数理解析研究所講究録
1017 巻 1997 年 104-115

104

On Group Topologies on the Group of Diffeomorphisms
HIROAKI SHIMOMURA and TAKESHI HIRAI(Kyoto University)(Fukui University)
平井 武 (京都大)下村 宏彰 (福井大)Introduction.
-manifold with $1\leq r\leq$
Let $M$ be a connected, non-compact, a-compact
its
the group of all diffeomorphisms and. Denote bysubgroup consisting of diff\‘eomorphisms with compact supports. Here we study. Usually, as seen in the beginninggroup topologies on the groupof [Ki], we have been considering on the topology given by the following way, converges to if supports of and of$k=1,2,$of convergence: a sequence
on $K$ uniformly together
are contained in a compact subset $K$ andallwith all derivatives.
This topology is normally understood as an inductive limit of topologies$narrow\infty$ , as follows.
First take an increasingof canonical subgroupsof relatively compact open subsets so thatsequence, is a manifold with, the closure ofand that eachboundary. Put$C^{r}$$\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}(M)$$\infty$$\mathrm{b}.\mathrm{y}\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}_{0}(M)$$G=\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}_{0}(M)$$G$$\tau$$\cdots$$g_{k},$$g$$g$$g_{k}arrow g$$g_{k}$$\tau$$G_{n}..\nearrow G,$$M_{0}\subset M_{1}\subset M_{2}\subset\cdots$$K_{n}:=\overline{M}_{n}$$\bigcup_{n=0^{M_{n}}}^{\infty}=M$$M_{n}$$G_{n}=\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}(K_{n}):=\{g\in G;\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{P}(g)\subset K_{n}\}$Then we have an increasing sequence$0\dot{\mathrm{f}}\mathrm{s}\dot{\mathrm{u}}\mathrm{b}\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{u}_{\mathrm{P}}\mathrm{s}$$G_{0}\subset G1\subset G2\subset\cdots$

,

.

as

$\bigcup_{n=0}^{\infty}cn=G$.

as a topological subgroup ofon
The topology
is given by considering, whereis the compact manifold obtained by
the Fr\’echet Lie groupthe boundary. For the Lie groupand its mirror image
patchingof a compact manifold $M$ , we refer [Le] or [Om].
structure of the groupis nothing but
and $M_{n}=\{x\in \mathrm{R}^{d};||x||<n\}$ , the topology
Whenand also of all derivatives as $karrow\infty$ .
the uniform convergence of, and as a topology on , we have $\tau=$
In an algebraic sense,. Since we will consider other topologies on later, we denote this induc.
tive limit topology as$\tau_{n}$$G_{n}$$G_{n}$$M_{n}^{\prime/}$$\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}(M_{n}^{\prime/})$$M_{n}$$M_{n}’\mathrm{t}\mathrm{h}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{g}\mathrm{h}$$\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}(M)$$M=\mathrm{R}^{d}$$\tau_{n}$$g_{k}\in G_{n}$$G$$G= \lim_{narrow\infty}c_{n}$$G$$\lim_{narrow\infty}\tau_{n}$$\tau_{ind}$105

proved, when a consistent increasing
On the other hand, as, is, with a group topology
sequence of topological groupsongiven, the inductive limitof topologies is not necessarily a group topology,
that is, it does not necessarily make the inductive limit groupa
topological group. This negative result is contrary to the affirmative statement
in [Iw, Article 75] or in [Enc, Article 210]. In fact, he gave a counter example
even in a case of simple abelian groups (Example 1.1).
It seems for us that this phenomenon is rather general for the case of nonlocally-compact topological groups.
In this paper, we prove that this is the case for diffeomorphism groupfor any non-compact $M$ . Thus our main theorem here is the following.
$\mathrm{T}\mathrm{a}\mathrm{t}\mathrm{s}\mathrm{u}\mathrm{u}\mathrm{m}\mathrm{a}[\mathrm{T}\mathrm{a}]$$(G_{n}, \tau_{n})$$\tau_{ind}$$G_{n}$$\tau_{n}$$\tau_{n}$$G= \lim_{narrow\infty}G_{n}$$\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}_{0}(M)$-manifold,
Theorem A. Let $M$ be a connected, non-compact, -compact$1\leq r\leq\infty$ . For the group, the product map $G\cross G\ni(g_{1},g_{2})\mapsto g_{1}g_{2}\in$, is not continuous with respect to the inductive limit topology . on .
$C^{r}$$\sigma$$\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}_{0}(M)$

$G$

$G$

$\tau_{nd}\dot{.}$This fact does not affect so much the theory of unitary representations of thegroup , because we can take, as our background, the topology. on which is(cf. \S 1).
-continuous positive definite
defined by means of the set of
However it has certainly some effects, for instance, for determining continuous$(g,p)\in G\cross M$ , depending on which continuity we choose (cf.
1-cocycles[HS] .
, isNote that if a sequence $g_{k}\in G,$ $k=1,2,$-convergent to $g\in G$ ,then there exists a compact subset $K$ of $M$ such that
and
are
$K$
contained in , and the convergence is as in [Ki]. To see this last assertion, weof the inductive limit
on isremark that the restriction on
be a -open subset of , then, for $k>n$ ,exactly the original . In fact, letofsuch that $o_{k}\mathrm{n}c_{k-1}=O_{k-1}$ ,we can choose inductively a -open subset, thenis equal to. Put
since the restriction of onto
-open in and $O\cap G_{n}=O_{n}$ .
isAcknowledgements. The authors express their thanks to Professors N. Tatsuuma and A. Yamasaki for their kind discussions and suggestions about many
subjects containing especially inductive limits of group topologies.
Here, on this oppotunity, the authors together express their deepest thanks toProfessor Yasuo Yamasaki for his long acquaintance with us through our studies.
$G$$\tau_{p.d}$$\tau_{nd}\dot{.}$$G$$\mathrm{f}.\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}\mathrm{i}_{0}\mathrm{n}\mathrm{S}$$\alpha(g,p),$$)$$\cdots$$\tau_{ind}$$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(g_{k})$$G_{n}=\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}(K_{n})$$O_{n}$$\tau_{n}$$\tau_{k}$$\tau_{ind}$$G_{k-1}$$\tau_{ind}$$G$$G_{n}$$\tau_{n}$$O_{k}$$\tau_{k}$$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(g)$$G_{k}$$\tau_{k-1}$$O= \bigcup_{k=n}^{\infty}O_{k}$

$O$

$G$

\S 1. Some generalities on inductive limits.
Let us consider an inductive system$G_{\alpha}(\alpha\in A),$ $\psi_{\alpha\beta}(\alpha,\beta\in A, \alpha\preceq\beta)$

, of

106

, are injective:
topological groups, where is a directed set andwithand we identify eachcontinuous homomorphisms. Put. Denote by the group topology on and by
its image in throughof istheir inductive limit. Note that, by definition, a sub.set
is -open-open in short) if and only if(oropen with respect to.
for anyin.
We see easily the following fact on$A$$G_{\alpha}arrow G_{\beta}$$\psi_{\alpha\beta}$$G_{\alpha}$$G= \lim_{arrow}G_{\alpha}$$G$$\psi_{\alpha\beta}’ \mathrm{s}$$G_{\alpha}$$\tau_{a}$$U\cap G_{\alpha}$$\tau_{ind}$$\tau_{ind}$$G_{\alpha}$

$G$

$U$

$\tau_{ind}=\lim_{arrow}\tau_{\alpha}$$\tau_{\alpha}$$\alpha\in A$$\tau_{ind}$, the following maps are
Lemma B. On the inductive limit group:
continuous with respect to(i) the inverse: $G\ni g\mapsto g^{-1}\in G$ ;(ii) the left and right translations: for a fixed $h\in G$ ,$G= \lim_{arrow}G_{\alpha}$$\tau_{ind}=\lim_{arrow}\tau_{a}$$G\ni g\mapsto gh\in G$,$G\ni g\mapsto hg\in G$.

However the product map $G\cross G\ni(g_{1},g_{2})\text{ }arrow g_{1}g_{2}\in G$ is not necessarily-continuous as the following example of Tatsuuma shows.
ト$\tau_{ind}$orwith the usual non. Let
Example 1.1
as. Then, for
into
discrete topology, and imbed, the product map is not-continuous. Or, there exists an open
is not containedneighbourhood of the identity element of such thatof .
infor any open neighbourhood, converges to , then there exists
Note that, if a sequence $g_{k}\in G,$ $k=1,2,$.
asuch thatfor all , and they converge in$G_{n}=\mathrm{Q}\cross F^{n},$$([\mathrm{T}\mathrm{a}])$$F=\mathrm{R}$$G_{n+1}$$G_{n}$$= \mathrm{Q}\cross\prod’\mathrm{R}$$\mathrm{Q}$$x\vdash\Rightarrow(x, 0)$$G= \lim_{narrow\infty}c_{n}$$\tau_{ind}$

$U$

$V$

$U$

$V^{2}$

$G$

$e$

$e$

$\cdots$$G_{n}$$e$$k$$g_{k}\in G_{n}$$G_{n}$He also proved the following affirmative fact.

, ofProposition. For an inductive sequence$n=1,2,$are locally compact. Then the inductive
topological groups, assume that all
gives a group topology on
limit topology.
$(G_{n}, \mathcal{T}_{n}),$$\mathrm{C}([\mathrm{T}\mathrm{a}])$$\cdots$$G_{n}’ s$$G= \lim_{narrow\infty}c_{n}$$\tau_{ind}=\lim_{narrow\infty}\tau_{n}$. Let $GL(\infty, F)$ with
Examplegroup of $G_{n}=GL(n, F),$ $n=1,2,$ , where$1.2([\mathrm{Y}\mathrm{a}])$$\cdots$or be the inductive limit
is imbedded intoas

$F=\mathrm{R}$$G_{n}$$g-$$\mathrm{C}$$G_{n+1}$.

Then, by the above proposition,
is a group topology on $GL(\infty, F)$ . A basis
for-neighbourhoods of is given by A.Yamasaki. Rewriting it in a different
form, we get another basis as follows. For $g\in.GL(\infty, F)$ , put $g=1+x,$ $x=$$\tau_{ind}$$\tau_{ind}$

$e$

107

$(x_{ij})_{i,j}\infty=1$. Take$\kappa=(\kappa_{ij})_{i,j=}^{\infty}1$

with

’

$\kappa_{ij}>0$, and put$V(\kappa)=\{g=1+x;|x_{ij}|<\kappa_{i}j(\forall i,j)\}$.

Note 1.3. Generally speaking, why
does not give a group topology is thathas too many open neighbourhoods of . So we should have some criterion
to decrease the number of these nighbourhoods. In this context, we can refer thecase of locally convex topological vector spaces. In that case the criterion is theconvexity of neighbourhoods.
$\tau_{ind}$$\tau_{ind}$$e$

As a group topology on
weaker than
, one can propose the topologyof all positive definite functions on. defined by means of the set
continuous with respect to. Note that a positive definite function is-continuous on
if it isuous at , because the topology
istranslation-invariant (by Lemma), and the positive definiteness of gives$f(e)\geq|f(g)|,$, and Krein’s inequality [Kr]$G$$\tau_{ind}$$P(\tau_{ind})$$\tau_{p.d}$

$G$

$f$

$\tau_{ind}$$G$$\tau_{ind}$$\tau_{ind}- \mathrm{C}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{i}\dot{\mathrm{n}}$$e$$\tau_{i\pi d}$$f$$\mathrm{B}(\mathrm{i}\mathrm{i}.)$$f(g^{-1})=\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{j}\{f(g)\}$$|f(g)-f(h)|^{2}\leq 2f(e)\{f(e)-\Re(f(gh^{-}1))\}$$(g, h\in G)$

By definition, an open neighbourhood of with respect tofollows. Take a finite number of$1\leq j\leq N$ , and an$e$$f_{j}\in P(\mathcal{T}_{ind}),$$\tau_{p.d}$$\tau_{p.d}$$G$. is given as$\epsilon>0$$U(f_{1}, f_{2}, \cdots, f_{N};\epsilon)=\{g\in G;|f_{j}(g)-f_{j}(e)|<\epsilon(\forall j)\}$

The topology. is also defined as a weakest topology on-continuous unitary.representations continuous.
Finally we note that $P(\tau_{ind})=P(\tau_{p.d}.)$ .

.
, then.

which makes all$\tau_{ind}$\S 2. Preparation for the proof of Theorem A.
Let $d=\dim M$ . To express
as an inductive limit, we choose
under the following additional condition.
$G=\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}_{0}(M)$$M_{0}\subset M_{1}\subset\cdots\subset M_{n}\subset\cdots$(Condition 1) There exists a coordinate neighbourhood
containing
the closureand such that, with respect to a -class Riemannian structure
$M,$
on
and
are open balls with the common center, and further that,
under the. coordinate map . , t.h.e Riemannian structure is of the canonical formon:$(V_{M}, \iota_{M})$$\overline{M}_{1}$$M_{0}$$C^{r}$$M_{1}$$l_{M}.$$M_{1}$$ds^{2}=dp\backslash - 21+dp_{2}^{2}+\cdot\cdot’+dp_{d}^{2}$for$p=(p_{i})_{i=1}^{d}\in M_{1}arrow \mathrm{R}^{d}lM$

.

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the distance of two points $q\in M$ . We fix the origin of
Denote by.
, and putofthe coordinates on the boundary
which are reinto
denotes the set of all maps from
Let
into
Take
of -maps from some open sets containing
strictions on, put alike a j.e at. For $1\leq k\leq r$ , finite, and$\rho(p, q)$$\mathrm{O}$$p,$$\partial(M_{0})$$\rho(p)=\rho(p, \mathrm{O})$$M_{0}$$C^{r}(\overline{M}_{0}, M_{1})$$C^{r}$$\overline{M}_{0}$.

$\phi$$\overline{M}_{0}$$M_{1}..\cdot$$\mathrm{t}$$p.\in\overline{M}_{0}$$\in C^{r}.(\overline{M}0, M1)$

with$\partial_{i}=\frac{\partial}{\partial p_{i}}$$j_{p}^{k}\phi$$=$$(\partial_{1}^{\alpha_{1}}\partial^{\alpha}2^{2}\ldots\partial\alpha_{d}\phi d(p))_{1}\alpha|\leq k$,$=$$(\alpha_{1}, \alpha_{2}, \cdots, \alpha_{d}),$$\alpha$

.

’

$p$

..

$\mathrm{s}$$|\alpha|=\alpha_{1}+\alpha_{2}+\cdots+\alpha_{d}$Considering this value as an element of a Euclidean space, we
its norm:
priate$N_{k}$$M_{1}$$\overline{M}_{0}$$(\mathrm{R}^{d})^{N_{k}}$.

for an appro-$\mathrm{t},\mathrm{a}\mathrm{k}\mathrm{e}$$||j_{p}^{k} \phi||:=(|\alpha|\sum_{k\leq}||\partial_{1}^{\alpha_{1}}\partial^{\alpha_{2}}2\ldots d\partial\alpha_{d}\phi(p)||^{2})^{1/2}$

and put for$\phi,$$\psi\in C^{r}(\overline{M}0, M1)\subset C^{r}(\overline{M}_{0}, \mathrm{R}^{d})$,,$d^{k}(\phi, \psi):=$$\sup_{\overline{M}_{0},p\in}||j_{p}^{k}(\phi-\psi)||$.

We put also, taking the k-th homogeneous part,$j_{p}^{(k)}\phi:=(\partial_{1}\alpha_{1}\partial^{\alpha}22\ldots\partial d\alpha_{d}\phi(p))|\alpha|=k$’$d^{(k)}( \phi, \psi):=\sup_{0p\in\overline{M}}||j_{p}^{(k)}(\phi-\psi)||$.

be
The next lemma is a key of our proof of Theorem A. Let
connected open sets, and $C^{r}(D_{1}, D_{2})$ be the set of all -class maps from. Consideringto. For $\phi=(\phi_{i})_{i=1}^{d}\in C^{r}(D_{1,2}D)$ , we have),canonically, we denote its operator norm by
it as a linear map on$||x||=(x_{1}^{2}+x_{2}^{2}+\cdots+X_{d}^{2})^{1}/2$.
as the norm of
where we take$D_{1},$ $D_{2}\subset \mathrm{R}^{d}$$C^{r}$$D_{1}$$\phi$$j_{p}^{(1)}\phi=(\partial_{j}\phi i)_{1}\leq i,j\leq d$$D_{2}$$\mathrm{R}^{d}$$||j_{p}^{(1}$$\phi||_{\varphi}$$x=(x_{i})_{i=1}^{d}\in \mathrm{R}^{d}$be an open ball and denote by id the identity map
Lemma 2.1. Let
on D. Assume for $\phi\in C^{r}(D, D)$ , the s\’upport
is compact, $a,nd$$D\subset \mathrm{R}^{d}$$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\phi):=\mathrm{C}1\{p\in D_{1;}\phi(p)\neq p=$$\mathrm{i}\mathrm{d}(p)\}$$||j_{p}^{(1)}(\phi-\mathrm{i}\mathrm{d})||_{\varphi}=||j_{p}(1)\phi-1d||_{\mathit{0}}P<1(\forall p\in D)$

where$1_{d}$

denotes theProof. Since$d\cross d$

identity matrix. Then$\phi$,

is a diffeomorphism on $D$ .

, by the theorem of implicit functions, we
see that is an open map and locally diffeomorphic.
, take $p-q\in$.
On the other hand, is globally 1-1. In fact, forand put $p_{t}=q+\mathrm{t}(p-q)(0\leq t\leq 1)$ , then$\det(j_{p}^{(1)}\phi)\neq 0(\forall p\in D)$$\phi$$\phi$$p,$$q\in D\subset \mathrm{R}^{d}$$\phi(p)-\phi(q)=\int_{0}^{1}\frac{d}{dt}\phi(pt)dt=\int_{0}^{1}(j_{p}^{(1}t\phi))(p-q)dt$$\mathrm{R}^{d}$

.

109

From the similar formula for$\psi=\phi-\mathrm{i}\mathrm{d}$, we have.

$|| \psi(p)-\psi(q)||\leq\int_{0}^{1}||j_{p_{t}}\psi||_{\varphi}||p-q||dt<||p-q||$

Hence $||\phi(p)-\phi(q)||\geq||p-q||-||\psi(p)-\psi(q)||>0$ .
Now let us prove that is onto. To do so, it is enough to prove that $\phi(D)$ is relatively closed, i.e., $D\cap \mathrm{C}1(\phi(D))=\phi(D)$ , because we know already that $\phi(D)$ isopen. Heredenotes the closure of $\phi(D)$ in . Take a $p\in D\cap \mathrm{C}1(\phi(D))$ .
Then there exists a sequence $q_{n}\in D$ such thatas $narrow\infty$ . Since is
1-1 andnear the boundary
has an cumulation point inside $D$ .
Thus we get $p=\phi(q)$ .
Q.E.D.
$\phi$$\mathrm{R}^{d}$$\mathrm{c}1(\phi(D))$$\phi(q_{n})arrow p$$\partial(D),$$=\mathrm{i}\mathrm{d}$$\phi$$q$$q_{n}$$\mathrm{a}\mathrm{c}_{l}$\S 3. Behavior of a diffeomorphism on$M_{0}$

and$\overline{M}_{0}$.

.

3.1. A basis of neighbouhoods of $e\in G_{0}$ We denote the identity map id
on $M$ also by , since it is the.identity element of . Put
$G$

$e$

$\Omega=\{g\in G;g\overline{M}0\subset M1\}\subset G$.

-open in , as is easily seen. Note that, foron.
belongs to
We define subsets
of as follows dependirig on the classThen$g|_{\overline{M}_{0}}$$\Omega$is$G$$\tau_{ind}$$g\in\Omega$, its restriction$C^{r}(\overline{M}0, M1)$$\overline{M}_{0}$$\Omega$$W_{k}$$C^{r}$$W_{k}$$:=$$\{g\in\Omega;d^{k}(g, e)\leq 1/k\}$in Case $r=\infty$ ,$W_{k}$$:=$$\{g\in\Omega;d^{r}(g, e)\leq 1/k\}$in Case $r<\infty$ .

:Then we have the following lemma.

Lemma 3.1. Put $W_{k,0}:=W_{k}\cap G_{0}$ for $k=1,2,$. Then they $fom$ a basis$te\inG_{0}$
with respect to the topology .
of neighbourhoods of the identity elemen$\cdots$$\tau_{0}$3.2. Convex combination of maps. Take
(3.1)

$g_{s}$$g\in\Omega$. For $0\leq s\leq 1$ , we can put$:=.s\cdot \mathrm{i}\mathrm{d}_{\overline{M}_{0}}+(1-s)\cdot g|_{\overline{M}0}\in C^{r}(\overline{M}0, M1)$

More generally we put, for$\phi\in C^{r}(\overline{M}0, M1)$.

,$\phi_{s}:=s\cdot \mathrm{i}\mathrm{d}_{\overline{M}_{0}}+(1-\mathit{8})\cdot\phi\in C^{r}(\overline{M}0, M1)$.

Further put$\alpha_{k}(\phi)$$:=$$\inf\{s;0\leq s\leq 1, d^{k}(\phi_{s},\mathrm{i}\mathrm{d})\leq 1/k\}$in Case $r=\infty$ ,$\alpha_{k}(\phi)$$:=$$\inf\{s;0\leq s\leq 1, d^{r}(\phi_{s}, \mathrm{i}\mathrm{d})\leq 1/k\}$in Case $r<\infty$ .

110

Since

$d^{k}( \phi_{s},\mathrm{i}\mathrm{d})=\sup_{p\in\overline{M}0}||j_{p}^{k}(\phi_{s}-\mathrm{i}\mathrm{d})||=(1-s)\cdot d^{k}(\phi,\mathrm{i}\mathrm{d})$$r=\infty$, we have according as

or $r<\infty$ ,(3.2)$\alpha_{k}(\phi)=0(1-\frac{1}{k\cdot d^{k}(\phi,\mathrm{i}\mathrm{d})})$in Case $r=\infty$ ,$(3.2^{})$$\alpha_{k}(\phi)=0(1-\frac{1}{k\cdot d^{r}(\phi,\mathrm{i}\mathrm{d})})$in Case $r<\infty$ .

Define further, for$\phi\in C^{r}(\overline{M}0, M1)$,

$P_{k}\phi=\phi\alpha_{k(\emptyset})=\alpha k(\phi)\cdot \mathrm{i}\mathrm{d}\overline{M}0+(1-\alpha_{k}(\phi))\cdot\phi\in C^{r}(\overline{M}0, M1)$.

Then we have the following facts.
(イ) Let$g\in W_{k}\subset\Omega$. Then$\alpha_{k}(g)=0$, whence$P_{k}g=g|_{\overline{M}_{0}}$. Assume $g\in W_{k,0}=W_{k}\cap G_{0}$ with $k\geq 2$ . Then, for(D) Let.
, and getas
any $0\leq s\leq 1$ , we can extend outside of$g\in G_{0}\subset\Omega$$s,$

Proof. Since$p\in M_{0}$$M_{0}$$g_{s}$$M_{0}$$g_{s}\in G_{0}\subset G$$g_{s}=\mathrm{i}\mathrm{d}$is an open ball, we have $g_{s}\in C_{0}^{r}(M_{00}, M)$ . Moreover, for any,$||j_{p}^{(1)}(gs-\mathrm{i}\mathrm{d})|.|_{\varphi}\leq d^{(1)}(g_{S},\mathrm{i}\mathrm{d})\leq d^{1}(g_{s},\mathrm{i}\mathrm{d})\leq 1/k<1$

By Lemma 2.1 applied to $D=M_{0}$ , we see

3.3. A crutial inequlity on(3.3)$M_{0}$.
.

$g_{S}\in \mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}_{0}(M_{0})\subset G_{0}\subset G$. Now put for$g\in\Omega$$\beta_{k}:=\inf_{g\in W_{k}0},\int_{\overline{M}_{0}}\rho(g(p))dp=\inf_{g\in Wk0},\int_{\overline{M}_{0}}||g(p)||dp_{1}d_{P}2\ldots dp_{d}$,where $p=(p_{i})_{i1}^{d}=’ dp=dp_{1}dp_{2}\cdots dp_{d}$ , and $||g(p)||=(\Sigma_{i=1}^{d}g_{i}(p)^{2})^{1}/2$ with $g(p)$$=(g_{i}(p))_{i=}^{d}.1$.

The inequlity in the following lemma reflects the fact thatcompact and is crutial for our proof of Theorem A.

$G_{0}$

is not locally

Lemma 3.2. Let $k\geq 2$ . Then, for any $g\in W_{k,0}=W_{k}\cap G_{0}$ , we have$\int_{\overline{M}_{0}}\rho(g(p))dp>\beta_{k}$Proof. STEP 1. Since

have, at the origin $0,$

$g\in G_{0},$$c^{r_{-\mathrm{C}}}1\mathrm{a}s.\mathrm{S}$$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(g)\subset\overline{M}_{0}$.

and so$g$and the identity map idcontact. Hence$j_{\mathrm{o}(g)}^{k’}=j_{\mathrm{o}(\mathrm{i}\mathrm{d}}^{k’})$($\forall k’\leq r$, finite).

111

We can consider$g-\mathrm{i}\mathrm{d}$as an element of$C^{r}(M_{1}, \mathrm{R}^{d})$($j_{0}^{k’}(g-\mathrm{i}\mathrm{d})=0$$\forall k’\leq r$, then, finite).

We fix $k\geq 2$ , and take $k’=k$ in Case $r=\infty$ , and $k’=r$ in Case $r<\infty$ .
Then there exists an open neighbourhoodof in $M$ such that
$U_{M}$

Now take an(3.4)$||j_{p}^{k’}(g-\mathrm{i}\mathrm{d})||$$<$$\frac{1}{2k}$$(\forall p\in U_{M}\cap M_{0})$$j_{p}^{k’}(g-\mathrm{i}\mathrm{d})$

$=$

$0$

$(\forall p\not\in M_{0})$$\eta=(\eta_{i})_{i=1}^{d}\in C_{0}^{r}(U_{M}\cap M_{0}, \mathrm{R}^{d})$$||j_{p}^{k’} \eta||<\frac{1}{2k}$where diam$\mathrm{O}$satisfying$||j_{p}^{0}\eta||=||\eta||<\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}(M_{1})-\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}(M_{0)}$and$\phi(\overline{M}_{0})\subset M_{1}$$||j_{p}^{k’}(\phi-\mathrm{i}\mathrm{d})||$

.

,

and

denotes the diameter of$(M_{1})$,$<$$M_{1}$. Put $\phi=g-\eta$ . Then,$\phi=\mathrm{i}\mathrm{d}$$\frac{1}{2k}+\frac{1}{2k}=\frac{1}{k}$on$M_{1}\backslash M_{0}$,$(\forall p\in U_{M}\cap M_{0})$.

Hence $\phi\in C^{r}(M_{1}, M_{1})$ and, for any $p\in M_{1}$ ,.

$||j_{p}^{(1)}( \phi-\mathrm{i}\mathrm{d})||_{op}\leq||j_{p}^{k}(’\phi-\mathrm{i}\mathrm{d})||<\frac{1}{k}<1$

Therefore we can apply Lemma 2.1 to. Since, we get$W_{k}\cap G_{\mathit{0}}$and $D=M_{1}$ , and see thatand so $\phi\in W_{k,0}=$$\phi\in$$\phi\in G_{0}=\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}(\overline{M}_{0)}$$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\phi)\subset\overline{M}_{0}$$\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}_{\mathrm{o}(M_{1})}$$\phi$.

STEP 2. Let us compair the following two values:

$A$

$:=$

$B$

$:=$

$\int_{\overline{M}_{0}}\rho(g(p))dp=\int\overline{M}0(_{i=}\sum_{1}^{d}g_{i}(p)^{2})dp1/2$,.

$\int_{\overline{M}_{0}}\rho(\phi(p))dp=\int_{\overline{M}}0(_{i=1}\sum^{d}(gi(p)-\eta i(p))2)dp1/2$

To get $A>B(\geq\beta_{k})$ , it is sufficient to have the following:1$g_{i}(p)|$$\geq$$|g_{i}(p)-\eta_{i}(p)|$$|g_{i_{0}}(p0)|$$>$$|g_{i_{0}}(p\mathrm{o})-\eta_{i0}(p_{0})|$$(\forall i,\forall p\in\overline{M}_{0})$,$(\exists i_{0}, \exists p\mathrm{o}\in\overline{M}0)$,

On the other hand, since the maps and id are sufficiently near to each other, there certainly exist and $p_{0}\in U_{M}\cap M_{0}$ such that
.
on
$g$

$U_{M}\cap M_{0}$$i_{0}$$g_{i_{0}}(p_{0})\neq 0$

112Then there exists a smalland some $\kappa>0$ ,hbourhood $U(p_{0})$ of$\mathrm{n}\mathrm{e}\mathrm{i}\acute{\mathrm{g}}$,$\kappa\geq\epsilon\cdot\eta_{i_{0}}(p)\geq 0$, and$i\neq i_{0}$$\eta_{i}=0$$\epsilon\cdot\eta_{i_{0}}(p_{0})>0$$\epsilon=1$

or-l

.

in such a way thatforsatisfies the condition (3.4) and$\eta=(\eta_{i})_{i=1}^{d}$$C_{0}^{r}(U(p_{0})\cap U_{M}\cap M_{0}, \mathrm{R}^{d})$such that, for$(\forall p\in U(p_{0}))$$\epsilon\cdot g_{i_{0}}(p)>\kappa$

We can choose$p_{0}$$(\forall p)$$\eta_{0}\in$.

Under this choice of the above sufficient condition for $A>B$ holds.
This gives that $A>\beta_{k}$ , which is to be proved.
.D.
$\eta$$\mathrm{Q}.\mathrm{E}$\S 4. A$\tau_{ind}$-neighbourhood of $e\in G$ .

4.1.
ourhood . We define awhich it will be proved thatfor any-neighbourhood, and put, for
,
Let
$U$

$\mathrm{N}\mathrm{e}\mathrm{i}\dot{\mathrm{g}}\mathrm{h}\dot{\mathrm{b}}$$V^{2}\not\subset U$$\tau_{ind}$$g\in\Omega\subset G$$M_{0}^{C}=M\backslash M_{0}$

.

(4.1)

$F_{k}(g):=| \int_{\overline{M}_{0}}\rho((P_{k}g)(p))dp-\beta k|+\int_{M_{0}^{c}}\rho(g(p), \mathrm{i}\mathrm{d}(p))dp$

where$\mathrm{i}\mathrm{d}(p)=p$. Then the following fact is a consequence of Lemma 3.2.

Lemma 4.1. Let $k\geq 2$ . Then,$F_{k}(g)>0(\forall g\in\Omega)$

Proof. Assume that the 2nd term in$M_{0}^{c}$of $e\in G$ , for$V$of $e\in G$ .

$\tau_{ind^{-}}\mathrm{n}\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{b}\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{h}\mathrm{o}\mathrm{o}\mathrm{d}U$, and so$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(g)\subset\overline{M}_{0}$whence$F_{k}(g)$.

is equal to zero. Then,. Then,$g=\mathrm{i}\mathrm{d}$on$g\in G_{0}\subset C^{r}(\overline{M}0, M1)$

$P_{k}g\in C^{r}(\overline{M}0, M1)$$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(P_{kg)}\subset \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(g)\subset\overline{M}_{0}$$\subset$and$C^{r}(M_{1}, M_{1})$,$d^{k^{l}}(P_{k}g, \mathrm{i}\mathrm{d})\leq 1/k<1$,

where $k’=k$ or $=r$ according as $r=\infty$ or $r<\infty$ . Therefore we can apply
Lemma 2.1 to $\phi=P_{k}g$ and $D=M_{1}$ , and see that
. Thenby Lemma 3.2 we get$P_{k}g\in \mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}(\overline{M}_{0)}=G_{0}$$\int_{\overline{M}_{0}}\rho((P_{kg})(p))dp>\beta_{k}$This means that the 1st term in (3.4) of$F_{k}(g)$.

is positive, and so $F_{k}(g)>0$ .

4.2. Proof of Theorem A. Choose non-empty open sets
thatfor $k\geq 2$ . Fix $\gamma>1$ , and for $k\geq 2$ , put$O_{k}$$O_{k}\subset M_{k}\backslash M_{k-1}$$U_{k}:= \{g\in\Omega;F_{k}(g)>\gamma\cdot\int_{\mathit{0}_{k}}\rho(g(p),\mathrm{i}\mathrm{d}(p))dp\}$.

in such a way

113

Since

$G_{n}=\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}(\overline{M}_{n})=\{g\in G;\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(g)\subset\overline{M}_{n}\}$$g=\mathrm{i}\mathrm{d}$$G_{n}$

on$O_{k}$. Then, by Lemma 4.1,

. In particular,$G_{0}=\Omega\cap G_{0}\subset U_{k}$Proof. For any$n\geq 2$$U$is$\tau_{ind}$$\tau_{n}$. Put$U= \bigcap_{k=2}^{\infty}U_{k}\subset\Omega$

Lemma 4.2. The subset, we see that, if $n<k$ , then, and this is -open in$U_{k}\cap G_{n}=\Omega\cap G_{n}$-open in, the intersection

$G$

.

.

$U\cap G_{n}$is$\tau_{ind}$-open in$U \cap G_{n}=\bigcap_{k=2}n(U_{k}\cap G_{n})\cap(\Omega\cap G_{n})$, because.

Now we come to the final stage of the proof of Theorem
for us to prove the following lemma.
Lemma 4.3. There does not exist any.
that$G_{n}$$\mathrm{A}$, and it is enough$\tau_{ind^{-}}neighb_{\mathit{0}}.u.rh,oodV$of $e\in G$ such$V^{2}\subset U$. Since $V\cap G_{0}$ isbe such thatform a basis of -neighbourhoods of $e\in G_{0}$ , there exsits a. Then. Put

Proof. Suppose the contrary and let$\tau_{0}$-open andsuch that$W_{k,0}’ \mathrm{s}$$W_{k,0}$$V^{2}\subset U$$V$$\tau_{0}$$V\cap G_{0}\supset W_{k,0}$$V_{k}=V\cap \mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}_{0}(o_{k})$$W_{k,0}V_{k}\subset V^{2}\subset U\subset Uk\subset\Omega$.

Hence, for any $g\in W_{k,0},$ $h\in V_{k}$ ,.

$F_{k}(g \mathrm{o}h)>\gamma\cdot\int_{\mathit{0}_{k}}.\rho((g\mathrm{o}h)(p),\mathrm{i}\mathrm{d}(p))dp$

Note that$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(g)\subset\overline{M}_{0},$$g\mathrm{o}h=g$on$\overline{M}_{0},$$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(h)\subset M_{k}\backslash M_{k-1}$$g\mathrm{o}h=h$on$O_{k},$, and that$g\mathrm{o}h=\mathrm{i}\mathrm{d}$anywhere else.

Hence$-| \int_{\overline{M}_{0}}\rho((Pkg)(p))dp-\beta_{k}|>(\gamma-1)\cdot\int_{\mathit{0}_{k}}\rho(h(p), \mathrm{i}\mathrm{d}(p))dp$.

Further, since $g\in W_{k,0}=W_{k}\cap G_{0}$ , we have $P_{k}g=g$ , and the above inequality
turns out to be$\int_{\overline{M}_{0}}\rho(g(p))dp-\beta k>(\gamma-1)\cdot\int_{\mathit{0}_{k}}\rho(h(p), \mathrm{i}\mathrm{d}(p))dp$

Taking the infimum over $g\in W_{k,0}$ , we get$0$on the left hand side and so$0= \int_{\mathit{0}_{k}}\rho(h(p), \mathrm{i}\mathrm{d}(p))dp$

Hence$h=\mathrm{i}\mathrm{d}$. This means that.

.

$V\cap \mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}_{0}(o_{k})=\{\mathrm{i}\mathrm{d}\}$. A contradiction.

114
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