Akira YOSHIOKA and Osamu KOBAYASHI. Science University of Tokyo, Keio University. Tokyo Metropolitan University and Keio University. It is well-known that ...
TOKYO J. MATH. VOL. 6, No. 2, 1983
On Regular Fr\’echet-Lie Groups VI Infinite Dimensional Lie Groups Which Appear in General Relativity
Hideki OMORI, Yoshiaki MAEDA Akira YOSHIOKA and Osamu KOBAYASHI Science University of Tokyo, Keio University Tokyo Metropolitan University and Keio University
It is well-known that Newtonian mechanics is an approximation for general relativity. However, mathematically, both mechanics can stand independently without any subordination by each other. That means, without experimental fact one cannot prove mathematically that general relativistic mechanics is more adequate than Newtonian mechanics. Nevertheless, beside physics, we insist in this paper that under the group theoretical point of view, general relativistic mechanics is a more closed system than Newtonian mechanics. Introduction
Hamiltonian mechanics is a beautifully organized mathematical expression of classical mechanics (cf. [1]) and is expressed as the triplet $(M, \Omega, H)$ , where (i) $(M, \Omega)$ is a symplectic manifold, called a phase pace. $M$ is a smooth symplectic structure even dimensional smooth manifold, and on it. (ii) $H$ is a smooth function on $M$, called a Hamiltonian. Mechanical motions governed by the above Hamiltonian $H$ are given by the , where is defined integral curve of the Hamiltonian vector field by $s$
$\Omega$
$X_{H}$
$\Omega-X_{H}=dH$
Let namely
$\varphi_{t}(x)$
$X_{H}$
.
with an initial point be the integral curve of is the unique solution of the following equation:
$x_{t}=\varphi(x)$
Received December 15, 1981
$X_{H}$
,
$x\in M$
218
H. OMORI, Y. MAEDA, A. YOSHIOKA AND
(1)
$\frac{d}{dt}x_{t}=X_{H}(x_{t})$
,
$x_{0}=x$
$0$
.
KOBAYASHI
.
an idealized, dynamically closed system, the mechanical state must exist for all $t\in R$ and $x\in M$. Thus, it is natural to assume that is a complete vector field on $M$, hence is a one parameter Now, in
$\varphi_{\ell}(x)$
$X_{H}$
$\varphi_{t}$
transformation group on $M$. It is widely known that the Hamiltonian equation (1) is equivalent to the following Poisson’s equation: Let be a smooth function on $M$, called an observable, and set $f(x)=f(\varphi_{t}(x))$ . Then, expresses the time evolution of the observed quantities, which satisfies $f$
$f$
(2)
$\frac{d}{dt}f_{t}=-\{H, f_{t}\}$
,
$f_{0}=f$
,
is the Poisson bracket, defined by $\{f, g\}=\Omega(X_{f}, X_{g})$ . Obviously, (1) implies (2), and the converse is also true if every local coordinate function is an observable, or equivalently if sufficiently many functions
where
$\{f, g\}$
are observables.
Now, remark that observables are, mathematically, non-defined concept, but it seems to be natural to assume that if $f$ and are observables, then all linear combinations $\mu\in R$ are observables. We $g$
$\lambda f+\mu g,$
denote by
$x,$
the linear space of all observables in a mechanical system . Obviously, all possible Hamiltonians should be observables. Moreover, every $f\in p$ will be able to be a Hamiltonian. In such a situation, it is plausible to assume that every Hamiltonian vector field is complete whenever $f\in P$, for $f\in p$ can be a Hamiltonian. In quantum mechanics, every observable $A$ is assumed to be a self-adjoint operator, and hence by Stone’s theorem it generates a unitary one parameter subgroup . This corresponds to the completeness of , $p$
$(M, \Omega)$
$X_{f}$
$e^{itA}$
$X_{f}$
.
$f\in P$
It is also natural to assume that every time-evolution of an $f$ observable is an observable, because otherwise one can not recognize $f_{t}$
such unstable functions as observables. Since for every , it is also natural to assume that $(d/dt)f_{t}\in pp$. Since every cf “ can be a Hamiltonian, Poisson’s equation (2) shows that $p$ must be closed under the Poisson bracket, that is, $p$ is a Lie algebra. Let be the set of all Hamiltonian vector fields $f\in P$. , is then a Lie algebra under the usual Lie bracket, because of the identity $d\{f, g\}=-\Omega-[X_{f}, X_{g}]$ . Let exp be the one parameter group generated by . Since every time-evolution is an observable, we have $(\exp tX_{f})^{*p}=d$ for every $f\in p$, and hence $(1/\delta)(f_{t+\delta}-f_{t})\in\rho$
$\delta>0$
$ f\in$
$\mathfrak{U}_{p}$
$X_{f},$
$tX_{f}$
$X_{f}\in \mathfrak{U}_{e},$
$\mathfrak{U}$
219
FR\’ECHET-LIE GROUPS
Ad
(3)
$(\exp tX_{f})\mathfrak{U}_{\mathcal{E}^{7}}=\mathfrak{U}_{6^{\mathcal{P}}}$
,
$\forall f\in P$
,
. ) is defined by (Ad where Ad By the above reasoning, we assume that cl must have the following $(\varphi)u$
$(\varphi)u$
$(x)=d\varphi u(\varphi^{-1}x)$
properties: (A1) (A2) (A3)
is a Lie algebra under the Poisson bracket. for $f\in P$ is complete. Each Hamiltonian vector field $ f\in\rho$ $(\exp tX_{f})^{*}p=\rho$ for every and for all $t\in R$ . df
$X_{f}$
Note at first that (A1) and (A3) aren ot necessarily independent. If topology), is closed under a suitable topology (for instance, the $g\in P$. However, then $ d/dt|_{t=0}(\exp tX_{f})^{*}g\in$ cf for every $C^{\infty}$
$\rho$
$f,$
$\frac{d}{dt}|_{t=0}(\exp tX_{f})^{*}g=X_{f}g=-\Omega(X_{f}, X_{g})=\{g, f\}$
,
hence (A3) yields (A1). Although the converse is not necessarily true (cf. [2] \S 0 and \S 2) it holds in many natural situations. If $M$ is a compact symplectic manifold without boundary, then the space of all smooth functions on $M$ satisfies $(A1)\sim(A3)$ , and is the Lie algebra of the group of all canonical transformations on $M$. In [4] p. 103, this group is known to be a strong ILB-Lie group, and hence it is a regular Fr\’echet-Lie group (cf. \S 6 in Th. 6.1 [7]). However, if $M$ is non-compact, the situation is quite different. For instance, let us consider Newtonian mechanics on a configuration space $N$. such The cotangent bundle $T^{*}N$ has a natural symplectic structure , where is a local coordinate system that $\Omega=-d\eta,$ a linear coordinate system corresponding to the on $N$ and we denote a smooth riemannian metric on basis $(dx^{1}, \cdots, dx^{n})$ . By $N$ and by its inverse matrix. Now, Newtonian mechanics can be understood as a Hamiltonian system $\{M, \Omega, H\}$ with $M=T^{*}N$, Hamiltonian $H(x, p)=(1/2)g^{ij}p_{i}p_{j}+V$, where $V$ is a smooth function on $N$, called a potential function, which is naturally regarded as a function on $M=T^{*}N$ . In Newtonian mechanics, potential through the projection functions can be chosen almost arbitrarily, while, the riemannian metric is understood to be fixed. However, there exists the following quite curious fact: $C^{\infty}(M)$
$\mathfrak{U}_{p}(\rho=C^{\infty}(M))$
$\Omega$
$(x^{1}, \cdots, x^{n})$
$\eta=p_{i}dx^{l}$
$(p_{1}, \cdots, p_{n})$
$(g_{ij})$
$(g^{ij})$
$\pi:T^{*}N\rightarrow N$
$(g_{ij})$
be a set of smooth functions on $T^{*}N$ satisfying THEOREM A. Let (A1) and (A2), and containing a smooth function $(1/2)g^{ij}p_{i}p_{j}+V$ for some at a point $x\in N$, then . If satisfies $\rho$
$V\in C^{\infty}(N)$
$ f\in C^{\infty}(N)\cap\rho$
$j_{x}^{2}f=0$
220
H. OMORI, Y. MAEDA, A. YOSHIOKA AND
$j_{x}^{k}f=0$
for all
$k\geqq 2$
, where
$j^{k}f$
is the k-th jet
of
$0$
.
$f$
KOBAYASHI
at
PROOF.
$x$
.
Assume there is $f\in C^{\infty}(N)\cap d$ such that but for some . One may suppose that without loss of generality. Let be a normal coordinate system at with respect to the given riemannian metric . We denote by ad $(f)g$ the Poisson $\{f, g\}$ bracket . Denoting $H=(1/2)g^{ij}p_{i}p_{j}+V$, we set $j_{x}^{2}f=0$
$k\geqq 3$
$j_{x}^{k}f\neq 0$
$j_{x}^{k-1}f=0$
$(x^{1}, \cdots, x^{n})$
$x$
$(g_{ij})$
$W=(-1)^{k-1}$
Since
ad
.
$(H)^{k-1}f$
$\{g, h\}=(\partial h/\partial p_{i})(\partial g/\partial x^{i})-(\partial h/\partial x^{i})(\partial g/\partial p_{i})$
, we obtain
$W=g^{i_{1}j_{1}}\cdots g^{i_{k-1}j_{k-1}}p_{i_{1}}p_{i_{2}}\cdots p_{i_{k-1}}\frac{\partial^{k-1}f}{\partial_{x^{1}}^{\dot{f}}\cdots\partial_{x}^{\dot{3}k-1}}+\Phi_{k-2}(f)$
,
where is a linear combination of the terms containing the derivatives up to order $k-2$ . Since of is a normal coordinate system at , we have $\Phi$
$f$
$(x^{1}, \cdots, x^{n})$
$x$
$dW|T_{x}^{*}N=\Gamma^{i_{1}\ldots.,i_{k}}p_{i_{1}}\cdots p_{i_{k-1}}dx^{i_{k}}$
where and fore the Hamiltonian vector field $\Gamma^{i_{1}\cdots i_{k}}=(\partial^{k}f/\partial_{x^{1}}^{i}\cdots\partial_{x}^{i_{k}})(0)$
$|T_{x}^{*}N$
$X_{W}$
means the restriction. There-
satisfies
$X_{W}|T_{x}^{*}N=\Gamma^{i_{1}\cdots i_{k}}p_{t_{1}}\cdots p_{t_{k-1}}\frac{\partial}{\partial p_{\iota_{k}}}$
is a tangent vector field on $T^{*}N$. point at which the homogenous polynomial
hence
$X_{W}|T_{x}^{*}N$
,
Let
be a attains the is a stationary
$(\overline{p}_{1}, \cdots,\overline{p}_{n})$
$\Gamma^{i_{1}\cdots i_{k}}p_{i_{1}}\cdots p_{i_{k}}$
maximum under the restriction point, we get
$\sum p_{i}^{2}=1$
.
Since
$\Gamma^{i_{1}\cdots i_{k}}\overline{p}_{i_{1}}\cdots\overline{p}_{i_{k-1}}dp_{i_{k}}=\frac{2}{k}\lambda\overline{p}_{i_{k}}dp_{i_{k}}$
By
$(\overline{p}_{1}, \cdots,\overline{p}_{n})$
,
$\lambda\in R$
.
a suitable linear change of coordinates one may assume that , hence we have
$(\overline{p}_{1}, \cdots,\overline{p}_{n})=(1, \cdots, 0)$
$\frac{\partial^{k}f}{\partial(x^{1})^{k}}(0)\neq 0$
Therefore,
and
$\frac{\partial^{k}f}{\partial(x^{1})^{k-1}\partial x^{i}}(0)=0$
.
,
$i\geqq 2$
.
defines a tangent vector field on a one dimensional linear subspace $(p, 0, \cdots, 0)$ , which is not complete because . Thus, is not complete contradicting the assumption (A2), because $W\in p$. $X_{W}|(p, 0, \cdots, 0)=cp^{k-1}(\partial/\partial p),$ $c\neq 0$
$k\geqq 3$
Thus
$X_{W}$
$X_{W}$
$\square $
221
FR\’ECHET-LIE GROUPS
The above result shows that there are few potential functions in Indeed, if $N$ is a real analytic riemannian manifold, then
dim $C^{\omega}(N)\cap P\leqq\frac{1}{2}(n+2)(n+1)$ ,
$n=\dim N$
.
$PP$
,
where is the space of all real analytic functions on $N$. The above curious fact is mainly caused by the assumption (A2). However, if we give up assuming (A2) then we have to struggle again to obtain reasonable criterions of observables and Hamiltonians without any group-theoretical stand point. To resolve the above difficulty, we have to consider relativistic Hamiltonians on $(T^{*}N, \Omega)$ , which will be discussed in the next section. $C^{\omega}(N)$
\S 1. Relativistic Hamiltonians and the
statement of Theorem.
In general relativity, the free motion of a particle is given by a timelike geodesic on an $(n+1)$ -dimensional Lorentzian manifold regarded as the space-time or the universe. However, the universe $N$ ’ is not an arbitrary Lorentzian manifold, but is under certain restrictions imposed by some cosmological reasons, such as the existence of a is a function on such that universal time , where is a timelike vector field. Suppose the vector field grad $t/|gradt|^{2}$ is complete. Then, it is not diffeomorphic to $R\times N$, on which the given hard to see that $N$ ’ is Lorentzian metric is written in the form $N^{\prime}$
$t$
$t$
$N^{\prime}$
$C^{\infty}$
$\partial/\partial t$
$C^{\infty}$
(4)
where
$d\tau^{2}=c(t, x)^{2}dt^{2}-h_{ij}(t, x)dx^{i}dx^{j}$
$c(t, x)$
is a
$C^{\infty}$
positive function and
,
is a
$h_{ij}(t, x)$
$C^{\infty}$
riemannian
metric on . In what follows, we impose the above restriction, and moreover we assume $N$ is compact without boundary. Consider a free motion of a particle. Since it is a timelike geodesic, it must be a solution of the variational equation . Since every timelike curve can be parametrized by , the above variational problem on $R\times N$ can be regarded naturally as that on $N$. Take the Legendre transform and compute the Hamiltonian $H(t, x, p)$ ([1], [3]). Then, we get easily $N$
for every fixed
$t\in R$
$\delta\int d\tau=0$
$t$
(5)
$H(t, x, p)=c(t, x)\sqrt{1+h^{ij}(t,x)p_{i}p_{j}}$
.
The motions of a particle are described by integral curves of
$X_{H}$
in
,
$T^{*}N$
222
H. OMORI, Y. MAEDA, A. YOSHIOKA AND
$0$
.
KOBAYASHI
is a time-dependent vector field on $T^{*}N$. Namely, the general relativity in our situation is nothing but a Hamiltonian mechanics with $H$ in (5) as a Hamiltonian. The above $H$ will be called a relativistic Hamiltonian. Note that in our situation, $c(t, x)$ and $h_{lj}(t, x)$ can be chosen almost arbitrarily just like potential functions in Newtonian mechanics. This is because the gravitational forces are involved in and in general where
$X_{H}$
$h_{ij}$
$c$
relativity.
Let
be an arbitrarily fixed riemannian metric on $N$ and set $S^{*}N$ . We denote by the unit cosphere bundle in $T^{*}N$ with $C^{\infty}$
$g_{ij}$
$r=\sqrt{g^{ig}p_{i}p_{j}}$
respect to
$g^{ij}$
.
LEMMA 1.1. Every relativistic Hamiltonian ing asymptotic expansion: (6)
has the
$H_{t}(x, p)$
follow-
,
$ H_{t}(x, p)\sim a_{1}r+a_{0}+a_{-1}r^{-\iota}+\cdots+a_{-m}r^{-m}+\cdots$
are smooth functions on $S^{*}N$. on $T^{*}N-\{0\}$ . is a positively PROOF. Set homogeneous function of degree , hence naturally identified with a function on $S^{*}N$. Since $G>0$ and
for
$\gamma\gg 0$
, where
$a_{j}’ s$
$G$
$G_{t}(x, p)=h^{i\dot{g}}p_{i}p_{\dot{f}}/g^{ij}p_{i}p_{j}$
$0$
$C^{\infty}$
$H_{t}(x, p)=r\cdot c(x, t)\sqrt{G}\sqrt{1+1/Gr^{2}}$
$H_{t}$
has an asymptotic expansion of type (6).
Now, let be the space consisting of which have asymptotic expansion of type . of all Hamiltonian vector fields every for and all relativistic Hamiltonians paper is to show the following: $\Sigma^{1}$
$X_{f}$
THEOREM B.
of
,
$\Sigma^{1}$
satisfies
$f\in\Sigma^{1}$
$(A1)\sim(A3)$
.
$\square $
all smooth functions on $T^{*}N$ (6) and let be the set of Obviously, contains fixed . The purpose of this $\mathfrak{U}_{\Sigma^{1}}$
$\Sigma^{1}$
$C^{\infty}(N)$
$t$
$Mor\cdot eove?\cdot,$
$\mathfrak{U}_{\Sigma^{1}}$
is a Lie algebra
a regular Fr\’echet-Lie group.
Remark that the above result shows that general relativity is more closed system than Newtonian mechanics from the Lie group theoretical point of view.
\S 2. A compactification of
, we have to compactify the cotangent on $N$ and we denote by riemannian metric
For the proof of Theorem
bundle
.
$T^{*}N$
Fix a
$C^{\infty}$
.
$T^{*}N$
$B$
$g$
223
FR\’ECHET-LIE GROUPS $M=D^{*}N$ (resp.
$\overline{M}=\overline{D}^{*}N$
.
disk bundle) in , or
$T^{*}N$
unit open disk bundle (resp. the unit closed Points in $T^{*}N$ will be denoted by $(x;\xi),$ $x\in N$,
) the
$\xi\in T_{x}^{*}N$
$(x;\xi)=(x;rz)$
,
$ r\in[0, \infty$
),
$z\in S_{x}^{*}N$
,
is the fibre at of the unit cosphere bundle . Let $c:S^{*}N\rightarrow T^{*}N$ $S^{*}N$ be the inclusion mapping. Then, carries canonically a smooth contact structure
where
$S_{x}^{*}N$
$S^{*}N=\partial\overline{M}$
$x$
(7)
$\omega=c^{*}\eta$
Note that
,
$\eta=\sum\xi_{i}dx^{i}$
$R_{+}=(0, \infty)$ is diffeomorphic to and the on $T^{*}N-\{0\}$ can be written In the form
$T^{*}N-\{0\}$
symplectic form (8)
$\Omega$
$R_{+}\times S^{*}N,$
$\Omega=-d(r\omega)=-(dr\wedge\omega+\gamma d\omega)$
Let
$\tau:M\rightarrow T^{*}N$
be the
(9)
$C^{\infty}$
,
$r=(g^{ij}\xi_{i}\xi_{j})^{1/2}$
diffeomorphism defined by
$\tau(x;\theta z)=(x;(\tan\frac{\pi\theta}{2})z)$
,
(cf. [6]). By using this diffeomorphism, everything on $T^{*}N$ can be transferred to that on $M,$ is a symplectic form on $M$, and we see
where
$\theta\in[0,1$
),
$z\in S_{x}^{*}N$
$e.g.,\tilde{\Omega}=\tau^{*}\Omega$
(10)
$C^{\infty}$
$\tilde{\Omega}=-d((\tan\frac{\pi\theta}{2})\omega)$
$=-\frac{\pi}{2}\frac{1}{(\cos\frac{\pi\theta}{2})^{2}}d\theta\wedge\omega-(\tan\frac{\pi\theta}{2})d\omega$
on
$M-\{0\}$
.
be the space of all functions on is said to be on if can be extended to a on a neighborhood of . Denote by a non-decreasing ) such on that Let function
$C^{\infty}(\overline{M})$
$f$
$\overline{M}=\overline{D}^{*}N$
$C^{\infty}$
$\overline{M}$
$C^{\infty}$
$f$
$\overline{M}$
$\phi(\theta)$
, where a
$C^{\infty}$
$C^{\infty}$
function function
$[0, \infty$
(11)
$\phi(\theta)=\left\{\begin{array}{ll}0, & \theta e[0,1/3)\\1, & \theta e[2/3, \infty).\end{array}\right.$
Every $f\in C^{\infty}(S^{*}N)$ can be naturally regarded as a , function on . and hence of has the following asymptotic Recall that each element expansion; $C^{\infty}$
$\phi f\in C^{\infty}(\overline{M})$
$h$
$\Sigma^{1}$
$\overline{M}-\{0\}$
224
H. OMORI, Y. MAEDA, A. YOSHIOKA AND
(12)
.
KOBAYASHI
$ h(x;\xi)\sim a_{1}r+a_{0}+a_{-1}r^{-1}+\cdots+a_{-m}r^{-n}+\cdots$
for
$\gamma=(g^{ij}\xi_{i}\xi_{j})^{1/2}$
LEMMA 2.1.
$\tau^{*}\Sigma^{1}=$
(
$\phi(\theta)$
PROOF. For every function on $S^{*}N$.
tan
$h\in\Sigma^{1}$
$r\gg O$
$(\pi\theta/2)$
)
,
.
$C^{\infty}(S^{*}N)\oplus C^{\infty}(\overline{M})$
, we see easily that
Hence, settIng has the asymptotic expansion: $C^{\infty}$
$0$
$a_{1}$
$\lim_{r\rightarrow\infty}(1/r)h$
$\tau^{*}k=\tau^{*}h-$
$ k\sim a_{0}+a_{-1}\gamma^{-1}+\cdots+a_{-m}r^{-n}+\cdots$
.
(
$\phi(\theta)$
$(r\gg 0)$
tan
exists as
$(\pi\theta/2)$
)
$a_{1},$
$k$
.
, and by the converse of the Taylor’s theorem at , we get that . Conversely, if , then Taylor’s theorem shows that has an above asymptotic expansion. This completes the proof.
Set
$\gamma=\tan\pi\theta/2$
$\theta=1$
$\tau^{*}k\in C^{\infty}(\overline{M})$
$\tau^{*}k\in C^{\infty}(\overline{M})$
$k$
$\square $
From now on we treat the unit disk bundle symplectic structure
$\tilde{\Omega}$
$\overline{M}=\overline{D}^{*}N$
.
and the
be the group of all diffeomorphisms of onto itself. Then, is a strong ILH-Lie group with the Lie algebra , is the totality of where vector field on which is tangent . (Cf. [41 \S II, [5] p. 279). Let to the boundary be the $M$ identity component of the group { on }. Then . We set is a closed subgroup of Let
$\mathcal{D}(\overline{M})$
$C^{\infty}$
$\overline{M}$
$\mathcal{D}(\overline{M})$
$\Gamma(T\overline{M})$
$\Gamma(T\overline{M})$
$C^{\infty}$
$\overline{M}$
$u$
$\partial\overline{M}=S^{*}N$
$\mathcal{D}_{\tilde{\rho}}$
$\varphi\in \mathcal{D}(\overline{M});\varphi^{*}\tilde{\Omega}=\tilde{\Omega}$
$\mathcal{D}_{\rho}^{\sim}$
$\mathcal{D}(\overline{M})$
(13)
$\Gamma_{\tilde{\rho}}=$
LEMMA 2.2. (i) (ii) (iii)
$\Gamma_{\rho}^{\sim}$
{
$u\in\Gamma(T\overline{M})$
; exp
for all
$tu\in \mathcal{D}_{\rho}^{\wedge}$
$t\in R$
}.
the following properties:
$\Gamma_{\rho}^{\sim}has$
a closed subalgebra of . Every is a complete vector field. $\Gamma(T\overline{M})$
$\Gamma_{\rho}^{\sim}is$
$u\in\Gamma_{\tilde{\rho}}$
Ad
$(\exp tu)\Gamma_{\tilde{\rho}}=\Gamma_{\tilde{\rho}}$
for
any
$u\in\Gamma_{\tilde{\rho}}$
.
PROOF. By 1.4.1 Theorem in [4] or Lemma 1.4 in [8], we see that is a closed Lie algebra of . Note that $\Gamma(T\overline{M})$
$\Gamma_{\rho}^{\wedge}=$
Then,
we get easily
{
$u\in\Gamma(T\overline{M});- \mathscr{G}_{u}\tilde{\Omega}=0$
on
$M$
}.
(iii) (cf. [2]).
$\square $
Our first purpose is to get the following, which will be proved In \S \S 4, 5 and 6. THEOREM 2.3. is a $r\cdot egular$ Fr\’echet-Lie group. finition of regular Fr\’echet-Lie group, see [7].) $\mathcal{D}_{\rho}^{\sim}$
(For
the de-
225
FR\’ECHET-LIE GROUPS
be the set of all Hamiltonian vector fields $\Omega-X_{f}=df,$ . We translate this Lie algebra to proof of Theorem , we shall show the following in \S 3: Let
$X_{f}$
$\mathfrak{U}_{\Sigma^{1}}$
of
$d\tau^{-1}\mathfrak{U}_{\Sigma^{1}}$
$f\in\Sigma^{1}$
, For the
$f\in\Sigma^{1},$
.
$i.e.$
$B$
THEOREM 2.4. $\dim H^{1}(N;R)+1$ , where
is a closed ideal of of codimension $H^{1}(N, R)$ is the first cohomology group of $N$ $\Gamma_{\rho}^{\sim}$
$d\tau^{-1}\mathfrak{U}_{\Sigma^{1}}$
over
$R$
.
Before proving the above theorems, we shall show at first how to by Theorems 2.3 and 2.4. obtain Theorem $B$
under Theorems PROOF OF THEOREM B. We shall prove Theorem satisfies $(A1)\sim(A3)$ . Since 2.3 and 2.4. At first, we shall show that is a Lie algebra under and hence is a Lie algebra, so is is a complete vector field, we the Poisson bracket. Since every is also a complete vector field on $T^{*}N$. see that . Then $w(t)$ satisfies Set $w(t)=Ad(\exp tv)w$ for $(d/dt)w(t)=[v, w(t)]$ , and hence by Lemma 4.5 [7], we have . Since $d\tau(Ad(\exp tv)w)=Ad(\exp td\tau v)d\tau w$ , Ad . Now, for all the above equality shows that Ad remark that $\Omega-Ad(\exp tu)X_{f}=d(\exp-tu)^{*}f$. Since Ad satisfies and hence $(\exp tu)^{*}f\in\Sigma^{1}$ . Thus we have $(\exp tu)^{*}f+const$ . $B$
$\Sigma^{1}$
$\Sigma^{1}$
$\mathfrak{U}_{\Sigma^{1}}$
$d\tau^{-1}\mathfrak{U}_{\Sigma 1}$
$u\in\Gamma_{\rho}^{\sim}$
$d\tau u$
$v,$
$w\in d\tau^{-1}\mathfrak{U}_{\Sigma^{1}}$
$(\exp tv)d\tau^{-1}\mathfrak{U}_{\Sigma^{1}}=d\tau^{-1}\mathfrak{U}_{\Sigma^{1}}$
$u\in \mathfrak{U}_{Z^{1}}$
$(\exp tu)\mathfrak{U}_{\Sigma^{1}}=\mathfrak{U}_{\Sigma^{1}}$
$(\exp tu)X_{f}\in \mathfrak{U}_{\Sigma^{1}}$
$\Sigma^{1}$
$\in\Sigma^{1}$
$(A1)\sim(A3)$
.
is a finite is a regular Fr\’echet-Lie group and , Theorem 4.2 in [8] shows that there is codimensional subalgebra of having as its Lie algebra. of a locally flat FL-subgroup group. Fr\’echet-Lie is a regular By Corollary 2.4 in [8], , we put diffeomorphism using the Now, As
$d\tau^{-1}\mathfrak{U}_{\Sigma^{1}}$
$\mathcal{D}_{\Omega}^{\sim}$
$\Gamma_{9}^{\wedge}$
$\tilde{G}_{\Sigma 1}$
$d\tau^{-1}\mathfrak{U}_{\Sigma^{1}}$
$\mathcal{D}_{\Omega}^{\sim}$
$\tilde{G}_{\Sigma^{1}}$
$\tau:M\rightarrow T^{*}N$
$C^{\infty}$
,
$\mathcal{D}_{\Omega}(\overline{T}^{*}N)_{0}=\tau \mathcal{D}_{\rho}^{\sim}\tau^{-1}$
$\Gamma_{\Omega}(T\overline{T}^{*}N)=d\tau\Gamma_{\rho}^{\sim}$
,
$G_{\Sigma^{1}}=\tau\tilde{G}_{\Sigma^{1}}\tau^{-1}$
Obviously subgroup.
.
is a regular Fr\’echet-Lie group and . is given by The Lie algebra of
$\mathcal{D}_{\Omega}(\overline{T}^{*}N)_{0}$
$G_{\Sigma^{1}}$
$\mathfrak{U}_{\Sigma^{1}}$
$G_{\Sigma^{1}}$
is an FL$\square $
\S 3. Proof of Theorem 2.4. First of all, we explain the structures on the closed unit disk bundle which will be useful through this paper. Remark that $M$ has two geometrical structures, that is, it is a symplectic manifold with symplectic structure and also is a manifold with boundary
$\overline{M}=\overline{D}^{*}N$
$\tilde{\Omega}$
$S^{*}N=\partial\overline{M}$
226
H. OMORI, Y. MAEDA, A. YOSHIOKA AND
$0$
.
KOBAYASHI
(cf. (7)). on which there is a canonical contact structure following, In the we will omit or in the notations for Lie subgroups or Lie subalgebras of , respectively, e.g., or . is a subgroup of , define the For the contact manifold vector field on $S^{*}N$ defined by $\omega-x_{w}=1$ . , (14) $\omega$
$T\overline{M}$
$\overline{M}$
$\Gamma(T\overline{M})$
$\mathcal{D}(\overline{M})$
$\mathcal{D}(\overline{M})$
$\mathcal{D}_{\rho}^{\sim}$
$(S^{*}N, \omega)$
$C^{\infty}$
$\chi_{\omega}$
$d_{\omega}-x_{w}=0$
is called the characteristic vector by the distribution $\chi_{\omega}$
field of the contact form
$\omega$
.
Denote
$E_{\omega}$
$E_{\omega}=\{X\in TS^{*}N;\omega-X=0\}$
the anihilator of and by (resp. the tangent subbundle of the codimension one. It is obvious that $E_{\omega}^{*}(\subset T^{*}S^{*}N)$
$TS^{*}N=R\chi_{w}\oplus E_{\omega}$
,
.
,
) (resp. cotangent) bundle
$\chi_{w}$
$E_{\omega}$
$E_{\omega}^{*}$
$T^{*}S^{*}N=R\omega\oplus E_{\omega}^{*}$
is a smooth of $S^{*}N$ of
.
, defines an isomorphism of onto The linear mapping . The following fact is . So, its inverse will be denoted by well-known in Hamiltonian mechanics (cf. [1] and [4] 8.3.1. Lemma): $E_{\omega}$
$d\omega|_{E_{\omega}}:X\mapsto d\omega-X$
$d\omega^{-1}$
$E_{\omega}^{*}$
on $S^{*}N$ is a contact vector LEMMA 3.1. A vector field and only if there is a smooth function $f$ on $S^{*}N$ such that . written in the form $u=f\chi_{\omega}-\{f\}$ , where
field if
$u$
$u$
can be
$\{f\}=d\omega^{-1}(df-(\chi_{\omega}f)\omega)$
and stated is diffeomorphic to $(0,1$ ] Since . We use the same notations above can be naturally extended to again obvious that for the extended ones. It is $\times S^{*}N,$
$\overline{M}-\{0\}$
$\chi_{\omega},$
$E_{\omega}$
$E_{\omega}^{*}$
$\overline{M}-\{0\}$
(15)
$\left\{\begin{array}{l}T\overline{M}-\{0\}=R\frac{\partial}{\partial\theta}\oplus R\chi_{\omega}\oplus E_{\omega}\\T^{*}\overline{M}-\{O\}=Rd\theta\oplus R\omega\oplus E_{\omega}^{*}\end{array}\right.$
For every vector field
$u$
on
, we set
$\overline{M}=\overline{D}^{*}N$
(16)
.
$\sim b(u)=\tilde{\Omega}-u$
Also, for
a smooth function
(17)
$f$
on
, define
$\overline{M}$
$\# f=b^{-1}df\sim\sim$
$\# f\in\sim\Gamma_{\rho}^{\sim}$
by
.
be the set of all Hamiltonian vector fields Let diffeomorphism in (9), it is not hard to see that $\mathfrak{U}_{\Sigma^{1}}$
$\tau$
$X_{f},$ $f\in\Sigma^{1}$
.
By the
$ d\tau^{-\iota}\mathfrak{U}_{Z^{1}}=b^{-1}d(\tau^{*}\Sigma^{1})\sim$
.
227
FR\’ECHET-LIE GROUPS
In accordance with the decomposition (15), we shall compute the com, there exists uniquely ponents of . Given such that and $a_{1}\in C^{\infty}(S^{*}N)$
$f\in\tau^{*}\Sigma^{1}$
$u\in d\tau^{-1}\mathfrak{U}_{\Sigma^{1}}$
$g\in C^{\infty}(\overline{M})$
(18)
because of Lemma 2.1. written in the form (19)
if
,
$f=(\tan\frac{\pi\theta}{2})\phi(\theta)a_{1}+g$
Since each vector field
$u=\lambda(x;\theta z)\frac{\partial}{\partial\theta}+\mu(x;\theta z)\chi_{\omega}+\hat{u}$
$df=\tilde{\Omega}-u$
(20)
,
on
$u$
$\overline{D}^{*}N-\{0\}$
$\hat{u}e\Gamma(E_{\omega})$
can be
,
, then we get by (10) and (19)
$\left\{\begin{array}{l}x(x;\theta z)=-\frac{1}{\pi}\phi(\theta)(\sin\pi\theta)\chi_{\omega}a_{1}-\frac{2}{\pi}(\cos\frac{\pi\theta}{2})^{2}(\chi_{\omega}g)\\\mu(x;\theta z)=(\phi+\frac{1}{\pi}\pi\theta\frac{\partial\phi}{\partial\theta})a_{1}+\frac{2}{\pi}(\cos\frac{\pi\theta}{2})^{2}\frac{\partial g}{\partial\theta}\\\text{{\it \^{u}}}=-\phi(\theta)\{a_{1}\}-(\cot\frac{\pi\theta}{2})\{g\}\end{array}\right.$
where
$\{g\}=d\omega^{-1}(d’ g-(\chi_{\omega}g)\omega)$
and
$d’ g$
is the derivative of
$g$
regarding
$\theta$
as an arbitrarily fixed parameter. LEMMA 3.2.
$\#(\tau^{*}\Sigma^{1})\sim$
is a closed subspace
of
$\Gamma_{\Omega}^{\sim}$
.
can be expressed . Then PROOF. Let be an element of . Hence, induces , then becomes by (19) and (20). If . Since a smooth vector field on $S^{*}N$. Therefore, . , we see easily that , converges to an Now, suppose that a sequence topology on . On 1/3, $1]\times S^{*}N$, the above in the element , and . Thus, convergence means those of each component of using (20) we have that there are smooth functions $C^{\infty}([1/3,1]\times S^{*}N)$ such that $ u=\# f\sim$
$\tau^{*}\Sigma^{1}$
$f$
$\theta=1$
$\# f\sim$
$\# f\sim$
$a_{1}\chi_{\omega}-\{a_{1}\}$
$\mathscr{L}_{u}\tilde{\Omega}=$
$\# f\sim\in\Gamma(T\overline{M})$
$\# f\sim\in\Gamma_{\Omega}^{\sim}$
$ d(\tilde{\Omega}\rightarrow u)=db(u)\sim$
$\{\# f_{m}\},$
$C^{\infty}$
$u\in\Gamma_{\rho}^{\sim}$
$f_{m}\in\tau^{*}\Sigma^{1}$
$\overline{M}$
$[$
$\partial/\partial\theta,$
$E_{\omega}$
$\chi_{\omega}$
$a_{1}^{\prime}\in C^{\infty}(S^{*}N),$
$ g^{\prime}\in$
$\lim_{m\rightarrow\infty}\#\sim f_{m}=\sim\#((\tan\frac{\pi\theta}{2})\phi(\theta)a_{1}^{\prime})+\# g^{\prime}\sim$
.
Choose an arbitrary extension of by it by the same notation . Define
on
$[$
1/3,
$1]\times S^{*}N$
$g^{\prime}$
$f^{\prime}\in C^{\infty}(\overline{M})$
$g^{\prime}$
onto
$\overline{M}$
and denote
228
H. OMORI, Y. MAEDA, A. YOSHIOKA AND
$f’=(\tan\frac{\pi\theta}{2})\phi(\theta)a_{1}^{\prime}+g^{\prime}$
Thus, we have
$u=\# f$ ’
on
1/3,
, hence
$0$
.
KOBAYASHI
.
converges to $on\sim[1/3\sim’ 1]\times S^{*}N$. Therefore $\#(1-\phi)(f_{n}.-f’)\sim$ converges in . Recall that $\# h=b^{-1}dh$ , and remark that the mapping induces a linear homeomorphism of , where onto is the closed 2/3neighborhood of zero-section of . Therefore, there exists such that supp and . Thus, we have $[$
$1]\times S^{*}N$
,
$\#(f_{n}-f’)\sim$
$0$
$\Gamma_{\rho}^{\sim}$
$b\sim$
$\Gamma(T\overline{D}(2/3))$
$\Gamma(T^{*}\overline{D}(2/3))$
$\overline{D}(2/3)$
$\overline{M}=\overline{D}^{*}N$
$ h’\in$
$C^{\infty}(\overline{M})$
$h‘\subset\overline{D}(2/3)$
$\lim_{n\rightarrow\infty\#(1-\phi)(f_{fn}-f^{\prime})=\# h}^{\sim}\sim$
.
$\lim_{n\rightarrow\infty}\# f_{m}=\# f’+\lim_{m\rightarrow\infty}\#\phi(f_{m}-f’)+\# h\sim\sim\sim\sim$
Note that the second term of the right hand side converges to . Hence, we have . This implies the closedness of $0$
$\lim,,\rightarrow\infty\# f_{m}=\#(f^{\prime}+h’)\sim\sim$
$\#(\tau^{*}\Sigma^{1})\sim$
.
$\square $
LEMMA 3.3.
$\#(\tau^{*}\Sigma^{1})=d\tau^{-1}\mathfrak{U}_{\Sigma 1}\sim$
PROOF. Since
is an ideal
$\sim bAd(\varphi)u=\varphi^{*1}-((\varphi^{*}\tilde{\Omega})-u)$
Ad
$(\varphi^{-1})\# f=\#\varphi^{*}f\sim\sim$
,
of
$\Gamma_{\rho}^{\sim}$
.
for every
$f\in\tau^{*}\Sigma^{1}$
$\varphi\in \mathcal{D}(\overline{M})$
, we see
,
for every . We shall show at first that for any . To do this, it is enough to show that , because of the expression in Lemma 2.1. Let $s=\cot\pi\theta/2$ . Then ) can be naturally identified with an -neighborhood of $S^{*}N$ in . By this identification, can be written as $\varphi\in \mathcal{D}_{\rho}^{\sim}$
$\varphi^{*}\tau^{*}\Sigma^{1}=\tau^{*}\Sigma^{1}$
$\varphi\in \mathcal{D}_{\rho}^{\sim}$
$\varphi^{*}(\tan\pi\theta/2)\in\tau^{*}\Sigma^{1}$
$[0, \epsilon$
$\times S^{*}N$ $\overline{M}$
$\epsilon$
$\varphi$
$\overline{s}=\overline{s}(s, z)$
(21)
$(s, z)e[0, \epsilon)\times S^{*}N$ $\overline{z}=\overline{z}(s, z)$
Since
.
, we see and on a sufficiently small neighborhood of $S^{*}N$, where $g(s, z)$ is a smooth positive function. Note that , hence it is contained in on a sufficiently small neighborhood of $S^{*}N$, but this shows immediately $\varphi(S^{*}N)=S^{*}N$
$\overline{s}(0, z)\equiv 0$
$\overline{s}=g(s, z)s$
$\varphi^{*}(\tan\pi\theta/2)=\overline{s}^{-1}$
$\varphi^{*}(\tan\pi\theta/2)\in\tau^{*}\Sigma^{1}$
$(\tan\pi\theta/2)C^{\infty}(\overline{M})$
.
The above result shows especially
Ad for every ideal of
$(\exp-tu)\#\tau^{*}\Sigma^{1}=b^{-1}d(\exp tu)^{*}\tau^{*}\Sigma^{1}=b^{-1}d(\tau^{*}\Sigma^{1})\sim\sim\sim$
$\Gamma_{\rho}^{\sim}$
.
Since is closed in , we see that by taking the derivative at $t=0$ .
$u\in\Gamma_{\rho}^{\wedge}$
,
$\#\tau^{*}\Sigma^{1}\sim$
$\Gamma_{\rho}\sim$
$\sim\#\tau^{*}\Sigma^{1}$
is an $\square $
FR\’ECHET-LIE GROUPS
Let
on
.
$S^{*}N$
229
be the Lie algebra of all smooth contact vector fields By Lemma 3.1 and by (20), it is clear that .
$\Gamma_{\omega}(TS^{*}N)$
$\Gamma_{\omega}(TS^{*}N)=\#(\tau^{*}\Sigma^{1})|_{S^{*}N}\sim$
However, we have in fact the following:
LEMMA 3.4.
Every
PROOF. For every
$u\in\Gamma_{\Omega}^{\sim}$
induces a contact vector
.
field on
$S^{*}N$
denote by the one parameter subgroup generated by . For the proof of , it is enough to show that is a contact transformation. For an element we set . On a neighborhood of the boundary $S^{*}N$ of can be written in the form (21). Since , we have and $u\in\Gamma_{\rho}^{\sim}$
$\varphi_{t}$
$u$
$u|_{S^{l}N}\in\Gamma_{\omega}(TS^{*}N)$
$\varphi_{t}|_{S^{*}N}$
$\varphi\in \mathcal{D}_{\Omega}^{\sim}$
$\hat{\varphi}=\varphi|_{S^{*}N}$
$\overline{M},$
$\varphi$
$\tilde{\Omega}=-d(s^{-1}\omega)$
$\varphi^{*}\tilde{\Omega}=\tilde{\Omega}$
$\overline{s}^{-2}d\overline{s}\wedge\hat{\varphi}^{*}\omega=s^{-2}ds\wedge\omega$
,
hence $\hat{\varphi}^{*}\omega=\frac{\overline{s}^{2}}{s^{2}}(\frac{\partial\overline{s}}{\partial s})^{-1}|_{*=0}\omega$
Since
$\overline{s}=g(s, z)s,$
because of
$g\neq 0$
$\overline{s}(0, z)\equiv 0$
$\hat{\varphi}^{*}\omega=g(0, z)\omega$
hence
$\hat{\varphi}$
.
, we see that ,
is a contact transformation.
Now, let is surjective, because by the kernel of
be the restriction mapping $p(u)=u|_{S^{*}N}$ . and . We denote By a standard diagram chasing, we see easily
$p:\Gamma_{\rho}^{\sim}\rightarrow\Gamma_{\omega}(TS^{*}N)$
$\Gamma_{D,b}^{\sim}$
$p$
$\#(\tau^{*}\Sigma^{1})|_{S^{s}N}=\Gamma_{\omega}(TS^{*}N)\sim$
$\Gamma_{\Omega}^{\sim}\supset\#(\tau^{*}\Sigma^{1})\sim$
$p$
.
$\Gamma_{\Omega}\sim/\#(\tau^{*}\Sigma^{1})\cong\Gamma_{\Omega,b}^{\sim}/\Gamma_{\Omega,b}^{\sim}\cap\#(\tau^{*}\Sigma^{1})\sim\sim$
On the other hand, by (20), we see have
.
$\Gamma_{\Omega,b}^{\wedge}\cap\#(\tau^{*}\Sigma^{1})=\# C^{\infty}(\overline{M})\sim\sim$
$\Gamma_{\Omega}^{\sim}/d\tau^{-1}\mathfrak{U}_{\Sigma^{1}}\cong\Gamma_{JJ,b}^{\sim}/\Gamma_{\Omega,b}^{\sim\sim}\sim\cap\#(\tau^{*}\Sigma^{1})\cong b\Gamma_{\Omega,b}^{\sim}/dC^{\infty}(\overline{M})$
Let have
$Z(\overline{M})$
be the space of all smooth closed l-forms on
LEMMA 3.5.
$\phi\sim$( log tan
$b(\Gamma_{Qb}^{\sim})=Z(\overline{D}^{*}N)+Rd$
,
$\pi\theta/2$
.
Thus, we
. .
Then we
$\overline{M}$
).
PROOF. For every is a closed l-form on $M$. First, we show that ). ( log tan is contained in Given , put decomposition in accordance with the (19). Since , one may set $u\in\Gamma_{J/,b}^{\sim},$
$\sim b(\Gamma\sim)$
$u\in\Gamma_{\Omega,b}^{\sim}$
$\sim b(u)$
$Z(\overline{M})\oplus Rd$
$\phi$
$\pi\theta/2$
$u=\lambda(\partial/\partial\theta)+\mu x_{\omega}+\hat{u}$
$u|_{S^{s}N}\equiv 0$
$\lambda=(\cos\pi\theta/2)\tilde{\lambda}$
,
$\mu=(\cos\pi\theta/2)\tilde{\mu}$
,
$\hat{u}=(\cos\pi\theta/2)\tilde{u}$
,
230
on
H. OMORI, Y. MAEDA, A. YOSHIOKA AND
(2/3,
field on
, where ] and are (2/3, ] $\times S^{*}N$. Thus, on (2/3, ]
$1$
$\times S^{*}N$
$1$
.
KOBAYASHI
functions and a we have
$C^{\infty}$
$\hslash$
$\tilde{\lambda},\tilde{\mu}$
$0$
$C^{\infty}$
vector
$\times S^{*}N$
$1$
$\tilde{\Omega}-u=-d((\tan\pi\theta/2)\omega)-u$
$=\frac{\pi}{2}(\frac{\tilde{\mu}d\theta}{\cos\pi\theta/2}-\frac{\tilde{x}\omega}{\cos\pi\theta/2})-(sin\pi\theta/2)d\omega-\tilde{u}$
Set
$a=\tilde{\lambda}|_{S^{*}N},$
Then,
$\tilde{\lambda}-a$
$b=\tilde{\mu}|_{S^{*}N}$
and
$\tilde{\mu}-b$
.
and regard these as functions on (2/3, ] , and hence can be devided by cos $1$
.
$\times S^{*}N$
$\pi\theta/2$
$\sim b(u)=\tilde{\Omega}-u$
$=\frac{\pi}{2}(\ovalbox{\tt\small REJECT}^{b}\theta-\frac{a}{\cos\pi\theta/2}\omega)+\alpha$
,
is a smooth l-form on (2/3, ] . , especially Since has no singularity on (2/3, ] $S^{*}N$, we $b\equiv const.$ see easily that and . Thus, where
$\times S^{*}N$
$1$
$\alpha$
$d(\tilde{\Omega}-u)\equiv 0$
$d(\tilde{\Omega}-u)$
$1$
$\times$
$a\equiv 0$
$\sim b(u)=\frac{b}{\cos\pi\theta/2}d\theta+\alpha=Cd$
where $C$ is a constant and Conversely, assume that
$\alpha^{\prime}$
$\tilde{\Omega}-u=Cd$
( log $\phi$
( log $\phi$
tan
$\pi\theta/2$
)
is a smooth l-form on
tan
$\pi\theta/2$
) $+\beta$
,
$+\alpha^{\prime}$
,
.
$\overline{M}$
$\beta\in Z(\overline{M})$
.
on , hence we only prove that $u|_{SN}=0$ . On (2/3, ] $S^{*}N$, the right hand of the above equality can be written as Then,
$d(\tilde{\Omega}-u)=0$
$\overline{M}$
$1$
$\frac{C}{\cos\pi\theta/2}d\theta+\beta$
Hence, by using (20), we
,
$\beta^{\prime}\in\Gamma(T^{*}\overline{M})$
see easily that
$u|_{SN}=0$
$\times$
.
.
$\square $
PROOF OF THEOREM 2.4. By Lemma 3.2 and Lemma 3.3, is a closed ideal . Since , Lemma 3.5 shows that it is isomorphic to ( log tan ). Since $H^{1}(N;R)\cong H^{1}(D^{*}N;R)$ , we have the desired results. $d\tau^{-1}\mathfrak{U}_{\Sigma^{1}}$
$\Gamma_{\rho}^{\wedge}$
$\Gamma_{\tilde{\rho}}/d\tau^{-1}\mathfrak{U}_{\Sigma^{1}}\cong b(\Gamma_{\tilde{\rho},b})/dC^{\infty}(\overline{M})\sim$
$H^{1}(\overline{M};R)\oplus Rd$
$\phi$
$\pi\theta/2$
$\square $
\S 4. A reduction of Theorem 2.3. To complete the proof of Theorem , it remains proving only that is a regular Fr\’echet-Lie group (cf. Theorem 2.3 in \S 2). In this section, we shall show how the proof will be done. Put $B$
$\mathcal{D}_{\rho}^{\sim}$
231
FR\’ECHET-LIE GROUPS (22)
$\mathcal{D}_{\Omega,b}^{\wedge}=\{\varphi\in \mathcal{D}_{\Omega}^{\sim};\varphi|_{S^{l}N}=identity\}$
.
We start with the following remark. LEMMA 4.1. is a closed normal subgroup of naturally is isomorphic to the identity component of contact transformations. of the
$\mathcal{D}_{\rho}^{\sim}and$
$\mathcal{D}_{\tilde{O},b}$
$\mathcal{D}_{\omega}(S^{*}N)$
$\mathcal{D}_{\rho}^{\sim}/\mathcal{D}_{\rho,b}^{\sim}$
, the group
$C^{\infty}$
PROOF. Obviously, . By the is a normal subgroup of . To prove same proof as in Lemma 3.4, we see that , we that is isomorphic to the identity component of remark at first that is a strong ILH-Lie group (cf. [4], 8.3.6 Theorem). Hence, for every element in that identity component there joining between the identity and is a smooth curve . Define by , . Then and by Lemma 3.1, $\mathcal{D}_{\rho}^{\wedge}$
$\mathcal{D}_{\rho b}^{\sim}$
$\mathcal{D}_{\rho}^{\wedge}/\mathcal{D}_{\rho,b}^{\sim}\subset \mathcal{D}_{\omega}(S^{*}N)$
$\mathcal{D}_{\omega}(S^{*}N)$
$\mathcal{D}_{\Omega}^{\sim}/\mathcal{D}_{\Omega,b}^{\sim}$
$\mathcal{D}_{\omega}(S^{*}N)$
$\varphi$
$\varphi=\varphi_{1}$
$\varphi_{t}$
$u_{t}\in\Gamma_{\omega}(TS^{*}N)$
$(d\varphi_{t}/dt)\cdot\varphi_{t}^{-1}$
$g_{t}eC^{\infty}(S^{*}N)$
Now, , hence
$u_{t}=g_{t}x_{\omega}-\{g_{t}\}$
.
$\phi(\theta)$
$\tau^{*}\Sigma^{1}$
$u_{t}$
being as in \S 2, (9), and
$f_{t}=(\tan\pi\theta/2)\phi(\theta)g_{t}$
$\# f_{t}|_{S^{r}N}=u_{t}\sim$
$\sim\# f_{t}\in\Gamma_{\rho}^{\wedge}$
.
Setting
is an element of , we solve the
$ X_{J_{t}}=\# f_{t}\sim$
equation $\frac{d}{dt}\psi_{t}=X_{f_{t}}\cdot\psi_{t}$
Then, it is easy to see
Since
$X_{f_{t}}|_{S^{*}N}=u_{t}$
$\psi_{t}\in \mathcal{D}_{\Omega}^{\sim}$
(cf. (20)),
by the uniqueness, we have
$\psi_{0}=id$
..
is a closed subgroup of . satisfies the same equation as . Hence
, for
$\psi_{t}|_{S^{*}N}$
,
$\mathcal{D}(\overline{M})$
$\mathcal{D}_{\Omega}^{\sim}$
$\psi_{t}|_{S^{*}N}=\varphi_{t}$
$\varphi_{t}$
.
$\square $
For the proof of Theorem 2.3, we shall show the following theorem
in \S 6.
THEOREM 4.2.
$\mathcal{D}_{0.b}^{\sim}$
is an
$FL$ -subgroup
of
$\mathcal{D}(\overline{M})$
.
In this section, we shall show the above theorem yields the desired result.
PROOF OF THEOREM 2.3. Using Theorem 4.2, we will prove that is a regular Fr\’echet-Lie group. Since is a regular Fr\’echetLie group, the above theorem and Proposition 2.4 [8] show that is a regular Fr\’echet-Lie group. Let be a local coordinate system at the identity $\xi(0)=e$ (the identity), where $U$ is an open convex neighborhood such that of in . For each $u\in U$, we set $\kappa(u, t)=(d\xi(tu)/dt)\cdot\xi(tu)^{-1}$ . Then, is a and mapping. By Lemma 3.1, can be written in the form $\mathcal{D}(\overline{M})$
$\mathcal{D}_{\Omega}^{\sim}$
$\mathcal{D}_{\Omega,b}^{\sim}$
$\xi:U\rightarrow \mathcal{D}_{\omega}(S^{*}N)$
$0$
$C^{\infty}$
$\Gamma_{\omega}(TS^{*}N)$
$\kappa(u, t)\in\Gamma_{\omega}(TS^{*}N)\subset\Gamma(TS^{*}N)$ $\kappa(u, t)$
$\kappa:U\times[0,1]\rightarrow\Gamma_{\omega}(TS^{*}N)$
$C^{\infty}$
232
H. OMORI, Y. MAEDA, A. YOSHIOKA AND
$0$
.
KOBAYASHI
,
$\kappa(u, t)=\sigma(u, t)x_{\omega}-\{\sigma(u, t)\}$
mapping of $U\times[0,1]$ into where is a . Let be the Hamiltonian vector field defined by the $C^{\infty}(S^{*}N)$
$C^{\infty}$
$\sigma$
on
$(\tan\pi\theta/2)\phi(\theta)\sigma(u, t)$
$\overline{M}$
function
$C^{\infty}$
$X_{(u,t)}$
i.e.,
$\tilde{\Omega}-X_{(u,t)}=d((\tan\pi\theta/2)\phi(\theta)\sigma(u, t))$
We see easily that the mapping into . Set
is a
$(u, t)\mapsto X_{(u,t)}$
$C^{\infty}$
. mapping of
$U\times[0,1]$
$d\tau^{-\iota}\mathfrak{U}_{\Sigma^{1}}(\subset\Gamma(T\overline{M}))$
.
$\Phi(u, t)=\prod_{0}^{t}(1+X_{(u,)})ds$
Then, by the second fundamental theorem of regular Fr\’echet-Lie groups (cf. [7]), we see that mapping. Since is a $X_{(u,t)}|_{S^{*}N}=\kappa(u, t)$ , we see $\Phi(u, t)|_{S^{*}N}=\xi(tu)$ by the same reasoning as in the proof of the above lemma. , and let Set be an open neighborhood of in , we define by such that . For every and $\Phi:U\times[0,1]\rightarrow \mathcal{D}(\overline{M})$
$\tilde{V}$
$\tilde{U}=\xi(U)$
$\tilde{V}^{2}\subset\tilde{U}$
$g\in\tilde{U}$
$\gamma(g)=\Phi(\xi^{-1}(g), 1)$
Obviously, 7 is a Moreover, since subgroup of
$C^{\infty}$
mapping of into , we have $\tilde{U}$
(cf.
such that $\gamma(g)|_{SN}=g$ . because is a closed
$\mathcal{D}(\overline{M})$
$\mathcal{D}_{\rho}^{\sim}$
\S 2). Define for
$r_{\gamma}(g, h)=\gamma(gh)^{-1}\gamma(g)\gamma(h)$
Then and the assumed theorem we see that
$r_{\gamma}:\tilde{V}\times\tilde{V}\rightarrow \mathcal{D}(\overline{M})$
$r_{\gamma}(g, h)\in \mathcal{D}_{O,b}^{\sim}$
$\gamma(g)$
.
$\gamma(\tilde{U})\subset \mathcal{D}_{\rho}^{\sim}$
$X_{(u,t)}\in\Gamma_{\rho}^{\sim}$
$\mathcal{D}(\overline{M})$
For
$\mathcal{D}_{\omega}(S^{*}N)$
$e$
$\tilde{V}^{-1}=\tilde{V}$
Lemma 2.3
$C^{\infty}$
$g,$
is a
$he\tilde{V}$
$C^{\infty}$
$r_{\gamma}:\tilde{V}\times\tilde{V}\rightarrow \mathcal{D}_{\rho b}^{\sim}$
.
mapping. Thus, by mapping (cf. is a $C^{\infty}$
(ii) [8]).
$n\in \mathcal{D}_{\rho,b}^{\sim}$
we set $\alpha_{\gamma}(g)n=\gamma(g)^{-1}n\gamma(g)$
,
$g\in\tilde{V}$
.
It is obvious that because is a normal subgroup of by Lemma 4.1. Also, the map mapping is a by the same reasoning as above. Consider the following exact sequence: $\alpha_{\gamma}(g)n\in \mathcal{D}_{\rho,b}^{\sim}$
$\alpha_{\gamma}:\tilde{V}\times \mathcal{D}_{\tilde{\rho},b}\rightarrow \mathcal{D}_{\rho,b}^{\sim}$
$\mathcal{D}_{\rho}^{\sim}$
(23)
$\mathcal{D}_{\rho,b}^{\sim}$
$1\rightarrow \mathcal{D}_{\rho,b}^{\sim}\rightarrow \mathcal{D}_{\rho}^{\sim}\rightarrow \mathcal{D}_{\omega}(S^{*}N)\rightarrow 1$
$C^{\infty}$
.
a local section which satisfies (Ext. ) $\sim(Ext. 3)$ in [8] \S 6, that is, (23) is an FL-extension. Hence, by Proposition 5.2 and Theorem 5.3 in [8], we have that is a regular Fr\’echet-Lie group. Taking the derivative of a smooth family at the identity, we get (23) has
$1$
$\gamma$
$\mathcal{D}_{\Omega}^{\sim}$
$\varphi_{\iota}\in \mathcal{D}_{\rho}^{\sim}$
FR\’ECHET-LIE GROUPS
, where is .
$\mathscr{L}_{u}\tilde{\Omega}=0$
of
$u=(d\varphi_{t}/dt)_{i=0}$
.
233
we see that the Lie algebra
Therefore,
$\Gamma_{\rho}^{\sim}$
$\mathcal{D}_{\Omega}^{\sim}$
$\square $
\S 5. Subgroups
$\mathcal{D}_{b}$
and
$\mathcal{D}_{\Delta}$
of
$\mathcal{D}(\overline{M})$
.
be the closed unit disk bundle over $N$ and the symplectic structure on $M$ defined by (10). Recall that is a strong ILH-Lie group with the Lie algebra (cf. \S 2), hence it is a regular Fr\’echet-Lie group (cf. [7] \S 6). Set Let
$\overline{M},\tilde{\Omega}$
$\mathcal{D}(\overline{M})$
$\Gamma(T\overline{M})$
(24)
.
$\mathcal{D}_{b}=\{\varphi\in \mathcal{D}(\overline{M});\varphi|_{\partial M}=id.\}$
By recalling how a local coordinate system on was given in \S II. 4 in [4], we easily see that locally is a flat FL-subgroup of . Indeed, has a structure, which one may call a strong ILH-Lie subgroup (cf. [4] I. 4 or [5]). (cf. (22)) is by definition a subgroup by However, . of the certain reason which will be mentioned in this section, it is convenient to regard this as a subgroup of the following subgroup : $\mathcal{D}(\overline{M})$
$\mathcal{D}(\overline{M})$
$\mathcal{D}_{b}$
$\mathcal{D}_{b}$
$\mathcal{D}_{\rho.b}^{\sim}$
$\mathcal{D}_{b}$
$\mathcal{D}_{\Delta}$
can be extended to a smooth 2-form on }.
(25)
$\mathcal{D}_{\Delta}=\{\varphi\in \mathcal{D}_{b};\varphi^{*}\tilde{\Omega}-\tilde{\Omega}$
$\overline{M}$
To prove Theorem 4.2, we shall show at first the following, which is indeed the goal of this section. THEOREM 5.1.
$\mathcal{D}_{\Delta}$
is a closed FL-subgroup of
$\mathcal{D}_{b}$
.
REMARK. The proof which will be given in this section shows also that has the structure which one may call a strong ILH-Lie subgroup of . To prove Theorem 5.1 we shall have some steps as below. First, we get the following lemma. $\mathcal{D}_{\Delta}$
$\mathcal{D}_{b}$
LEMMA 5.2.
$\mathcal{D}_{\Delta}$
is a closed subgroup of
$\mathcal{D}_{b}$
.
We use the notations in the proof of Lemma 3.3. Near every the boundary can be written in the form (21). We write in the form
PROOF.
$S^{*}N=\partial\overline{M}$
$\varphi\in \mathcal{D}(\overline{M})$
$\varphi\in \mathcal{D}_{\Delta}$
(26)
where
$\left\{\begin{array}{l}\overline{s}=\overline{s}(s, z)\\\overline{z}=\overline{z}(s, z)\end{array}\right.$
$\overline{s}(0, z)=0$
and
$s=\cot\pi\theta/2$
$\overline{z}(0, z)=z$
.
Put
,
$(s, z)e[0, \epsilon)\times S^{*}N$
,
234
H. OMORI, Y. MAEDA, A. YOSHIOKA AND O. KOBAYASHI
(27)
is a smooth function on where and also remark by (26) and (27),
$[0, \epsilon$
$a_{\varphi}$
(28)
)
.
Recall that
$\times S^{*}N$
$\varphi^{*}(\frac{\omega}{s})-\frac{\omega}{s}=\frac{s}{\overline{s}}(\frac{\overline z^{*}\omega-\omega}{s}+\frac{a_{\varphi}}{s}ds+\frac{1-(\overline{s}/s)}{s}\omega)$
Since
$[0, \epsilon$
)
,
is smooth on
$s/\overline{s}$
$[0, \epsilon$
)
$\times S^{*}N$
$[0, \epsilon$
$\times S^{*}N$
$d(\varphi^{*}(\omega/s)-(\omega/s))$
$\overline{s}/s\equiv 1$
$[0, \epsilon$
$\times S^{*}N$
$\varphi e\mathcal{D}_{b}$
$a_{\varphi}|_{S^{*}N}=const.$
if and
Obviously, $\mathcal{D}_{b}$
$C^{\infty}$
$(1/s)(\overline{z}^{*}\omega-\omega)$
$(1/s)da_{\varphi}\wedge ds$
of
.
$\times S^{*}N$
$(z^{*}\omega-\omega)|_{SN}=0$
smooth on Thus, $\mathcal{D}_{\Delta}$
$\tilde{\Omega}=-d(\omega/s)$
and never vanishes. As l-form is a . Hence we see (28) is right of hand side . Also, the third term of the . on $S^{*}N$. Compute ) because , hence we see that ) must be smooth on is an element of . It is now not to hard to see that $S^{*}N$ is constant on $S^{*}N$. and only if on is a closed subgroup these are closed conditions and hence
$\varphi\in \mathcal{D}(\overline{M}),$
$\varphi|_{S^{*}N}=id.$
on
,
$\varphi^{*}\omega=\overline{z}^{*}\omega+a_{\varphi}ds$
$a_{\varphi}=\omega-d\varphi(\partial/\partial s)$
$\overline{s}/s\equiv 1$
$\mathcal{D}_{\Delta}$
.
$\square $
Set $\Gamma_{\Delta}=$
By Lemma 1.4 [8]
if
{
$u\in\Gamma(T\overline{M})$
we see that
LEMMA 5.3. An element it satisfies the following: (i) (ii)
$u|_{SN}=0$
$u$
$\Gamma_{\Delta}$
of
; exp
$tu\in \mathcal{D}_{\Delta}$
}.
is a closed Lie algebra of
$\Gamma(T\overline{M})$
is contained in
$\Gamma_{\Delta}$
$\Gamma(T\overline{M})$
if and
.
only
.
can be extended to a smooth 2-form on . is a smooth 2-form on PROOF. For every $\overline{M}$
$d(\tilde{\Omega}-u)$
$u\in\Gamma_{\Delta},$
$(\exp tu)^{*}\tilde{\Omega}-\tilde{\Omega}$
use.
at $t=0$ and Take the derivative with respect to is the Lie derivation. Then, we see where to a smooth 2-form on . Obviously, $u|_{SN}=0$ . To get the converse, remark that $t$
$d(\tilde{\Omega}-u)$
$\mathscr{L}_{l}$
.
$\overline{M}$
, can be extended $\mathscr{G}_{u}\tilde{\Omega}=d(\tilde{\Omega}-u)$
$\overline{M}$
$(\exp tu)^{*}\tilde{\Omega}-\tilde{\Omega}=\int_{0}^{t}(\exp su)^{*}\mathscr{L}_{u}\tilde{\Omega}ds$
$Since_{-}\mathscr{G}_{u}\tilde{\Omega}=d(\tilde{\Omega}-u)$
we have
is extended to a smooth 2-form on is a smooth 2-form on M.
and exp
LEMMA 5.4.
(19)
$\sim b(\Gamma_{\Delta})=\Gamma(T^{*}\overline{M})+Rd$
if near the
defined by for
$\sim b:\Gamma(T\overline{M})\rightarrow\Gamma(T^{*}\overline{M})$
\S 2) and also use the notation
and only
$\overline{M}$
$tu\in \mathcal{D}_{b}$
boundary
and
( log $\phi$
$S^{*}N=\partial\overline{M},$
(20)
,
$\square $
$(\exp tu)^{*}\tilde{\Omega}-\tilde{\Omega}$
Now, recall the notation (cf.
.
$\sim b(u)=\tilde{\Omega}-u$
$u\in\Gamma(T\overline{M})$
.
). Moreover, tan if can be written in the form $\pi\theta/2$
$u$
$u\in\Gamma_{\Delta}$
235
FR\’ECHET-LIE GROUPS $u=(\cos\pi\theta/2)^{2}a^{\prime}\frac{\partial}{\partial\theta}+$
(
$c$
cos
$\pi\theta/2+(\cos\pi\theta/2)^{2}g’$
)
$\chi_{\omega}+(\cos\pi\theta/2)\hat{u}^{\prime}$
where are smooth functions on a real constant and regarded naturally as a subbundle of smooth section of $a’,$
$\overline{M},$
$g^{\prime}$
$c$
$\hat{u}^{\prime}$
is a
$T(\overline{M}-\{0\})$
$E_{\omega}$
PROOF.
,
Use (19) and compute
.
$\tilde{\Omega}-(\lambda(\partial/\partial\theta)+\mu x_{\omega}+\hat{u}),\tilde{\Omega}=-d((\tan\pi\theta/2)$
, etc.. The first statement follows immediately from the second one, and second one is obtained by the same proof as in Lemma 3.5. $\omega)$
by putting
$\lambda=\lambda_{0}+\lambda_{1}\cos\pi\theta/2+\lambda_{2}(\cos\pi\theta/2)^{2}+\cdots$
$\square $
Now, recall that naturally contained in
$\sim b(\Gamma_{\rho,b}\sim)=Z(\overline{D}^{*}N)+Rd$
$b(\Gamma_{\Delta})$
( log $\phi$
tan
$\pi\theta/2$
and that it is
)
.
To prove Theorem 5.1, we shall recall how we define a structure of a strong ILH-Lie group (cf. [4], [5]). Put a smooth riemannian metric on such that the boundary is a totally geodesic. We denote by Exp the exponential mapping with respect to the metric on . For , put $\xi(u)(x)=Exp_{x}u(x)$ . If every is in a sufficiently small neighborhood $U$ of -topology, then in the is a smooth diffeomorphism on . The mapping gives a local coordinate system at the identity which satisfies the axioms (N. ) $\sim(N. 7)$ in \S 6 of [7] (cf. [4] \S 1 and [5] chap. 3). Set $S^{*}N=\partial\overline{M}$
$\overline{M}$
$\overline{M}$
$u\in\Gamma(T\overline{M})$
$u$
$0$
$C^{1}$
$\overline{M}$
$\xi(u)$
$\xi:U\cap\Gamma(T\overline{M})\rightarrow \mathcal{D}(\overline{M})$
$1$
$\Gamma_{b}=\{ue\Gamma(T\overline{M});u|_{\partial\overline{M}}\equiv 0\}$
.
Then, it is obvious that $\xi(U\cap\Gamma_{b})=\xi(U)\cap \mathcal{D}_{b}(\overline{M})$
,
gives a local coordinate system of hence which also satisfies (N. ) $\sim(N$ . 7 in \S 6 of [7]. Since linearly is imbedded , one may call in a strong ILH-Lie subgroup of . $\xi:U\cap\Gamma_{b}\rightarrow \mathcal{D}_{b}(\overline{M})$
$1$
$\mathcal{D}_{b}$
$U\cap\Gamma_{b}$
$)$
$\Gamma(T\overline{M})$
$\mathcal{D}(\overline{M})$
$\mathcal{D}_{b}$
REMARK. We have not necessary to use an exponential mapping. By an inverse mapping theorem in [4] \S III or [5] Chap. 1, it may be replaced by any smooth mapping of a neighborhood of zero-section satisfying the following: of into , Ye , $g(y, Y)=(y, \mathscr{G}_{y}(Y))$ and For is a smooth diffeomorphism of a neighborhood of on onto a neighborhood of in $T^{*}N$. If $y\in S^{*}N$ and Ye $T_{y}S^{*}N$, then . $(0,1$ diffeomorphic ] Recall that is . We fix a smooth to , it is a direct product riemannian metric on such that on (1/2, ] of (1/2, 1] and $S^{*}N$. Then for is a totally geodesic $\mathscr{G}$
$T\overline{M}$
$(^{*}1)$
$\overline{M}\times\overline{M}$
$y\in\overline{M}$
$T_{y}\overline{M}$
$\mathscr{G}_{y}$
$0$
$T_{y}\overline{M}$
$y$
$(^{*}2)$
$\mathscr{G}_{p}(Y)\in S^{*}N$
$\times S^{*}N$
$\overline{M}-\{0\}$
$\overline{M}$
$1$
$1/20$ such that $||\phi g||_{k- 2}+||\phi f\Vert_{k-2}+||\phi\hat{\alpha}\Vert_{k-2}\geqq C_{k}\Vert\phi\alpha||_{k-2}$
Thus, using
$\Vert\phi\hat{\alpha}\Vert_{k-2}\leqq\Vert\phi\hat{\alpha}\Vert_{k-1}$
.
we get
$||b^{-1}(\phi\alpha)\Vert_{k}\geqq C_{k}^{\prime}\Vert\phi\alpha||_{k-2}\sim$
Note that there is a constant
$C_{k}^{\prime\prime}>0$
.
such that
$\Vert\sim b^{-1}(\alpha)||_{k}\geqq C_{k}^{\prime\prime}\{\Vert b^{-1}((1-\phi)\alpha)\Vert_{k}+\Vert b^{-1}(\phi\alpha)\Vert_{k}\}\sim\sim$
Then,
we get
.
$\hat{\phi}/\phi$
and
$\overline{\phi}/\phi_{\alpha}^{\tau}$
are
243
FR\’ECHET-LIE GROUPS $\Vert\sim b^{-1}(\alpha)\Vert_{k}\geqq\overline{C}_{k}\{\Vert(1-\phi)\alpha\Vert_{k-2}+\Vert\phi\alpha\Vert_{k-2}\}$
$\geqq C_{k}\Vert\alpha\Vert_{k-2}$
.
$\square $
by putting on we define a new series of norms and ) for all , and ( log tan , where we put the com$Rd$ ( ) are assumed to be orthogonal. Denote by tan is then an . pletion of with respect to Now,
$\Vert b^{-1}(u)\Vert_{k}\sim$
$\Vert d$
$\pi\theta/2$
$\phi$
$\Vert u||_{k}=$
$\Gamma_{\Delta}$
$\{||\cdot\Vert_{k}\}$
$\Gamma(T^{*}\overline{M})$
$k$
$\Vert_{k}\equiv 1$
$\hat{\Gamma}_{\Delta}^{k}$
$\phi\log$
$\pi\theta/2$
$\Vert\cdot\Vert_{k}$
$\Gamma_{\Delta}$
$\{\Gamma_{\Delta},\hat{\Gamma}_{\Delta}^{k}, k\geqq\dim\overline{M}+7\}$
ILH-chain. is in fact a strong ILH-Lie group modeled on an REMARK. . However, it may not for every ILH-chain . In fact, a group can have many be modeled on strong ILH-Lie group structures modeled on various ILH-chains. It is very likely that is a strong ILH-Lie group modeled on dim }. However, at this moment this is only a conjecture. This is a strong ILH-Lie is the reason why we could not conclude that subgroup. To apply an implicit function theorem in [4] \S III, we have . to use the ILH-chain $k_{0}=\dim\overline{M}+5$ , such in Now, let $U$ be an open neighborhood of that defined in Lemma 5.5 gives a local coordinate system at the identity . Since of $\mathcal{D}_{\Delta}$
$\tilde{k}_{0}\geqq\dim\overline{M}+5$
$\{\Gamma_{\Delta}, \Gamma_{\Delta}^{k};k\geqq k_{0}\}\sim$
$\{\Gamma_{\Delta},\hat{\Gamma}_{\Delta}^{k}, h\geqq\dim\overline{M}+7\}$
$\{\Gamma_{\Delta},\hat{\Gamma}_{\Delta}^{k},$
$ k\geqq$
$\mathcal{D}_{\Delta}$
$\overline{M}+7$
$\mathcal{D}_{\rho}^{\sim}$
$\{\Gamma_{\Delta},\hat{\Gamma}_{\Delta}^{k}, k\geqq\dim\overline{M}+7\}$
$0$
$\Gamma_{\Delta}^{k_{0}},$
$\xi^{\prime}\ddagger U\cap\Gamma_{\Delta}\rightarrow \mathcal{D}_{\Delta}$
$\mathcal{D}_{\Delta}$
$e$
$\sim b^{-1}:\Gamma^{k_{0}+2}(T^{*}\overline{M})\oplus Rd$
( log $\phi$
tan
is bounded there is an open neighborhood . Define such that $\sim b^{-1}(U’)\subset U$
$\Psi_{\Omega}^{\sim}:$
$\pi\theta/2$
of
$U$ ’
$\rightarrow\Gamma_{\Delta^{0}}^{k}$
$0$
$U’\cap\Gamma(T^{*}\overline{M})\rightarrow E_{\Omega}^{\sim}$
$\Psi_{\Omega}^{\sim}(\alpha)=\Phi_{\tilde{\Omega}}(\xi^{\prime}(b^{-1}(\alpha)))\sim$
Then,
)
in the source space by
.
we have
ILB -normal mapping is LEMMA 6.4. (cf. [4] \S III), namely into (1) be extended to a smooth mapping of $\Psi_{\Omega}^{\sim}:$
$U’\cap\Gamma(T^{*}\overline{M})\rightarrow E_{\tilde{\Omega}}$
$a$
$C^{\infty}$
$C^{2}$
$U’\cap\Gamma^{k}(T\overline{M})$
$\Psi_{\Omega}^{\sim}can$
$E^{k-1}$
for every
$k\geqq\dim\overline{M}+8$
.
independent of and a polyThere are a positve constant such that depending on with positive coefficients for every
(2)
$C$
nomial
$P_{k}(t)$
$h\geqq k_{0}+1$
,
$k$
$k$
$\Vert(d\Psi_{\tilde{\rho}})_{\alpha}\beta||_{k-1}\leqq C\{\Vert\alpha||_{k}\Vert\beta\Vert_{k_{0}}+\Vert\beta\Vert_{k}\}+P_{k}(\Vert\alpha\Vert_{k-1})\Vert\beta\Vert_{k-1}$
$\Vert(d^{2}\Psi_{\tilde{\rho}})_{\alpha}(\beta_{1}, \beta_{2})\Vert_{k- 1}\leqq C\{\Vert\alpha\Vert_{k}\Vert\beta_{1}\Vert_{k_{0}}\Vert\beta_{2}\Vert_{k_{0}}+\Vert\beta_{1}\Vert_{k}\Vert\beta_{2}\Vert_{k_{0}}$
$+\Vert\beta_{1}\Vert_{k_{0}}\Vert\beta_{2}\Vert_{k}\}+P_{k}(\Vert\alpha\Vert_{k- 1})\Vert\beta_{1}\Vert_{k- 1}\Vert\beta_{2}\Vert_{k-1}$
.
244
H. OMORI, Y. MAEDA, A. YOSHIOKA AND
$0$
.
KOBAYASHI
PROOF. It is known that is a ILB -normal mapping of (cf. [4] Lemma 2.5.3). By the into second inequality in Lemma $6.3,$ is a ILB -normal mapping. Hence so is the composition (cf. [5] $\Psi’(u)=\Phi_{9}^{\sim}(\xi^{\prime}(u))$
$C^{\infty}$
$C^{2}$
$E_{\tilde{\rho}}=\tilde{\Omega}\oplus d\Gamma(T^{*}\overline{M})$
$U\cap\Gamma_{\Delta}$
$\sim b^{-1}:U^{\prime}\cap\Gamma(T^{*}\overline{M})\rightarrow U\cap\Gamma(T\overline{M})$
$C^{2}$
$C^{\infty}$
$\Psi_{\rho}^{\wedge}=\Psi^{\prime}\cdot b^{-1}\sim$
Chap. I).
$\square $
For simplicity of notations, we set $F=\Gamma(T^{*}\overline{M})\oplus Rd$
tan log tan
( log $\phi$
$F^{k}=\Gamma^{k}(T^{*}\overline{M})\oplus Rd$
(
$\phi$
$\pi\theta/2$
)
$\pi\theta/2$
).
Note that , and that is isomorphic to is given by for ( log tan and ) $=0$ . Thus, to apply the implicit function theorem (cf. Theorem 3.3.1 in [4] or [5] \S 6 in Chap. I), we have only to see the following: $F^{k}$
$\hat{\Gamma}_{A}^{k}$
$(d\Psi_{\tilde{\rho}})_{0}:F\rightarrow d\Gamma(T^{*}\overline{M})$
$(d\Psi_{\rho}^{\sim})_{0}\beta=d\beta$
$\beta\in\Gamma(T^{*}\overline{M})$
$(d\Psi_{\tilde{\rho}})_{0}d$
, LEMMA 6.5. For every , and there is a right inverse , that for $k\geqq 1$
$d\Gamma^{k}(T^{*}\overline{M})$
$\Gamma^{k-1}(\wedge^{2}\overline{M})$
$\pi\theta/2$
$\phi$
is a closed subspace of such
$Bofd:\Gamma(T^{*}\overline{M})\rightarrow d\Gamma(T^{*}\overline{M})$
$k\geqq 2$
$|1B\beta\Vert_{k}\leqq C\Vert\beta\Vert_{k-1}+D_{k}||\beta\Vert_{k-2}$
where
$C,$
$D_{k}$
are positive constants, and
$C$
,
does not depend on
$k$
.
The above fact is an immediate consequence of Hodge theory on a compact manifold with boundary, and rather widely known. Roughly speaking it is proved by using Stokes’ theorem and a regularity
PROOF.
theorem in [5] combined with Garding’s inequality in [5].
$\square $
THEOREM 6.2. $U$’ was an open neighborhood of in $k_{1}=k_{0}+2$ , and was a ILB -normal mapping of $U’\cap F$ into . By the above lemma has a right $B$ satisfying the inequality stated in the above lemma. Let inverse PROOF
OF
$0$
$F^{k_{1}},$
$C^{\infty}$
$\Psi_{\rho}^{\sim}$
$C^{2}$
$\tilde{\Omega}\oplus d\Gamma(T^{*}\overline{M})$
$(d\Psi_{\tilde{\rho}})_{0}:F\rightarrow d\Gamma(T^{*}\overline{M})$
$\hat{Z}=Z(\overline{M})+Rd$
Using
$B$
( log $\phi$
tan
$\pi\theta/2$
).
, we have the following splitting: $F=\hat{Z}\oplus Bd\Gamma(T^{*}\overline{M})$
,
$\alpha=(\alpha-Bd\alpha)+Bd\alpha$
.
, we consider Let be the closure of in . For a mapping of $U\cap F$ into . By Lemma 6.4, is a ILB -normal mapping, and by Lemma 6.5, $\hat{Z}^{k}$
$\hat{Z}$
$F^{k}$
$\tilde{\Psi}(\alpha+\beta)=(\alpha, \Psi_{\tilde{\rho}}(\alpha+\beta))$
$\tilde{\Psi}$
$C^{\infty}$
$C^{2}$
$\alpha\in\hat{Z},$
$\beta eBd\Gamma(T^{*}\overline{M})$
$\hat{Z}\oplus E_{\rho}^{\sim}$
$(d\tilde{\Psi})_{0}:F\rightarrow\hat{Z}\oplus$
245
FR\’ECHET-LIE GROUPS
has the inverse
$d\Gamma(T^{*}\overline{M})$
satisfying
$A$
$||A(\alpha+d\beta)\Vert_{k}\leqq C\{\Vert\alpha\Vert_{k}+\Vert d\beta\Vert_{k-1}\}+D_{k}\{\Vert\alpha\Vert_{k-1}+\Vert d\beta\Vert_{k-2}\}$
,
$k\geqq 2$
.
Thus, rewriting the suffix $k-1$ by , in the ILH-chain }, one can apply the inverse mapping theorem of [4] \S III. Hence, there are open neighborhoods , of zeros in respectively and a ILB -normal mapping of $\{d\Gamma(T^{*}\overline{M})$
$k$
$d\Gamma^{k}(T^{*}\overline{M}),$
$k\geqq k_{1}$
$W_{1},$
$C^{\infty}$
into
$d\Gamma(T^{*}\overline{M})$
$\hat{Z}^{k_{1}},$
$W_{2}$
$C^{2}$
$ W_{1}\cap Z\oplus W_{2}\cap$
$\lambda=\lambda_{1}+\lambda_{2}$
$F=\hat{Z}\oplus Bd\Gamma(T^{*}\overline{M})$
$d\Gamma^{k_{1}}(T^{*}\overline{M})$
such that
$\tilde{\Psi}(\lambda_{1}(\alpha, \omega)+\lambda_{2}(\alpha, \omega))=(\alpha,\tilde{\Omega}+\omega)$
.
, . Then, is a we set ILB -normal diffeomorphism at the origin, namely and are ILB -normal
$i.e.,$
$\lambda_{1}(\alpha, \omega)\equiv\alpha,$
$\Psi_{\tilde{\rho}}(\alpha+\lambda_{2}(\alpha, \omega))\equiv\tilde{\Omega}+\omega$
Now, for
$\nu(\alpha+\beta)=\alpha+(\lambda_{2}(\alpha, 0)+\beta)$
$\alpha\in\hat{Z},$
$\nu$
$C^{\infty}$
$\nu^{-1}$
$\nu$
$\beta\in Bd\Gamma(T^{*}\overline{M})$
$C^{2}$
$C^{\infty}$
$C^{2}$
mappings. , and that : Note that can be regarded Fr\’echet-Lie system group as a local coordinate . at of regular Through the linear mapping defines a smooth change of coordinates. Namely, is a smooth diffeomorphism of a neighborhood $W$ of in onto another neighborhood of . Remark that is a local coordinate system of at . Since the above implicit func$\sim\sim b^{-1}\hat{Z}=\Gamma_{\rho}^{\wedge},bF=\Gamma_{\Delta}$
$\xi$
$U\cap\Gamma_{\Delta}\rightarrow \mathcal{D}_{\Delta}$
$e$
$\mathcal{D}_{\Delta}$
$\sim b,$
$\nu$
$\nu^{\prime}=b^{-1}\nu b\sim\sim$
$0$
$0$
$\Gamma_{\Delta}$
$\mathcal{D}_{\Delta}$
$\xi^{\prime}=\xi\nu^{\prime}:W\rightarrow \mathcal{D}_{\Delta}$
$e$
tion theorem shows that $\xi(W)\cap \mathcal{D}_{\Omega}^{\sim}=\xi^{\prime\prime}(W\cap\Gamma_{\rho}^{\sim})$
gives an FL-subgroup structure on proof of Theorem 6.2. $\xi^{\prime\prime}|W\cap\Gamma_{\rho}^{\sim}$
,
$\mathcal{D}_{\Omega}^{\sim}$
.
This completes the $\square $
References
[81
R. ABRAHAM and J. MARSDEN, Foundation of Mechanics, Benjamin N.Y., 1967. A. KORIYAMA, Y. MAEDA and H. OMORI, On Lie algebras of vector fields, Trans. Amer. Math. Soc., 226 (1977), 89-117. L. D. LANDAU and E. M. LIFSHITZ, The Classical Theorey of Fields, Pergman Press,
[41
1975. H. OMORI, Infinite Dimensional Lie Transformation Groups, Lecture Notes in Math.,
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427, Springer, 1974.
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$39\ovalbox{\tt\small REJECT}$
.
Present Address: DEPARTMENT OF MATHEMATICS FACULTY OF SCIENCE AND TECHNOLOGY SCIENCE UNIVERSITY OF TOKYO NODA-SHI, CHIBA 278 DEPARTMSNT OF MATHEMATICS FACULTY or SCIENCE AND TECHNOLOGY KEIO UNIVERSITY YOKOHAMA, 223
DEPARTMENT OF MATHEMATICS FACULTY OF SCIENCE TOKYO METROPOLITAN UNIVERSITY FUKAZAWA, SETAGAYA, TOKYO 158 AND
DEPARTMENT OF MATHEMATICS FACULTY OF SCIENCE AND TECHNOLOGY KEIO UNIVERSITY YOKOHAMA, 223
