ON THE GEOMETRY OF THE SHEAF OF
FRAMES OF A VECTOR SHEAF
Efstathios Vassiliou
A b stra ct
G iven a v ector sh eaf E we define a corresp on d in g prin cipal sh eaf o f fram es
V{£)
and we sh ow that Λ -c o n n e c tio n s o f £ (in the sen ce o f [3], [4]) corresp on d
b ije ctiv e ly to p rin cip al con n ection s o f V{S) (in the sense o f [5], [6]). Details and
co m p le te p ro o fs will b e given in [7].
A M S S u b je c t C la s s ific a t io n : Primary 53C05. Secondary 18F20
K e y w o r d s : sheaves and presheaves, connections.
1
Introduction
In [3] and [4], A. M allios has considered ^ -co n n e ctio n s on vector sheaves. Roughly
speaking, a vector sheaf € is a locally free .A -m odu le o f finite rank over a topological
space X , where A is sheaf o f commutative, unital and associative C -algebras. An
.4 -con n ection on £ is defined, accordingly, by an appropriate morphism on S, as
explained in Section 2.
On the other hand, the present author has defined connections on principal sheaves,
i.e. connections on sheaves which locally look like certain “geom etric” sheaves o f
groups.
In both cases, the approach is purely algebrotopological, without any notion o f
differentiability (which is, o f course, meaningless in this context), a fact which allows
to extend a great part o f the classical differential geom etry to n on -sm ooth spaces.
M otivated by the relationship between linear connections on vector bundles and
their counterparts on the corresponding frame bundles, we associate a vector sheaf
with a principal sheaf o f frames and we show that ^ -co n n e ctio n s on the former
correspond bijectively to principal connections on the latter.
The paper contains an outline o f the main results in this direction, whereas full
details and proofs will appear in [7].
E ditor Gr.Tsagas Proceedings of the Workshop on Global Analysis, Differential Geometry and Lie
Algebras, 1995, 141-146
© 199 7 Balkan Society o f Geometers, Geometry Balkan Press
142
2
E. Vassiliou
Preliminaries
The starting point is an algebraized space (Λ ',.4 ), where λ' is a topological space and
A a sheaf o f com m utative, associative and unital (linear) C -algebras over X . This
is an abstraction o f the notion o f differential manifolds containing also many cases o f
differential spaces.
In order to develop geom etric structures along the main lines o f traditional dif­
ferential geometry, we also attach to X a differential triad ( Λ , ^ , Ω 1), where Ω 1 is an
.4-m od ule over X and d : A — ►Ω 1 is a C -linear morphism satisfying condition
d (s-t) = s-d (t) + t-d{s),
(1)
for every s ,t € A ( U ) and U C X open.
R e m a r k . T h e m orphism d in (1.1) is the one induced between sections. In fact, we
apply the identification o f a sheaf with the sheaf o f germs o f its sections, as well as the
identification o f a sheaf m orphism with that between the corresponding presheaves o f
sections (see e.g. [1], [2] for relevant details).
If we denote by A the sheaf o f units (: invertible elements) o f A , i.e. the sheaf
generated by the (com plete) presheaf o f abelian groups U i— ►( A ( U ) ) , U running
over the open sets o f X , then d induces the morphism o f sheaves o f (abelian) groups
d : A — ►Ω 1 given by
d(s) : = s ' 1 ■d(s),
seA (U ),
(2)
for every open U C X .
If M n( A ) (n > 1) denotes the matrix algebra sheaf generated by the com plete
presheaf U i— ►M n ( A { U ) ) o f matrices with entries in the algebra A ( U ) , for all
U C X open, i.e.
M n ( A ) ( U ) a M n(A (U ) ),
we may define the ^4-m odule
-Μη^Ω1) : = M n ( A ) 0.A Ω 1 = (Ω 1) " 3,
on which d induces the m orphism (using the same sym bol)
d : M n ( A ) — * M n i t t 1)
given by
d{a) := (5 (a ,j)),
if a = (a ,j) e M n ( A ) ( U ) = M n{ A ( U ) ) , U C X open.
Analogously, d extends to a logarithmic differential
d : g C ( n , A ) — ^ M n{Q1)
by setting
d (a ) := a-1 · d(a),
a = (a ,j) 6 QC{n, A ) ( U ) .
1
O N TH E G E O M E T R Y O F T H E SH EAF O F F R A M E S
143
Here ζ / £ (η ,Α ), is the sheaf o f units o f λ Λ η(Α ) , generated by the (com plete) presheaf
U y— > G L ( n , A ( U ) ) = M n( A { U ) ) ,
U C X open.
Hence, G C { n , A ) { U ) Ξ G L ( n , A ( U )) and G C ( n , A ) = M n{ A ) , n > 1.
Since the adjoint representation A d : Q C ( n , A ) — ► E
n
d
[*4<%)](α) : = g ■a ■g ~ x, induces the representation
A d -.g C (n ,A )—
, given by
( Λ ΐ ^ Ω 1)) := £n d (Λ Ί ,^ Ω 1))
with
A d ( g ) ( a ® Θ) : = (g · a ■g ~ l ) ® Θ,
for every g G ( j £ ( n , A ) ( U ) , a G A i n( A ) (U ) , θ G Ω 1( ί /) and U C X open, we easily
verify that
d(a ■b) = A d { b ~ l ) ■d(a) + d(b),
for every a, b G G £ (n , A ) ( U ) , ω G A 4 n(Q,l ) (U ) and U C X open.
3
Connections on Vector Sheaves
In this section we consider a vector sheaf £ = ( £ , X , 7t.e) o f finite rank, say n. This
means, by definition, that £ is an ,4 -m od u le such that there exist ^ -isom orp h ism s
Φα '-£\ua - ^ A n \ua = {A\ua) n ,
over an open cover C = { U a | a G 1} o f X .
(3)
A n A —connection on £ is a C -linear
morphism
V
£ — > £ <SU Ω1 = : Ω *(£)
satisfying the Leibniz condition
y (a· · s) = a ■V s + 5 ® da,
(4)
for every a G A ( U ) , s G £ { U ) and U C X open.
For the existence o f ^ -co n n e ctio n s, examples and related topics we refer to [3]
and [4], where V is denoted by D (the latter is used here for the principal connections
below ).
Now, each isom orphism (2.1) determines a natural basis e a := (e“ , . . . , e “ ) o f
£ ( U a) by
e ° ( x ) :=
-
1*, - . . , 0 , ) ;
x G Ua ,
if 0T and l x (in the i- t h entry) are the zero and unit elements o f the stalk A x
respectively. A s a result, any section s G £{U a ) can be written in the form
n
*= £ > ? ·< ;
t= l
s ? € A ( U a)
E. Vassiliou.
144
and (2.2) leads to the expression
+ei 0 d s °'} = Σ e°0 f ds° +Σ
vw =
i= l
«=1
\
sf
·ωο·
J=1
from which we determine the local connection matrix (with respect to Ua )
ω β := (ω ?·) € M n( ^ { U a )) “ M „C 4 (tfe )) ® ^ β) Ω 1^ ) .
.
As a matter o f fact, the previous m atrix com pletely determines the restriction o f v
on £\Ua.
The main result here is the follow ing
T h e o r e m 1 A n A -c o n n e c i io n V on £ corresponds bijectively to a family o f local
matrices (u/“ ) a g /, with repect to C, satisfying the compatibility condition (: local
gauge equivalence)
ωβ = A d ( g ~ l ) · u a + dgaP.
4
The Sheaf of Frames
For a given vector sheaf ( £ , Χ , π ) as in .th e previous section, we consider the basis
o f topology B on X the elements o f which are the open subsets V o f X such that
7 C [/a , for som e a £ I (recall that C = { U a | a G / } is an open cover o f X over
which (2.1) holds).
We verify that if IsoA\Ua (A n\Ua, £ \Ua) is the group o f all ^4|f/a-m od u le isom or­
phisms, then
U^
Iso Alu(A n\U,£\U),
U running in B , is a com plete presheaf. The resulting sheaf V { £ ) is defined to be the
sheaf o f frames o f £ .
GC(n, A ) acts on the right o f V { £ ) by means o f the local actions
„
Su ■IsoAlu(A n \U,£\U) x GL(n,A\U) — - lso Alu(A n\U, £\U),
given by
for every U 6 B.
Moreover, the mappings
φα : V{€)(ua) = ΐ80Λχϋα{Αη\υ*,ε\υ*)·-+
G £ (n ,A )(U a)
defined by Φα ( / ) :== φα o f , are QC(n, .^ (i/c^ -eq u iv a ria n t isom orphism inducing the
corresponding £ £ (n ,.4 )| t/a-equivariant sheaf isomorphisms
V{£)\Ua £ gC(n,A)\U a , Ua G C.
As a result, we conclude that
ON TH E G E O M E T R Y O F THE SH EAF O F FR A M E S
145
the sheaf o f frames V { £ ) o f £ is a principal sheaf (in the sense o f [2]), o f structure
type Q C ( n , A ) and with structure sheaf also Q C (A ).
It is quite standard to show that V { £ ) is fully determined by the 1-cocycle (gQp) €
Z l (C, G £ { n , >1)) coinciding with the coordinate cocycle o f £. Moreover, € is associated
with V ( £ ) , i.e.
£ ^ V { £ ) x x A n/ G £ ( n ,A ) .
A ccording to the general definition o f a connection on a principal sheaf (see [5]
and [6]), a connection on V { £ ) is a m orphism o f sheaves o f sets D : V ( £ ) — ►Λ /ί „ ( Ω 1)
such that
D ( a · g) = A d { g ~ l ) · D { a ) + dg,
(5)
for every σ £ V ( £ ) ( U ) , g € QC(n, A ) ( U ) and any U C X open.
If σ α := Φ ~1(ϊά\Αη ( υ α )) are the natural sections o f V ( £ ), with respect to C, then
the evaluation o f D on them yields the corresponding local connection matrices o f D
θα := D ( a a ),
a £ I.
A s a direct consequence o f (3.1) and σβ =■ σ α ■gQp, under appropriate restrictions
on υ αβ Φ 0, we see that the local connection matrices o f D satisfy the com patibility
condition
θβ = Α ά ( 9- 1 ) · θ α + δ 9αβ.
(6)
T h e follow ing result may be thought o f as the principal sheaf counterpart o f The­
orem 1.
T h e o r e m 2 A connection D on V ( £ ) corresponds bijectively to a fam,ily o f local
matrices { θα € Λ /ίη (Ω 1)(ί7α)| a t / }
satisfying the compatibility condition (3.2).
C om paring conditions (2.3) and (3.2), the previous theorems lead to the last main
result o f the note, nam ely we have
T h e o i'e m 3 There exists a bijeciion between A -con n e c tion s on £ and connections D
on V { £ ) .
References
[1]
[2]
R.
G odem ent, Topologie algebrique et theorie des faisceaux, Hermann,Paris, 1973.
A. G rothendieck, A General Theory o f Fibre Spaces with Structure Sheaf, Kansas
University, 1958.
[3] A. M allios, On an abstract form o f W eil’s integrality theorem, Note di Mat.ematica,
12(1992), pp. 167-202.
[4] A. Malllios, Geometry o f Vector Sheaves (book in press).
E. Vassiliou
146
[5] Ε. Vassiliou, From principal connections to connections on principal sheaves, Anal.
Stiint. Univ. “Al. I. Cuza” (in press).
[6] E. Vassiliou, Connections on principal sheaves (manuscript).
[7] E. Vassiliou, On M allios’s A -con nection s as connections on principal sheaves,
Note di M atem atica (to appear).
Author’s address:
Efstathios Vassiliou
University o f Athens
Department o f Mathematics
Panepistimiopolis
GR-15784 Athens
Greece