ∗ FY = {g ◦ f |g ∈ FY } ⊂ FX . The class of such maps makes
differential spaces into a category DSP. A map in DSP will be called
a differentiable map.
In the definition of differential spaces one would like to think of the
scalar fields in F as smooth maps. There is only partial control though,
over this property. To motivate our subsequent definition of Frölicher
space, we note heuristically that one often considers curves on a physical
surface where scalar fields (e.g., temperature or pressure) vary in a
smooth way as a function of position. If such curves, as maps from IR
to the surface, are smooth, one expects of course their compositions
with such scalar fields to be smooth maps IR → IR. These intuitive
notions are refined, in the case of IRn , by the following result of Boman
[1] which shows the important fact that the converse implication holds
as well.
THEOREM 0.1. Let f : IRn → IR be a function. Then, f is smooth
on IRn if and only if f ◦ c : IR → IR is smooth for each smooth curve
c : IR → IRn .
These ideas motivate the notion of Frölicher space which has been used
in papers [3] by the author in smooth homotopy. Traditionally, Frölicher
spaces were called by Frölicher and Kriegl [4] smooth spaces. Because
such smooth spaces
− are determined by contours (inputs) into them and scalar fields
(outputs) on them,
− as point subsets of the plane, they need not be smooth,
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− Frölicher has worked extensively with them,
we now call them Frölicher spaces but call the maps smooth. Their
definition, where M is the set of smooth maps from IR to IR follows.
DEFINITION 0.2. The triple (X, F, C) is a Frölicher space (assumed
to possess the initial topology T defined by F) if
− F ◦ C = {f ◦ g|f ∈ F, g ∈ C} ⊂ M.
− ΦC = {f : X → IR|f ◦ c ∈ M for all c ∈ C} = F.
− ΓF = {c : IR → X|f ◦ c ∈ M for all f ∈ F} = C.
The elements of C are called contours.
Frölicher and Kriegl [4] is our main reference for Frölicher spaces.
Example 1 : The most important Frölicher spaces are finite dimensional smooth manifolds where if X is such a manifold, then CX =
{c : IR → X|c is smooth } and FX = {f : X → IR|f is smooth }.
Although there may at first glance seem little connection between
Frölicher and differential spaces, one can show:
THEOREM 0.2. Let (X, C, F) be a Frölicher space. Then, (X, F) is a
differential space.
Proof. We use the notation in the definition of differential space. Let
c : IR → X be a contour. As hi ◦ c is smooth, so is g|Ui ◦ c|c−1 (Ui ) . Since
the sets c−1 (Ui ) for i ∈ K cover IR, g ◦ c must be smooth. But, since
(X, C, F) is a Frölicher space, g ∈ F .
Suppose that c is again a contour. Since fi ◦c, i = 1, · · · , n, is smooth,
the composite (f1 , · · · , fn ) ◦ c is smooth. But, then g ◦ (f1 , · · · , fn ) ◦ c is
smooth. This implies that g ◦ (f1 , · · · , fn ) belongs to F.
DEFINITION 0.3. A map of Frölicher spaces or just smooth map
is a map arising from the underlying map of differential spaces. The
resulting category of Frölicher spaces is denoted FRL.
It follows that on identifying a Frölicher space with its corresponding
differential space that one obtains:
PROPOSITION 0.1. The category FRL is a full subcategory of DSP.
Let C be a category and U : C → SET S be a faithful functor to
the category of sets. Here, the category C will be either DSP or FRL
and U will be the forgetful functor. Let S = {hi |B → U (Ai )}i∈I be a
collection of set maps. Suppose that B = U (A). Then, A will be called
an initial object for S if
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1. there are (unique) maps {gi : A → Ai }i∈I lying in C such that
U (gi ) = hi for i ∈ I and
2. given an object A0 and maps {ki |A0 → Ai }i∈I such that U (ki ) = hi
for i ∈ I, there is a necessarily unique map α : A0 → A in C such
that U (α) is the identity map on B.
Dually, one can define the notion of final object (for U and a set of
maps S0 = {hi |U (Ai ) → B}i∈I ). If any set S has an initial object, then
every set S0 has a final object, U is said to be a topological functor
and C is called topological over SET S. For more detail, see Brümmer
[2]. Now, for the forgetful functor U : FRL → SET S, Frölicher and
Kriegl [4] have shown that FRL is topological. They also show that
FRL is Cartesian closed. We will show to start with (Section 1) that,
for the forgetful functor U : DSP → SET S, the category DSP is
topological. If C is topological, it has all limits and colimits. One merely
lifts the limit in SET S to C. We will examine these limits and colimits
and also initial subobjects in FRL and DSP. We will see (Section
2) that initial subobjects in FRL compared to DSP better reflect
any singularities which the subobject possesses. Furthermore, in FRL,
tangent cones can be defined for Frölicher spaces and tangent cones can
be more appropriate than tangent spaces for the study of singularities in
a Frölicher space. For instance, in algebraic geometry, tangent cones are
used in “blowing up” singularities. See [5]. In Section 3 we compare the
products in FRL with those of DSP. This enables us to see in Section
5 that unlike FRL, if DSP is Cartesian closed (which seems unlikely),
the Cartesian closedness does not arise from that of sets. In Section
4 we show that colimits of objects in FRL are the same irrespective
of whether taken in DSP or FRL. The Cartesian closedness of FRL
enables us to construct in Section 6 the tangent bundle, cotangent
bundle and associated type (p, q)-tensor bundles over an object in FRL
in a natural way. We can also construct corresponding cone bundles in
FRL. Finally, in our last section, we show how to write down the
Einstein tensor for an arbitrary Frölicher differential space.
Although much of the theory is abstract, it leads to many classical
type questions.
The author would like to thank the referee for useful advice.
1. The category DSP is topological over sets
We will show that, for the forgetful functor U : DSP → SET S, the
category of differential spaces is topological over sets.
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∗ is a collection of maps X → IR, then F ∗ generates a collection
If FX
X
∗ of maps X → IR such that (X, ΣF ∗ ) is a
of functions denoted ΣFX
X
0 ) is a differential space with F ∗ ⊂ F 0 ,
differential space and if (X, FX
X
X
∗ ⊂ F 0 . More precisely, one can show, see [11] , that ΣF ∗ is
then ΣFX
X
X
the set of all g : X → IR such that, for each P ∈ X and on some open
∗ ),
set U with P ∈ U (for the initial topology given by FX
g|U = h ◦ (f1 |U , · · · , fn |U )
∗ and n ∈ IN .
for some smooth function h : IRn → IR, f1 , · · · , fn ∈ FX
Let now {hi : Y → Xi }i∈I be a collection of set maps where Xi is a
differential space with structure functions FXi . Set
FY∗ = {g : Y → IR|g = f ◦ hi with f ∈ FXi and i ∈ I} .
Then, we show:
LEMMA 1.1. The differential space (Y, ΣFY∗ ) is the initial object for
the collection of maps {hi : Y → Xi }i∈I .
Proof. Clearly, the maps hi : (Y, ΣFY∗ ) → (Xi , FXi ), where i ∈ I,
are maps of differential spaces.
Let (Y, F 0 ) be a differential space such that
hi : (Y, F 0 ) → (Xi , FXi )
is a map of differential spaces for each i ∈ I. Then, for f ∈ FXi ,
f ◦ hi ∈ F 0 for each i ∈ I. But then FY∗ ⊂ F 0 and, from the above
discussion, ΣFY∗ ⊂ F 0 .
As we mentioned in the introduction, from the lemma it follows that
DSP must thus also have final objects and be topological for the
forgetful functor U : DSP → SET S.
2. Initial subobjects and tangents in FRL and DSP
Initial subobjects of an object in FRL exist. Extracting from the more
general construction in Frölicher and Kriegl [4], let (X, CX , FX ) be a
Frölicher space and A a subset of X. Then, the inclusion iA : A → X
places an initial structure on A where the resulting Frölicher space is
(A, CA , FA ) with
− CA = {c : IR → A|iA ◦ c ∈ CX }.
− FA = ΦΓ {f ◦ iA |f ∈ FX }.
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With this structure A is called a Frölicher subspace of X.
Here the description of CA is most useful. Initial here means that if
iA : (A, C 1 , F 1 ) → (X, CX , FX )
is a map of Frölicher spaces, then the identity map
IA : (A, C 1 , F 1 ) → (A, CA , FA ) defines a smooth map.
Examples 2 : We use the above notation.
| denote the rationals. Then, C
− Let X = IR and A = Q
A consists of
|
the constant maps and FA thus consists of all functions. Since Q
then has the discrete topology, we then call it a discrete Frölicher
space. It is reasonably clear that there are examples of fractal
curves which are Frölicher subsets of IR2 and having the discrete
Frölicher structure.
− Let C be the curve and subset of IR2 defined in polar coordinates
by r = 1θ , θ ∈ (0, ∞) but also containing (0, 0). A straight forward
computation of the arc length l(θ) of the curve r = 1θ from θ0 to
θ (θ > θ0 ) shows that l(θ) ≥ ln θθ0 . Since any smooth curve
c : [0, 1] → IR2 must have finite arc length, it follows that there is
no smooth curve connecting ( θ10 , θ0 ) to (0, 0). Although C is a
connected subspace of IR2 in the usual topology, a contour into C
thus maps either to (0, 0) or to the complement of (0, 0). Hence, a
scalar on C can take any value at (0, 0) irrespective of values at
other point. The Frölicher subspace (A, CA , FA ) thus is
disconnected topologically.
Before we present other examples we define the tangent space at a
point of a differential space according to the usual definition and an
alternate definition of tangent at a point of a Frölicher space. This last
definition follows a common way for defining tangents in the theory of
differentiable manifolds.
DEFINITION 2.1.
1. Let (X, F) be a differential space. A tangent vector V at P ∈ X is
a derivation V : F → IR at P , i.e. a linear map such that
V (f g) = f (P )V (g) + g(P )V (f ).
The set of such vectors form the tangent space T XP of X at P .
2. Let (X, F) be a differential space and P ∈ X. Let c ∈ ΓF and
suppose that c(a) = P and f ∈ F. Suppose that Vc is the derivation
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defined by setting
Vc (f ) = lim
t→a
f ◦ c(t) − f ◦ c(a)
.
t−a
Set T CXP = {Vc |c ∈ ΓF, c(a) = P }. It is easy to see that
T CXP ⊂ T XP . We call T CXP the tangent cone to X at P .
In fact, as examples below show, T CXP need not be a vector space.
Examples 3 :
1. The rationals as a Frölicher subspace of IR have trivial tangent
spaces equal to their tangent cones: Since contours must have
| . The
constant values, the tangent cones must be trivial. Let q ∈ Q
function f : IR → IR such that f (q) = 1 and f (r) = 0 if r 6= q
2
belongs to FQ
| . Since f = f , one can show that, for any
0
derivation D at q, D(f ) = 0. Let g ∈ FQ
| and g = f g. Then,
D(g 0 ) = f (q)D(g). Since g(q)g 0 = (g 0 )2 , D(g 0 ) = 0 and thus
D(g) = 0. Hence, the tangent space at q is trivial.
2. Except at (0, 0) the Frölicher curve C above has a one-dimensional
tangent space(= tangent cone). At (0, 0) the tangent space(=
tangent cone) is trivial(as in 1.).
3. Let B be the Frölicher subspace of IR2 defined by xy = 0. The
tangent cone to B agrees with the tangent space and is
1-dimensional except at (0, 0) where the tangent cone is B and
the tangent space is IR2 . A scalar on B is a function f : B → IR
such that f is smooth on the x and y-axes, respectively.
4. Let V be an algebraic subvariety of IRn defined by polynomial
equations f1 , · · · , fm and P = (a1 , · · · , an ) ∈ V . Let pi be the
degree of the first non-vanishing homogeneous term in fi regarded
as a polynomial in the xj − aj for i = 1, · · · , m and j = 1, · · · , n.
Set p = min {pi }i=1,···,m . If g is a polynomial in the polynomial
ring R generated by the xj − aj for j = 1, · · · , n, let (g)k denote
the term of degree k in the same variables. Consider the ideal I
generated by (f1 )p , · · · , (fm )p in R. The tangent cone to V at P is
then the zero set of I in IRn . One can show that the tangent space
to V at P is the zero set of (f1 )1 , · · · , (fm )1 . Thus, if m = 1 and
f1 = x2 + y 2 , then T CV(0,0) = {(0, 0)} and T V(0,0) = IR2 .
Let now X be a differential space, A ⊂ X and FA∗ = {f ◦ iA |f ∈ FX }
where iA : A → X is the inclusion. Then A with scalars Σ(FA∗ ) is called
a differential subspace of X.
Examples 4 : We consider examples given above in DSP.
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| → IR which are
| as a differential subspace of IR has scalars f : Q
1. Q
locally in the usual topology the restrictions of locally smooth
| is the
functions on IR. Thus, the tangent space to a point q ∈ Q
same as the tangent space when q is regarded as a point of IR and
one dimensional.
2. The curve C in Examples 2, as a differential subspace of IR2 , is
connected. Scalars at (0, 0) are the restrictions locally of smooth
functions on IR2 and hence cannot take on arbitrary values at
(0, 0). For f ∈ FC , let
∂
f (0 + h, 0) − f (0, 0)
lim
|C,(0,0) f =
.
∂x
h
h→0,(h,0)∈C
Since f is the restriction locally of an element in FIR2 , this
∂
definition makes sense. One readily shows that ∂x
|C,(0,0) is a
∂
derivation on FC . The derivation ∂y |C,(0,0) is defined similarly. As
∂
∂
∂
∂x |C,(0,0) x|C = 1, ∂x |C,(0,0) y|C = 0, ∂y |C,(0,0) y|C = 1 and
∂
∂y |C,(0,0) x|C = 0, the tangent space of C at (0, 0) is at least
2-dimensional. To each derivation D on FC , one can associate a
derivation D on FIR2 by setting D(g) = D(g|C ). As D must have
∂
∂
|(0,0) + b ∂y
|(0,0) , it follows that
the form D = a ∂x
∂
∂
D = a ∂x |C,(0,0) + b ∂y |C,(0,0) . Hence, the tangent space of C at (0, 0),
as a differential space, is 2-dimensional.
3. Products in FRL and DSP
spaces
Suppose that one is given a collection {(Xi , Fi )}i∈I of differential
Q
spaces. Let i∈I Xi be the
or a collection {(Xi , Ci , Fi )}i∈I of Frölicher
Q
set product of the sets {Xi }i∈I and πj : i∈I Xi →
X for j ∈ I denote
Q j
the projection map. The initial structure on P = i∈I Xi in both DSP
and FRL is generated by the set
∗
FP
=
[
i∈I
{fi ◦ πi |fi ∈ Fi , i ∈ I} .
In smooth spaces one obtains
− CP = {c : IR → P| if c(t) = (ci (t))i∈I , then ci ∈ Ci }.
∗
− FP = ΦΓFP
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Here, the requirement that each component of c is a smooth map is
most useful.
We now provide an example that shows that finite products in FRL
and DSP need not be the same.
IN denote the Frölicher space and thus
Example 5 : Let IR⊕
differential space whose underlying set is IRIN and whose Frölicher
space structure is generated by the set C 0 is the set
{(xi (t))i∈IN ∈ CIRIN | except for finitely many i, xi (t) is identically 0}.
IN → IR defined by setting
It is clear that the function l : IR⊕
l((xi )i∈IN ) =
∞
X
xi
i=1
IN . Let IRIN denote a second copy of IRIN with
is a scalar on IR⊕
⊕⊕
⊕
coordinates (yi )i∈IN . Then, we show:
LEMMA 3.1. The function k(xi , yj ) =
P∞
i=1 xi yi
is
IN × IRIN for the Frölicher space product structure
1. a scalar on IR⊕
⊕⊕
but
2. not for the differential space product structure where the scalars
are of the form f (g1 , · · · , gn , h1 , · · · , hm ) with f : IRn+m → IR a
smooth function, g1 , · · · , gn ∈ FIRIN and h1 , · · · , hm ∈ FIRIN .
⊕
⊕⊕
IN × IRIN has the form
Proof. Since every contour c : IR → IR⊕
⊕⊕
c(t) = (xi (t), yj (t))i,j∈IN where all but finitely many xi (t) and yj (t)
are identically 0, the first statement is clear. To see the last assertion,
suppose that
f (g1 , · · · , gn , h1 , · · · , hm ) =
∞
X
xi yi .
(1)
i=1
One can write down power series expansions at (0, 0, · · ·):
gl =
∞
X
ail xi + · · ·
∞
X
bjm yj + · · ·
i=1
and
hk =
j=1
for l = 1, · · · n and k = 1, · · · m. Here, one lets xi = yi = 0 for i > M
and then lets M → ∞. Substituting into f and equating the result to
the right hand side of expression (1), since we can assume that each gl
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10
can pair up with at most one hm and vice-versa, one can assume
n = m. From the resulting equation, equating coefficients, one also
obtains a matrix equation
AB = I
(2)
where A is the ∞ × n matrix
c1 a11 · · · c1 an1
c2 a12 · · · c2 an2 ,
..
..
..
.
.
.
the cp are the appropriate coefficients in the expansion of f at
(0, · · · , 0), B is the n × ∞ matrix
b11 b12 · · ·
..
..
..
.
.
.
bn1 bn2 · · ·
and I is the ∞ × ∞ identity matrix
1
0
0
...
0
1
0
...
0
0
1
..
.
···
···
··· .
...
Since the invertibility of the n × n matrix consisting of the first n
rows of A implies that all columns in B after the first n columns are
zero, it is clear that equation (2) has no solution. Hence, k(xi , yj ) is
IN × IRIN .
not a scalar for the differential structure on IR⊕
⊕⊕
Let IR1 denote the reals but with the smooth structure where
FIR1 = ΦΓ(FIR ∪ {|t|}).
p
Clearly, |x| ∈ FIR1 . It seems likely that x2 + y 2 is a scalar for the
Frölicher product IR1 × IR1 but not the differential product IR1 × IR1 .
No proof of this fact has been found.
4. Coequalizers and coproducts in FRL and DSP
Let {Xi }i∈I be a set of differential or Frölicher spaces with Fi the set of
S
S
scalars on Xi (i ∈ I), b i∈I Xi the disjoint union and iXi : Xi → b i∈I Xi
the inclusion map. Place the differential or Frölicher final structure on
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11
S
b
i∈I Xi
given by the set {iXi }i∈I . The resulting differential or Frölicher
space
is the coproduct of {Xi }i∈I in the respective category, denoted
`
by i∈I Xi and
F`
i∈I
Xi
=
(
f:
a
i∈I
Xi → IR| for each i ∈ I, f |Xi ∈ Fi
)
is the set of scalars for the coproduct. See [4], [11]. Thus, from the
above description, it follows that the inclusion of FRL into DSP is
coproduct preserving.
Examples 6 : In Examples 2.1 (the setting is the category FRL)
| can be identified with a countable union of points.
1. Q
2. C can be identified with a coproduct of IR and a point.
A similar analysis holds for coequalizers. Let f, g : X → Y be
two differentiable maps between differential or Frölicher spaces and
Q = Y / ' where ' is the smallest equivalence relation with the property that y ' y 0 if y = f (x) and y 0 = g(x) for some x ∈ X. Furthermore,
suppose that q : Y → Q is the set quotient map and Q acquires a final
structure in differential spaces or Frölicher spaces via this quotient map.
In both differential spaces, as it is easy to see, and Frölicher spaces
(see [4]) FQ = {f : Q → IR|f ◦ q ∈ FY }.Thus, the inclusion functor
of FRL into DSP preserves coequalizers and, because every colimit
can be formed from coproducts and coequalizers, the inclusion functor
preserves colimits.
Now, for a differential space X̂ = (X, FX ) let
Υ(X̂) = (X, ΓFX , ΦΓFX ) be the associated Frölicher space. One can
prove:
LEMMA 4.1. The association of a Frölicher space Υ(X̂) to a differential space X̂induces a functor DSP → FRL.
Proof. Let f : X̂ → Ŷ be a map of differential spaces. We need to
show that f is a smooth map of the associated Frölicher spaces. The
rest of the proof is transparent. Let c ∈ ΓFX . We must then show that
f ◦ c ∈ ΓFY . But, f ◦ c ∈ ΓFY if and only if h ◦ f ◦ c is smooth for all
h ∈ FY . But, h ◦ f ∈ FX and this is thus the case.
For a differential space X̂, the identity set map IX : Υ(X̂) → X̂ is
a map of differential spaces which one can use to satisfy the solution
set condition in Freyd’s Adjoint Functor Theorem (see [6]). One thus
obtains:
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12
PROPOSITION 4.1. The inclusion functor FRL → DSP has a right
adjoint.
It is, in fact, easy to show that the right adjoint is the functor Υ.
Note that, for differential spaces, tangent cones are defined to be the
tangent cones of the associated Frölicher spaces.
5. The non-Cartesian closedness of DSP
The category of topological spaces is not Cartesian closed; so, in algebraic topology, where Cartesian closedness is useful, the category
of compactly generated spaces was considered as an alternative (see
[6]). The rationals form a non-compactly generated space and in the
| , IR)
discussion below we try to see why hom-sets such as HomDSP (Q
should provide examples where the Cartesian closedness of DSP fails.
We will show that two possible ways to begin constructing Cartesian
| plays a role) fail and that the set mappings
closedness for DSP (where Q
defining sets and FRL to be Cartesian closed do not work for the
category of differential spaces.
− Let X be a differential space and X̂ the associated smooth space.
The identity map I : X̂ → X is a map of differential spaces which
induces a map
I : HomDSP (X, IR) → HomDSP (X̂, IR) = HomF RL (X̂, IR).
The set HomFRL (X̂, IR), since FRL is Cartesian closed, can be
given an FRL structure and, via I, one can put an initial structure
on
G = HomDSP (X, IR) in DSP. Will G with this initial structure be
under set adjunction a Cartesian closed object in DSP? In other
words, is there a natural bijection
θ : HomDSP (Z, G) → HomDSP (Z × X, IR)
| be viewed as a differential
where θ(f )(z, x) = f (z)(x)? Let X = Q
| , IR) consists
|
subspace of IR. Since Q̂ is discrete, Ĝ = HomDSP (Q̂
| → IR. Also, I : G → Ĝ is the inclusion map. The
of all maps Q̂
scalars FG on G are generated by functions of the form f ◦ I where
f ∈ FĜ . However, f ∈ FĜ if and only if
f ∈ HomDSP (
a
|
q∈Q
q, IR) ∼
=
Y
ˆ
IR = IR,
|
q∈Q
correctdiffeo2.tex; 4/05/2001; 11:30; p.12
13
ˆ Let Z = IR. A map l : IR →
where f is identified with (f (q)) ∈ IR.
G sending t to lt is smooth if and only if I ◦ l is smooth and using
our earlier statement about the functions on G if and only if
I(lt ) = (lt (q))q∈Q
|
(3)
is smooth for each t in each coordinate. However, the corresponding
| → IR in differential spaces is differentiable if and only
map ˆl : IR×Q
ˆ
if l(t, q) = lt (q) is smooth in an IR2 -neighborhood of each point
| . Clearly, however, smoothness of the expression in
(t, q) ∈ IR × Q
(3) just above does not imply the smoothness required for ˆl. Thus,
Cartesian closedness in DSP cannot be obtained by putting an
initial structure on G.
− We now attempt to make G, as above, into a differential space by
identifying G with the set of maps f : IR − I → IR where I ranges
over the discrete subsets of IR consisting exclusively of irrationals.
This identification is possible since every f ∈ G is smooth in a
neighborhood of each rational. Let I be a discrete subset of IR
consisting exclusively of irrationals. Then, set
HI = {f : IR → IR|f is smooth except at point in I} .
Notice that every map in HI restricts to a unique map in G and
that every map in G that extends to a map in HI does so uniquely.
One can thus view G as a direct limit in sets of the HI . Let
Ie = {(i, j)|i, j ∈ I, i < j and no l ∈ I is strictly between i and j} .
It is easy to see that HI , since FRL is Cartesian closed, can be
endowed with the Frölicher structure of the Frölicher space
HomFRL (
a
IRj , IR)
j∈Ie
where IRj is a copy of IR. Since DSP is topological over sets,
G can be viewed as a differential space and the direct limit of
the HI in DSP, where I < I 0 if I ⊂ I 0 . We then ask if G with
this structure is a Cartesian object in DSP. Since coproducts and
coequalizers are the same taken over Frölicher spaces in DSP, as
we have seen earlier, G will also be an Frölicher space. Now take a
path c : IR → G where c(a) ∈ HI in DSP. A scalar g : G → IR is
essentially a collection of scalars gI : HI → IR such that if I ⊂ I 0 ,
then the restriction of gI 0 to HI is gI . This implies that if c(t) ∈ HI
for t in an open interval O containing a, then the associated map
correctdiffeo2.tex; 4/05/2001; 11:30; p.13
14
ĉ : O × Q → IR extends to a map O × IR → IR which is smooth
except possibly on lines y = i, i ∈ I. However, without writing
down the detail, it is not difficult, for a rational number q, to find
a sequence rn of irrational numbers converging to q and a sequence
of numbers an converging to a such that c(an ) is not extendable
to a smooth function at rn for each n ∈ IN . Then, the associated
map ĉ will not be smooth at (a, q) which is impossible if one wishes
G to be a Cartesian object under the map making sets Cartesian
closed.
IN defined earlier. Suppose that
− Consider now the Frölicher space IR⊕
DSP is Cartesian closed under the usual set adjunction and K deIN , IR)
notes the differential space with underlying set HomDSP (IR⊕
and arising from the Cartesian closedness of DSP. Let K be the
associated Frölicher space and g : IR → K a smooth map. Since
IN → IR
g : IR → K is also a differentiable map, the map ĝ : IR × IR⊕
defined by setting ĝ(t, x) = g(t)(x) is also a differentiable map
(using the assumed Cartesian closedness of DSP). Suppose that
IN taken in DSP is also the product in FRL.
the product IR × IR⊕
Then, ĝ is a smooth map. Hence, using the Cartesian closedness
of FRL, g must also define a smooth map g : IR → L where L
IN , IR)
denotes the Frölicher space with underlying set HomFRL (IR⊕
arising from the Cartesian closedness of FRL. Thus, every contour
of K is a contour of L.
IN → IR is
Let g : IR → L be a contour of L. Then ĝ : IR × IR⊕
IN
a smooth map with the product in FRL. Again, since IR × IR⊕
is also the product in differential spaces, g : IR → K must be a
differential map and induce a smooth map g : IR → K. Thus, L
and K must have the same contours and are thus equal.
Finally, if DSP is Cartesian closed we have, using the natural
bijection from sets:
IN
IN
HomDSP (IR⊕
× IR⊕
, IR)
I
N
∼
= HomFRL (IR , K)
⊕
∼ HomDSP (IRIN , K)
=
⊕
IN
∼
, L)
= HomFRL (IR⊕
IN
∼
= HomF RL (IR × IRIN , IR)
⊕
⊕
where the products are taken in the appropriate categories. Since
IN with itself is different in DSP and FRL, it
the product of IR⊕
is clearly impossible to have all these bijections. Thus DSP is not
Cartesian closed under the canonical set adjunction.
correctdiffeo2.tex; 4/05/2001; 11:30; p.14
15
IN taken in DSP is not the product in FRL,
If the product IR × IR⊕
following similar arguments, one sees that DSP is not Cartesian
closed under the canonical set adjunction.
− In a Cartesian closed category, products commute with coequalizers and it would be useful to show that DSP is not Cartesian
closed using this fact.
6. Tangent and cotangent bundles on a Frölicher space
Let X and Y be Frölicher spaces. Then, (X, Y ) will denote the Frölicher
space whose underlying set is HomFRL (X, Y ) and with structure arising from the Cartesian closedness of FRL. We then write FX = (X, IR)
to denote FX with its associated Frölicher structure. Using the notation
of Definition 2.1, the sets
DX = {D : FX → IR|D is a derivation at some P ∈ X}
and
DX,C = {Vc ∈ DX |c ∈ CX , c(0) = P and P ∈ X}
can be viewed as Frölicher subspaces of (FX , IR) and the product
X × DX is a Frölicher space. We are now able to make the following
definition:
DEFINITION 6.1. The tangent bundle T X (resp., tangent cone
bundle TCX) on X is the Frölicher subspace of X × DX (resp.,
X ×DX,C )consisting of all (P, D) such that D is a derivation at P . The
projection map π : T X → X (resp., Π : T CX → X) is the smooth map
sending (P, D) to P . A vector field on X is (most properly) a section
of Π or (more generally) π.
Remarks : The notion of tangent bundle can clearly be extended to
differential spaces. In fact, for differential spaces, there are other, not
quite so canonical ways to define tangent bundles. See [9]. However,
| , it is clear that the tangent bundle of a
looking at the rationals Q
differential space need not coincide with the tangent bundle of the
associated Frölicher space. The tangent cone bundle of a differential
space can be defined as the tangent cone bundle of the associated
Frölicher space. This may be useful in some circumstances, but, as
| , information might be lost in certain cases.
in the case of Q
Example 7 : Let X = IRn . Then, there is a bijection
δ : T X → IRn × IRn
correctdiffeo2.tex; 4/05/2001; 11:30; p.15
16
P
where if (P, D) ∈ T X and D = ni=1 di ei , using the notation in
Nakahara [7], then δ(P, D) = (P, (di )). A map ζ : IR → T X is a
contour if and only if, in
ζ(t) = (c(t),
n
X
di (t)ei,c(t) ),
i=1
P
c(t) is a contour of X and ni=1 di (t)ei,c(t) is a contour of DX where
P
∂
ei,c(t) denotes the partial ∂x
taken at c(t). But, ni=1 di (t)ei,c(t) is
i
smooth
if and only if the map µ : IR × FX → IR sending (t, f ) to
Pn
d
(t)e
i (f )(c(t)) is smooth. Taking f = xi , one sees that di (t)
i
i=1
must be a smooth function of t for i = 1, · · · , n. The map µ is smooth
if and only if, for any smooth map IR → FX sending s → fs , di (t) and
ei (fs )(c(t)), for i = 1, · · · , n are smooth in s and t. But,
ˆ
ei (fs )(c(t)) = ∂ f (s,c(t)) and fˆ(s, t) = fs (t) is smooth in s and t. It
∂xi
follows that ζ is smooth if and only if c(t) and di (t), i = 1, · · · , n are
smooth. In that event, the bijection δ is a smooth isomorphism.
Let X be an n-dimensional smooth manifold and U an open coordinate
neighborhood of X isomorphic as a smooth manifold to IRn . One can
extend the above arguments to show that π −1 (U ) is isomorphic to
IRn × IRn as a Frölicher space. It is then clear that the tangent bundle
defined for X as a smooth manifold is smooth isomorphic to TX when
X is viewed as a Frölicher space.
Let E be a Frölicher space and π : E → X be a smooth map such
that π −1 (x) is a given vector space or a vector space aside from the
fact that the addition is only partially defined, for each x ∈ X. Then,
by LIN(E, R) we mean the Frölicher subspace of (E, IR) consisting of
all f such that f restricted to π −1 (x) satisfies f (ca) = cf (a)(c ∈ IR)
and f (a + b) = f (a) + f (b), whenever a + b is defined, for each x ∈ X.
Define a relation ' on
LIN = X × LIN(T X, IR)
by setting (P, f ) ' (Q, g) if and only if P = Q and, for some neighborhood U of P, f |π−1 (U ) = g|π−1 (U ) .
DEFINITION 6.2. The cotangent bundle T ∗ X to a Frölicher space is
LIN/ '. The projection map π10 : T ∗ X → X is the smooth map induced
from the projection of LIN onto its first factor. Replacing T X by T CX,
one obtains the notion of cotangent cone bundle, denoted T ∗ CX and
where the projection is Π10 . A section s of π10 or Π01 will be called a
(smooth) 1-form provided that, for each point P ∈ X, P has a neighborhood U such that s(x) = [(x, s0 )] for some fixed s0 ∈ LIN(T X, IR)
and all x ∈ U . Thus, s is represented locally by a global section.
correctdiffeo2.tex; 4/05/2001; 11:30; p.16
17
Remarks : Notice that the fiber (π10 )−1 (P ) above P has an induced
vector space structure. One commonly writes T ∗ XP = (π10 )−1 (P ). We
would prefer to omit the condition given on the section s but it is
not clear whether this is possible and, in practice, one needs to work
at P ∈ X with an element of LIN(T X, IR) restricted to a suitable
neighborhood of P .
Example 8 : Let X = IRn and q : LIN → T ∗ X be the quotient
map. A map τ : IR → T ∗ X is smooth if and only if f ◦ τ is smooth for
each smooth map f : T ∗ X → IR. But f is smooth if and only if f ◦ q
is smooth on
LIN. Let [(P, L)] be an equivalence class in T ∗ X and
Pn
L|π−1 (P ) = i=1 ai,P dxi |P . There is then a bijection
υ : T ∗ X → IRn × IRn sending [(P, L)] to (P, (ai,P )). We wish to show
that this bijection is a smooth isomorphism. Let c : IR → T ∗ X be a
contour and write υ(t) = υ ◦ c(t) = (Pt , (ai,Pt (t))).
Using the fact that q is a quotient, one sees that there is a smooth
map ψ : T ∗ X → X such that ψ ◦ q = λ1 where λ1 is projection onto
the first factor of LIN. As λ1 (Pt , Lt ) = Pt is smooth in t and
ψ = P1 ◦ υ = π10 , where P1 is the projection of IRn × IRn onto the first
of two factors, υ is smooth in its first coordinate.
Define
Θi : LIN → IR
P
by setting Θi (P, L) = ai,P where L|π−1 (P ) = ni=1 ai,P dxi |P . Again, as
q is a set quotient, there is a map ρi : T ∗ X → IR such that ρi ◦ q = Θi .
We wish first to show that Θi is smooth. Let d : IR → LIN be a
smooth map and d(t) = (Pt , Lt ). Then, Lt is smooth in t if and only if
the associated map L̂ : IR × T X → IR defined by L̂(t, v) = Lt (v) is
smooth. Identifying T X with IRn × IRn , as above, one can write
L̂(t, P, w) = A(t, P ) · w
where A(t, P ) is a matrix with entries which are smooth functions of t
and P , w is written as a column vector and · is matrix multiplication.
∂
. Then, with w viewed as a column vector,
Let w = ∂x
i
ai,Pt (t) = Θi ◦ d(t) = A(t, Pt ) · w . Hence, Θi and thus also the ρi are
smooth. Since ρi = Pn+i ◦ υ, where Pn+i is projection on the n + i-th
coordinate of IRn × IRn , υ must be smooth.
Conversely, suppose that the ai,PP
t (t) for i = 1, · · · , n and Pt are smooth
functions of t. Let β(t) = (Pt , ni=1 ai,Pt (t)dxi |Pt ). Then the second
component of β is smooth since the induced map sending (t, (Pt , bi ) to
P
aiPt bi , arising from the Cartesian closedness of FRL and using the
identification of T X with IRn × IRn , is clearly smooth. As β is then
smooth, so is q ◦ β. Hence, υ −1 is smooth and the conclusion sought
follows.
correctdiffeo2.tex; 4/05/2001; 11:30; p.17
18
Let E1 , · · · , En be Frölicher spaces and πi : Ei → X be a smooth
map for i = 1, · · · , n such that πi−1 (P ) is a given vector space or a vector
space aside from the fact that the addition is only partially defined, for
each P ∈ X. Let Π∗ : E1 ⊕E2 · · ·⊕En → X be the pullback of π1 ×π2 ×
· · · × πn : E1 × E2 × · · · × En → X n along the diagonal map δ : X → X n
which is clearly smooth. The set E1 ⊕ E2 · · · ⊕ En is usually called the
Whitney product of E1 , E2 , · · · , En . By MULTILIN(E1 , · · · , En ; IR)
we mean the Frölicher subspace of
(E1 ⊕ E2 · · · ⊕ En , IR)
consisting of all f such that
1.
f (a1 , · · · , cai , · · · , an ) = cf (a1 , · · · , ai , · · · , an )
for c ∈ IR, (a1 , · · · , an ) ∈ (π ∗ )−1 (P ) and P ∈ X.
2.
f ((a1 , · · · , ai−1 , bi , ai+1 , · · · , an ) + (a1 , · · · , ai−1 , ci , ai+1 , · · · , an ))
= f (a1 , · · · , ai−1 , bi , ai+1 , · · · , an ) + f (a1 , · · · , ai−1 , ci , ai+1 , · · · , an )
for
(a1 , · · · , ai−1 , bi , ai+1 , · · · , an ), (a1 , · · · , ai−1 , ci , ai+1 , · · · , an )
in (π ∗ )−1 (P ) and P ∈ X but only if
(a1 , · · · , ai−1 , bi , ai+1 , · · · , an ) + (a1 , · · · , ai−1 , ci , ai+1 , · · · , an )
is defined.
Define a relation ≈ on
MULTILIN(E1 , · · · , En ) = X × MULTILIN(E1 , · · · , En ; IR)
by setting (P, f ) ≈ (Q, g) if and only if P = Q and there is a neighborhood U of P such that
f |(Π∗ )−1 (U ) = g|(Π∗ )−1 (U ) .
DEFINITION 6.3. The higher level tensor bundle T pq X of type (p, q)
is equal to the Frölicher space
MULTILIN(T ∗ X, · · · , T ∗ X , T X, · · · , T X )/ ≈
|
{z
p
} |
{z
q
}
correctdiffeo2.tex; 4/05/2001; 11:30; p.18
19
where T ∗ X appears in the first p factors and T X appears in the last q
factors. Replacing T X by T CX and T ∗ X by T ∗ CX, one obtains the
tensor cone bundle. The projections π qp and Πpq are the evident ones. A
section s of π pq or Πpq is called a (p, q)-tensor field if, for each P ∈ X,
there is a neighborhood U of P such that s(Q) = [(Q, K)] for some
K ∈ MULTILIN(T ∗ X, · · · , T ∗ X, T X, · · · , T X; IR) and all Q ∈ U . A
pointwise cone Riemannian metric g is a (0, 2)-tensor field such that,
if gP is the restriction of g to (Π02 )−1 (P ), then
1. gP (U, V ) = gP (V, U ).
2. The induced fiber preserving smooth map
(g∗ ) : T CX → LIN(T CX, IR),
where (g∗ )P (X)(Y ) = gP (X, Y ), X, Y ∈ π −1 (P ) and P ∈ X, is a
smooth isomorphism onto its image. A pointwise Riemannian
metric is defined by taking π 02 instead of Π02 , T X instead of T CX
and requiring, in addition, (g∗ )P to be bijective.
Remarks : In the case of finite dimensional smooth manifolds, condition (2) of Definition 6.3 is usually replaced by the condition:
gP (U, V ) = 0 for all U ∈ T XP implies V = 0. However, a Frölicher
space may have infinite dimensional tangent spaces and this necessitates the revised requirement. Since the fibers of π : T X → X need
not be finite dimensional, T 10 need not be the same as T X. Thus, we
distinguish between ordinary and higher level tensor bundles, which
involve taking duals. Application of g∗ in tensor analysis is lowering of
indices.
Let X be an Frölicher space and c : I = [0, 1] → X be a smooth
map. Suppose that c(a) = P . Then, P has a neighborhood U such that
g(x) = [(x, g ∗ )] for each x ∈ U . As g ∗ ∈ LIN(T X, T X; IR) and
d : IR → T X × T X
defined by setting d(t) = (c0 (t), c0 (t)) is smooth, g ∗ (c0 (t), c0 (t)) is smooth
for t in a neighborhood of a. Letting g P = gP |π−1 (P )×π−1 (P ) if
g = [(P, gP )],
g c(t) (c0 (t), c0 (t)) = g ∗ (c0 (t), c0 (t))
is a real
R valued smooth function for t near a and hence for all t ∈ I.
Thus, c ds can be defined as
Z 1q
0
g c(t) (c0 (t), c0 (t))dt.
correctdiffeo2.tex; 4/05/2001; 11:30; p.19
20
Example 9 : Let X ⊂ IR2 be the union of the x and y axes. Let
α : IR → IR be a smooth function such that α(t) = 0 for |t| ≤ 81 and
α(t) = t for t ≥ 21 . Define c : IR → X by setting c(t) = (α(t), α(−t)).
The curve c(t) starts at (0, 1) and passing through (0, 0) proceeds to
(1, 0). Define the Riemannian metric in the usual way except at (0, 0).
Thus, g = dx along the x axis and g = dy along the y axis. At (0, 0),
let g = dx + dy. Then,
Z
c
ds = −
Z 0
α‘(−y)dy +
−1
Z 1
α‘(x)dx.
0
It is now clear that the notion of geodesic can be extended to
Frölicher spaces.
7. Derivation of Einstein tensor
We will show how one can formally define the Einstein tensor by looking
at the tensor bundles of a Frölicher space. On singular spaces it would
be better to look at the tangent cone bundle but the definitions require
more care. Our references are [7],[13].
DEFINITION 7.1. Let A be an Frölicher space, f ∈ FA and X [A]
be the set of smooth sections of the projection π : T A → A,i.e., the
smooth vector fields of A. The covariant derivative ∇X (f ) of f or
the directional derivative X[f ] of f is the element in FA defined by
setting ∇X (f )(a) = X[f ]a = q(X[a])(f ) where, if X(a) = (a, Da ), then
q(X(a)) = Da . A smooth map
∇ : X [A] × X [A] → X [A]
is a covariant derivative or connection if
1. ∇ is bilinear.
2. ∇f X Y = f ∇X Y for f ∈ FA .
3. ∇X (f Y ) = X[f ]Y + f ∇X Y (the Leibniz rule).
4. ∇ is induced from an smooth map
∇0 : X [A] × T A → T A,
mapping fiber to fiber, on setting (∇X )Y (P ) = ∇0X (Y (P )).
correctdiffeo2.tex; 4/05/2001; 11:30; p.20
21
Remark: The last condition in our definition of connection is normally
implicit in the theory of connections on a finite dimensional smooth
manifold.
Let F (0,1) be the collection of smooth (0, 1)-tensor fields or, in other
words, smooth 1-forms. Define h : F (0,1) → LIN(T A; IR) by letting
h(ω)(X(Q)) = ω P (X(Q)), for Q in a neighborhood of P , where ω(P ) =
[(P, ω P )] and ω P defines ω in a neighborhood of P . Here, we write
X(P ) merely to indicate that X(P ) ∈ T XP . As h ◦ q(P, ω P )(X(Q)) =
ω P (X(Q)), for Q in a neighborhood of P , h ◦ q and hence h is smooth.
In order for ∇Y , where Y is a smooth vector field, to satisfy the Leibniz
rule
∇Y (h(ω)(X)) = ∇Y h(ω)(X) + h(ω)(∇0Y X)
(4)
for X ∈ T A. Using condition (4) in the definition of connection (Definition 7.1), ∇Y h(ω) : T A → IR, defined by equation (4) directly
above, is smooth. Thus, (P, ∇Y h(ω)) ∈ A × LIN(T A; IR) and hence
the assignment
P → [(P, ∇Y h(ω))]
defines a smooth 1-form ∇Y ω.
In a similar way, one defines
h : F (0,2) → MULTILIN(T A, T A; IR).
The relation
∇Z (h(g)(X, Y )) = (∇Z h(g))(X, Y ) + h(g)(∇0Z X, Y ) + h(g)(X, ∇0Z (Y ))
implies that the map ∇Z h(g) : T A × T A → IR is smooth and finally
one lets ∇Z g be the map sending P to [(P, ∇Z h(g))].
We note, for the following definition, that the set of vector fields X [A]
on a Frölicher space A has a Lie product. Let X, Y ∈ X [A], X(P ) =
(P, DPX ) and Y (P ) = (P, DPY ). Since multiplication on IR is smooth,
the map A → T A defined by setting [X, Y ](P ) = (P, DPX DPY − DPY DPX )
is smooth and we have our Lie product.
DEFINITION 7.2.
1. The covariant derivative ∇ is a metric connection if and only if
∇X g = 0 for each X ∈ X [A].
2. The smooth map T : X [A] × X [A] → X [A] defined by setting
T (X, Y ) = ∇X Y − ∇Y X − [X, Y ]
is called the torsion tensor.
correctdiffeo2.tex; 4/05/2001; 11:30; p.21
22
3. The smooth map R : X [A]×X [A]×X [A] → X [A] defined by setting
R(X, Y, Z) = ∇X ∇Y Z − ∇Y ∇X Z − ∇[X,Y ] Z
is called the Riemannian curvature tensor.
Let A be a smooth finite dimensional manifold. Let
LIN(X [A]; X (A)) be the set of smooth linear maps X [A] → X [A] induced from pointwise linear maps of the tangent bundle. One can show
that there is a trace function TR : LIN(X [A]; X [A]) → FA , additive
and invariant under coordinate change. The Riemannian curvature tensor R : X [A] × X [A] × X (A) → X [A] induces, because of the Cartesian
closedness of FRL, a map R : X [A] × X [A] → LIN(X [A]; X [A]). The
map TR ◦ R : X [A] × X [A] → FA is a contraction of R and one calls
RIC = TR ◦ R the Ricci tensor. Let g be a Riemannian metric on a
smooth space.
DEFINITION 7.3.
1. Suppose that R : X [A] × X [A] → LIN(X [A]; X [A]) is the map
induced from R (using the Cartesian closedness of FRL and condition 4 of Definition 7.1) and one possesses a trace function
TR : LIN(X [A]; X [A]) → FA . Then, the Ricci tensor
RIC : X [A] × X [A] → FA is defined by setting RIC = TR ◦ R.
2. Because of the Cartesian closedness of FRL, the Ricci tensor
RIC(X, Y ) induces a smooth linear map
[ : X [A] → LIN(X [A], FA ).
RIC
3. If the smooth map g∗∗ : X [A] → LIN(X [A], FA ) (defined by setting
g∗∗ (Q)(X)(Y ) = h(X(Q), Y (Q)) where g = [(Q, h)] for Q in a
neighborhood of P ) is a smooth isomorphism, then g is called a
global Riemannian metric.
4. For a global Riemannian metric, the scalar curvature R on a
smooth space A is defined by setting
−1 [
◦ RIC).
R = TR(g∗∗
5. Finally, for a global Riemannian metric, G = RIC − 21 Rg is called
the Einstein tensor.
Remark: In the case of finite dimensional smooth manifolds, a pointwise Riemannian metric is a global Riemannian metric. The same
assertion is unclear for general smooth spaces.
correctdiffeo2.tex; 4/05/2001; 11:30; p.22
23
In conclusion, one sees that categorical language surrounding the
category FRL can fix many of the terms in mathematical physics in a
natural way and provide some clues to the direction of future endeavors.
References
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
J. Boman, Differentiability of a Function and of its Compositions with
Functions of One Variable, Math. Scand. 20(1967),249-268.
G. C. L. Brümmer, Topological Categories, Topology and its Applications
18(1984),27-41.
P. Cherenack, Smooth Homotopy, Bolyai Society Mathematical Studies: Topology with Applications,Szekszárd (Hungary),1993, pp. 47-70,4.
A. Frölicher and A. Kriegl, Linear Spaces and Differentiation Theory. Wiley–
Interscience,New York, 1988.
R. Hartshorne, Algebraic Geometry. Springer–Verlag,New York, 1977.
S. Mac Lane, Categories for the Working Mathematician. Springer–Verlag,New
York, 1971.
M. Nakahara Geometry, Topology and Physics. Adam Hilger,Bristol, 1990.
M. Heller and W. Sasin, Anatomy of the Elementary Quasi-regular Singularity,
Acta Cosmologica 21(1995),47-59.
A. Kowalczyk, Tangent Differential Spaces and Smooth Forms, Demonstratio
Math. 23(1980),893-905.
R. Penrose and W. Rindler, Spinors and Space-Time, vol. 1: Two-Spinor
Calculus and Relativistic Fields. Cambridge University Press,Cambridge, 1984.
W. Sasin, Differential Spaces and Singularities in Differential Space Time,
Demonstratio Math. 24(1991),601-634.
R. Sikorski, Differential Modules, Colloq. Math. 24(1971),45-79.
R. M. Wald, General Relativity. The University of Chicago Press,Chicago,
1984.
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