Green's operators on multi-almost
product manifolds
TAKUYASAEKI
(Received
Iutroduction.
multi-almost
of an almost
a multifoliate
In the
present
product manifold.
product structure
structure
defined
paper,
July
4, 1969)
we shall
A multi-almost
defined
in [1,3],
by K. Kodaira
study
about
Green's
operators
on a
product structure is one of generalizations
and it seems to be closely
related
and D. C. Spencer [2].
to
B. L. Reinhart
[3] studied
harmonic integrals
on an almost product manifold.
In
[3],
operators
d', 3' and A' are given with respect to x alone and a Green's operator G'
is defined
and an analogue to Hodge's theorem is proved, i. e., it is proved that the d!
cohomology is isomorphic
to the space of A' harmonic forms. But the Green's operator
G is not completely
continuous
and the kernel of A' is infinite
dimensional.
In § 1, we shall state the definition
of multi-almost
product structure,
which is
defined
by giving m fields
operators
d* (a=l,2,--,m)
In § 2, we shall construct
of subspaces of tangent
spaces of the given manifold.
And
and A*= d^S... + dad* (8* is the adjoint
operator
of d^).
Green's operators
for AK.
§ 1
Multi-almost
product
structure
Z,et M be a connected paracompact
(real)
w-dimensional
of class O. Denote its tangent
bundle by T(M),
the tangent
{p=\,l,--,n)
or TP(M),
and a bundle
of Grassmann
algebras
differentiahle
P-vectors
by AT(M)
manifold
by APT(M)
=2
r*(M),
where AaT(.M) is trivial
bundle of real numbers, and AXT(M)
=T(AQ. Let 0 (M)
the bundle of cotangents,
with other notations,
Ap@ (M), A0 (M), etc., analogous
those for T(M).
Let d denote the operation
of exterior
differentiation
on A0 (M).
A multi-almost
product
(we assume that the integer
subspaces
of Tx.
structure is defined
m is in Q<m^nX
If we set m*=dim
on M by giving m fields
S1, S2,---, Sm
ot class C°°, of complementary
proper
S' («=1,
where we assume nx=£ 0 for all a. Let P* (a=1,
SI
at every
point
x e M, where S% denotes
be
to
2,å å å ,m),
l,---,m)
the value
then
we have w=2
be the projection
of
n<t,
of Tx onto
S" at x.
In particular,
if m = 2, the multi-almost
product
structure
on M reduces to an
almost produet structure
in the sense of [1,3].
A manifold
with multi-almost
product
structure
is said to be a multi-almost
product manifold. Let M be a mulii-almost
product
manifold.
A multi-almost
product
sturcture
on M induces
the
projection
operators
17* ,,,,,, ...,,, (resp. 17 ,,,,,, ...,,,,) in A T(IV) (resp. A@). A vector fiefd which is a cr'osssection of n*,,,,,, ...,, AT(M) = T"'"'-.-"(M) is said to be of type (s,, SZ;..,~,) and
similarly for A@ (M). A map f of M into itself is said to be of type (t,, t,,...,t,) with
respect to the strncture if
where f * : A T ( M ) + A T (M) is the map induced by f; an operator L on A T (M) or
A@(M) may be decomposed into various types according to the definion:
Since the operator d of exterior differentiation is decomposed, so we find that
where dd.? is of type (0,
,o,
@P
-;,0 ,...,(I) for
a
... ,0, -J, 0, ...,0,2,0, ... ,0) for
lp
> p , and
a
< P , of
of type (O;.. ,0,1,0;.. ,0) for a
=
type (0;-.,0,2,0;.-
p respectively.
-
According to Guggenheim and Spencer [I], we may define an operator da :
(M) @p+l(M)by the axioms:
(1)
(2)
(3)
If $€AO@x(M),v € T X ( M ) ( x M~) , then
(da$) (v)= (Pal)) ($1.
If $E A0@,(M), then
(d,d+dda) $= 0.
If $ ~ A p d i ~ ( Manb
l + E ~ @ ~ ( Mthen
),
(da ($A$)=da$A$+ (-l)p$Adac$.
m
It is easily verified that 2
C dpaP=l
m
C daQ,for fixed
P=l
a , satisfieds these axioms hence we
may set
m
da = 2 C dP5fi=1
m
C dsP for
P-l
fixed a,
and find that d = $ dn.
a-1
In general, the operator d, is not differential, i. e., d i f 0. It is, however, differential if and only if thn mllti-alm3st product structur? is integrable, by which we
mean that the Poisson bracket of two vector fields of type P a is again a vcctor field
of the same type. In this case, the PoincarS lemma is true for d5 and forms of type
( ~ 1 , s~,.-.,s~,.-.,s,)with S a > O .
Let I, be the increasingly orclered n , - tuple (il, iz;..,in5) ( a = 1, 2
,m). If a
multi-almost product structure on M is integrable, we can apply the Frobenius' theorem
;.a
to obtain local coordintates (x,I';..,~~",...,x&~)
in the neighbourhood of any point such
that P5 T (M) is spanned by ( d / d x?) . This local coordinates is said to be a local multi-
almost product coordinate system, In terms of such a coordinate system, a differential
form $ of type (sl, sz,...s,;..,sm) may be written as
where I, (Su) is the increasingly ordered s,-tuple
dx-"("L dx$ A dx2
(il, iz,...,is,) of integers in I, and
...A &","OD
We notice that an integrable multi-almost prodect structure is a special case of
a (real) multifoliate structure given by Kodaira and Spencer [2]. In fact, if a multifoliate structure on M has a finite totally ordered set instead of a finite partially
ordered set P which defines a P-multifoliate set of integers and then if a local coordinate system (x,I1,...,xF,..., xLm) has a peoperty ax;/an;=o for a # p . The multifoliate
structure on M in this case becomes an integrable multi-almost product structure.
We may introduce on a multi-almost prodct manifold M a Riemannian metric
such that the subspaces P U T( M ) ( a = 1,2, .-.,m) are mutually orthogonal. If a multialmost product structure on M is integrable and if local multi-almost product coordlnates are used, the metric will have the form
where components of metric teusor gi,")( a
=
1 , 2 , ...,m) are C- functions of (x1I1,x,'"-.-,
xim). This metric is said to be a multi-almost product metric.
Hereafter we assume that the manifold M has an integrable multi-almost product
strucfure and a multi-almost product metric.
We may define an operator 6, for a form $ with totally degree q by
where
*
is the duality operator deilned by the given metric. Since
a#= (-l)"q+"+'*d*+,
m
we have 6 = C 6,. Let K be the increasingly ordered p tuple (R,,u:..,v)
a=i
(0 < p S m ; I?,
,u,-..,v
we denote the increasingly ordered (m-p) -tuple consisting of
and by
the subset of (1, 2, ...,m} complementary to K. We may define special Laplacian operators by
E {1,2;..,m})
-
-
It is easily showed that A,, A K and A preserve types, i.e., they are of type (0,0;..,0).
We denote a usual Laplacian operator by A.
If the manifold M is compact, the inner product ($, 4 ) is defined in usual way
for $, $ E @ ( M ) and the norm $=v'($=
is defined.
Let M be a c o m ~ a c tmulti-almost product manifold with integrable structure. A
Riemannian metric on M is said to be to?siouless if g-l;) ( a = 1 , 2 , ... ,m) are C" functions
of xi" ( a
=
1,2, ... ,m) alone. Setting
and
we have the following proposition.
PROPOSITION
1. If the metric on M is torsionless, then i r r = d ~ holas good and
-
moreover A = A.
Since the metric is torsionless we ean prove by the method similar to Reinhart
C3l.
PROPOSITION
2 . If M is an integrctblc multi-almost product manifold, we have
d2=0 and d i = ~ .
In fact, dL=O hence d~d,=O for all 2, p E ( 1 , 2;..,m}. On the other hand, di =O
for all a € {1,2;..,rn} since the structure is integrable. Then the proposition is proved.
Then we have
for a torsionless integrable multi-almost product manifold M.
§ 2
Green's operators for AK
Let 2; be a sheaf of germs of square integrable fo-ms of degree r which depend
F the direct
on Xrc alone, wherex K = IX.), xp,...,~?},and 2; is defined similarly. Let ~ K be
sum
C
8& A 2;. By the method similar to Reinhart [3], we can construct a coherent
T,s
subsheaf 23, of 2 , ~such that $,, #,,...,#t are sections of %K, where $1, $ ~ ; - . , $ t are also
sections of 2Kz. And any $ is a stalk of BK is expressed as C #, A r, in some neighbourhood, where y, is one of finite set of sections of 2;.
Next we shall define the second inner product of forms on M. It is defined by
and the corresponding norm by
11 # 11 D =V (\$, $))u, where (#,
$) is the usual inner
product and U a finite covering {U,}
of M.
The space L, ( B K ) is the completion in the usual norm of the set of Cm sections
of BK; the space P, ( ' 2 3 ~ ) is the completion in the second norm. Let Ha ( % K ) ( a € K )
be a subspace consisting of forms w in Pz ( 2 3 ~ )satisfying dffi3=6ffi,s=0
and HK ( B K ) =
HA ( B K ) n H , ( B K ) n ... n H, ( B K ) . We have the following theorems by the method
similar to [3] and the proof is omitted.
THEOREM
1. Let M be a compact multi -almost product manifold with torsionless
mctric and let B K be a coherent svbsheaf of &I. Let wo be a form i n LZ (88)orthogonal
to HK(@K)such that
the from o, to Bo is a bounded linear transformation
for all 6 E P, (BK). F~~rthermore,
from Lz ( B K ) into P, ( B K ) .
THEOREV
2. Let M be a compact multi-almxt product manifold with torsionless
with suffi) (uo, 5) for all
metric, aud BR bs a coherent subsheaf of 2gn. If (Be, A K ~ =
ciently small support i n L, ('13K) which are C", and if w., is i n L2 ( B K ) and Cm, then 520
is Cm.
Combining Theorem 1 and 2 , we can construct the Green's operator for As. By
the method of M. H. Stone we can sho,v that dffi=dffiam-t8~dffi
is a self-adjoint operator,
in particular is also closed, hence so is AK.
THEOREM
3. There is defiucd on L2 ( B K ) a bounded symmetric opdrator G K ,BK
such that AK GK %K$= GK,B~AK$=$-HK,BK$
and G K , B K HK,BK$=HK,BKGK,BK$=o,
whtre HK,BK is the projection to the space HK( 2 3 ~ ) .
PROOF,Let oo be orthogonal to HK(BK)= H A( B K )n Ha ( B K )n ... n Hv ( B K ) . B y
Theorem 1, We can find Do orthogonal to HKCBK)
that C (dancl, dxc) + C (6ffiB0,
a,<)
(w0,5) for all 6 E P2()3K).Moreover if w,, is Cm b y Theorem 2 , Bo is Cm,whence
AKQo=oo. Define G K B K ( W=BO,
~ ) then AKGK,BKCOO
=wO. Let HK,BK be a projection on
H K ( B K ) . If HK,%8+ is Cm for any $, so that #-HK,%K is C- if $ is. Let $ be an
arbitrary form in L2(23e), and let {#,,) be a sequence of CP forms approximating $.
Then {w,} = {$%- HK,BK$z}is a sequence of C- - forms approximating w = $ - HK,%K+
and each w, satisfies A ~ G ~ , % K ~ , =Since
( o , . I GK,%K110 1 S /I GK,BK00 /I S C I wo I , G:(,%Kis
bounded. Hence G K , B K ~ , GK,9K11, while AKGK,BKU,=J~,
-- w. Since AK is closed, we
= #NOW
) = let
~ . us define GK,BK$= GK,BK?~.Then AK,BKGK,@K#
have A K G ~ , B ~ ~ L
=
-+
HK,BK$. If HK(%K), we have GK,BKQ=O,hence G ~ , b e G r i , @=~ 0#. Also HK,BKGK,BK$
=(I. Hence GK,8K is symmetric on L2 (BK) and GK,BKAK$= $ -HK,%K$ for all $ in
the domain of Ax.
q.e.d.
By a method similar to Reinhart, me can show that the Green's operator is
pendent from the choice of coherent subsheaves of % K K .
THEOREM
4, Let M be a compact mulfi-airnost product manvold with torsionless
metric. Let $ be a 6- section o f the sheaf ZKK. Then therg is a Cm se~tion GK$ of ZKK
such that dKGK$ = GKAK$=$- HK$ and GKHK$=HKGK$=O. In these equations, GK$=
GK,BK$ and HK#=HK,BK$, where BK is d n arbitrary coherent subsheaf of 2gl1 having
# as a section.
The operator GK appeared in Theorem 4 is called a Green's operator for AK. GK
is not in general a bounded operator on the Hilbert space of all square integrable forms
on M, since GK may be densely defined without being everywhere defined. We can
easily show that d ~ d ~ = d K Aand
x GK commutes wlth d~ and BK. Hence we have an
analogue to Hodge's theorem.
THEOREM
5. Let M be a compact multi-almost product manifold with torsionless
metric. Then the d~ cohomdogy of sections of the sheaf 2 ~ is
2 isomorphic to the space of
dg harmonic sections of ZKs, the isomorphism being given by assigning to each cohomology
class the unique harmonic form contained i n it.
Next we shall consider a special case. We consider a Green's operator for 2. In
this case, We can construct the Green's operator by a method similar to Reinhart [31,
and we can prove the fo1:owing result.
PROPOSITION
3. Let M be a compact multz-almost prodz~ct manifold. There exi.~ts
a symmetric and completely continuous operator which satisfies
p=iGp+Hp=Go"p+Hp
and GI?= H6= 0, where fip is the projection of /3 on the space H = { $ i#= O} and /3 is
a Cmform i n Bz(M).
The Green's operator G for 2 fails to commute with d, da and related operators.
[I]
[21
[3]
[4]
V. K. A. M. Guggeheim and D. C. Spencer, Chain homotopy and the de Rham theory,
Proc. Amer. Math. Soc., 7 (1956) 144-152.
K. Kodaira and D. C. Spencer, Multifoliate structures, Ann. Math., 74 (1961) 52-100.
B. L. Reinhart, Harmonic integrals on almost product manifolds, Trans. Amer. Math.
sot., 88 (1958) 243-276.
T. Saeki, Chain homotopy in q-almost product manifolds (in Japanese), Annual Report,
Lib. Arts, Univ. Iwate, 25 (1965) Pt. 3 , 1-4.