C OMPOSITIO M ATHEMATICA
M. D E W ILDE P. L ECOMTE
Algebraic characterizations of the algebra of functions and of the Lie algebra of vector fields of a manifold
Compositio Mathematica, tome 45, n
o2 (1982), p. 199-205
<http://www.numdam.org/item?id=CM_1982__45_2_199_0>
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199
ALGEBRAIC CHARACTERIZATIONS OF THE ALGEBRA OF FUNCTIONS AND OF THE LIE ALGEBRA OF VECTOR
FIELDS OF A MANIFOLD
M. De Wilde and P. Lecomte
© 1982 Martinus Nijhoff Publishers - The Hague Printed in the Netherlands
1. Introduction
Let M be a
smooth, connected,
Hausdorff and second countable manifold. Denoteby H(M)
the Liealgebra
of smooth vector fields ofM and
by C~(M)
the space of smooth functions on M. Recall that the Lie derivativeJX(X
EH(M))
is defined on~p C~(M) by
If L :
~pC~(M) ~ C~(M)
is ap-linear
map, theadjoint
action ofJX
on
L, ad(:£x)L
is the commutatorJX °
L - L 0Yx.
We intend to show that theonly symmetric p-linear
mapsL:~pC~(M) ~ C~(M)
for whichad(Yx)L
= 0 for each X EH(M)
are themultiples
of theprod-
uct of p factors:
f @ ’ ’ ’ ~fp~f1··· fp.
Itprovides
thus a charac-terization of the
product by
means of its derivations.A similar
question
isinvestigated
for the Lie bracket of vectorfields,
which is shown to be determined up to a constantby
the factthat its derivations are the Lie derivatives. This
question
has beensolved
by
Van Strien in[4]
understronger assumptions
which will be discussed in§3.
2. Characterization of the
algebraic
structure ofC~(M)
LEMMA 2.1: There exist a
finite partition of
theunity 03BBt(t ~ r) of M,
vector
fields Xt(t - r)
EH(M)
andfunctions 03BCt(t ~ r)
EC~(M)
suchthat
Àt
=ÀtXt ’
1£tfor
each t ~ r.0010-437X/82020199-07$00.2010
200
PROOF: It is well known from dimension
theory [2],
p. 20 that M admits an open coverUt(t ~ r)
such that eachUt
is thedisjoint
unionof domains of charts
Uit(i E N),
as well as alocally
finitepartition
ofthe
unity
03C1it(t S r,
i EN)
such that each pi, hascompact support
inUit.
We set
Vit = f x
EUit : 03C1it(x)
>01
and choose apartition
of theunity 03C1’it(t
~ r; i EN)
such that supppit
iscompact
inVit
for alli,
t.For each t ~ r and i E
N,
we may find 03B1it,f3it
EC~(M)
withcompact supports
inUit
such that ait1 Vit =
1 and03B2it
= 1 in someneighborhood
of supp ait. If
(x1it, ..., xnit)
are local coordinates inUi,,
have the
required properties.
Hence the lemma.Denote
by Ap
thesubspace
ofOP C~(M) spanned by
the tensorswhich are
antisymmetric
in at least twoarguments.
LEMMA 2.2: For all
fi ~ C~(M) (i ~ p),
there existN ~ N,
vectorfields Xi
E7Ie(M) (i
~N)
andfik
EC~(M) (i
~N,
k -p),
such thatMoreover the vector
fields Xi(i ~ N)
and thefunctions fik (i
~N,
k ~p)
can be choosenindependently of fi.
PROOF: We have
Since
03BBt = Arxr
iit,Let us consider for instance the term i = 2.
Setting Xt . f2
= g2, it readsThe last term is
antisymmetric
in the two firstarguments,
hence may beneglected.
The other one can bewritten, setting 192Xt
=X t,
In the terms we had to consider in
(**),
one of thearguments
was another onehad 03BBt
as a factor. In the terms which we are left toconsider,
we now have twoarguments equal to ILt
and one more divisibleby Àt.
This shows the outline of theproof,
which will now beachieved
by proving
thefollowing, by
induction on k : if one of thefi’s
is divisible
by 03BBtf (f
EC~(M))
andp - k
others areequal
to it,, thenfor suitable
Xi
EH(M) and fij
EC~(M), independent
off.
For k =
1, assuming
for instance thatfi = 03BBtfg
andfi
=03BCt(i
>1),
has the
required
form.In
general,
if theproperty
holds truefor k, assuming
forsimplicity
that
fk+1 = 03BBtfgk+1 and fi
= 03BCt for i > k +1,
andsetting fi 0... ~fk
=T,
hence the
conclusion, by
induction.Applying
this to(*) for f = fi and k
= pyields
the lemma.202
PROPOSITION 2.3: Let P :
C~(M) ···
xC~(M) ~ C~(M)
be a p- linearsymmetric
map.If ad(JX)
P = 0for
each X EH(M),
thenfor
some k E R.PROOF: Since P is
symmetric,
P vanishes onAp. Using
lemma2.2.,
where the
Xi’s
andfij’s
do notdepend
onf1.
ThereforeIt is clear that
P(1, f2, ..., fp ) = P’(f2, ..., fp)
satisfiesagain ad(5Ex)P’ == 0, thus, by induction,
The condition
and(JX)P
= 0clearly
shows thatP(1, ..., 1)
is con- stant, hence the result.RESULT 2.4: The
symmetry assumption
on P is necessary.Indeed,
consider acompact, connected,
oriented manifold M of dimensionp - 1
and take the map P :It is clear that p is such that
ad(JX)P
= 0 for all X EH(M).
However,
P ~0,
thus P is not even local.3. Characterization of the
algebraic
structure ofH(M)
PROPOSITION 3.1: Let B :
H(M)
xH(M) ~ H(M)
be a bilinear map such thatThen there exists k E R such that
A similar result is
proved by
Van Strien in[4].
Theassumption (*)
is
replaced by
the"naturality"
ofB,
which means thefollowing:
B issupposed
to be defined on every manifold Mand,
for every smooth openimbedding
cp, thediagram
commutes
(see [3]).
It is clear that thenaturality implies
thelocality
of
B and, using
thepseudo-group
ofX,
it alsoimplies (*).
LEMMA 3.2:
If
X EH(M)
vanishes in an open subset Uof M, for
each x e
U,
there exists aneighborhood w of
x and vectorfields Xi, X’i(i
SN)
such that X== 2: [Xi, X’i]
andXi, X’i
vanish in w.i:5N
PROOF: Fix 03C9
relatively compact
in U. Choose pit as in lemma 2.1 and ~it EC~(M)
withcompact support
inUitB03C9
andequal
to 1 in aneighborhood
of supp03C1itX. Then,
if(x 1, ..., xn)
are local coordinates inUit,
where
X k being
the k’scomponent
of thecorresponding
local form of X. Inthe
integral,
the function is extendedby
0 outside theimage
ofsupp
p,tX.
The vector fields
204
are
vanishing
in w and moreover,hence the lemma.
PROOF QF PROPOSITION 3.1: We first prove that B is a local map.
By
Peetre’s theorem(for
the multilinearversion,
see[1]),
it will then be a differential bilinearoperator.
Suppose
that X EYe(M)
vanishes in an open subset U. For each xo EU,
there exist w such that Xo E w C U andXi, Xi vanishing
in msuch that
Choose Z E
H(M)
withsupport
in W and in a domain of chart V ofM,
such thatZx0
= 0 andDx0Z
= 1 for a coordinatesystem
of V. Thenin w and
because
JZZ’ =
0 whenever Z’ = 0 in (ù. Thus X = 0 on Uimplies
thatB (X, Y)
= 0 on U and B is local.The end of the
proof
consists incomputations
on the differentialoperator
B. Thearguments
of Van Strien wouldeasily adapt
to thepresent
situation. Aslightly
differentapproach
isgiven
here for thesake of
completeness.
Let us
decompose
B in itssymmetric
andantisymmetric parts,
which bothverify
theassumption
of prop. 3.1.Fix a coordinate
system (U, x l, ..., xn)
of M andcompute
B in these coordinates.Taking
for X the fieldsDxi and I X’Di
showsthat,
for some A
independent
of(x’, ..., x"),
(everything
is now written in localcoordinates) according
to B issymmetric
orantisymmetric.
The
assumption
on B reads thenIf we choose
X,
Y EH(M)
such thatDxiDxjX
= 0 at x andYx
=0,
it follows thatand thus that the bilinear form A : Rn
gl(n, R) -
Rn verifiesIn other
words, Q ~ A(., Q)
EL(gl(n, R)) belongs
to the centralizer of theadjoint
action ofgl(n, R).
It is theneasily
seen thatfor some
k,
1 E R.Substituting
this in(*),
it follows that k = 1 = 0 if B issymmetric
and that k = 1 if B isantisymmetric.
Thushence the result.
REFERENCES
[1] M. CAHEN, M. DE WILDE and S. GUTT: Local cohomology of the algebra of C~
functions on a connected manifold. Lett. in Math. Physics 4 (1980), 157-167.
[2] W. GREUB, S. HALPERIN and R. VANSTONE: Connections, curvature and cohomology, Vol. I, Academic Press, New York and London (1973).
[3] R. PALAIS: Natural operations on differential forms. Trans. A.M.S. 92 (1959),
125-141.
[4] S. VAN STRIEN: Unicity of the Lie Product. Comp. Math. 40 (1980), 79-85.
(Oblatum 5-1-1981 & 27-IV-1981) University of Liège Institute of Mathematics 15, Avenue des Tilleuls B-4000 Liège, Belgium.