数理解析研究所講究録
1531 巻 2007 年 10-19

10

GROUPS OF MEASURE-PRESERVING HOMEOMORPHISMS AND
VOLUME-PRESERVING DIFFEOMORPHISMS OF NONCOMPACT
MANIFOLDS AND MASS FLOW TOWARD ENDS
矢ケ崎達彦 (TATSUHIKO YAGASAKI)

京都工芸繊維大学

KYOTO INSTITUTE OF TECHNOLOGY

1. SPACES OF MEASURES AND GROUPS OF MEASURE-PRESERVING HOMEOMORPHISMS
Suppose $M$ is a connected -manifold possibly with boundary. The symbol$n$$\mathcal{B}(M)$de-notes the -algebra of Borel subsets of $M$ .
$\sigma$

Definition 1.1. A Radon measure on $M$ is a Borel measure on $M$ such that $\mu(K)<\infty$
for any compact subset $K$ of $M$ . A Radon measure is said to be good if
$\mu$

$\mu$

for any point of $M$ and(ii) $\mu(U)>0$ for any nonempty open subset(i)$\mu(p)=0$

$p$

$U$

of $M$ .

Definition 1.2.
(1)$\mathcal{M}_{g}^{\partial}(M)$denotes the set of good Radon measures on $M$ wth $\mu(\partial M)=0$ .

(2) The weak topology

$w$

on

$\mathcal{M}_{g}^{\partial}(M)$$\Phi_{f}$:is the weakest topology such that the function$\lambda 4_{g}^{\partial}(M)arrow \mathbb{R}:\Phi_{f}(\mu)=\int_{M}fd\mu$

is continuous for any continuous function$f$:$Marrow \mathbb{R}$with compact support.

denote the group of homeomorphisms of $M$ with the compact-open topology.
Letdenote theandAny subgroup of $H(M)$ is equipped with the subspace topology.
connected component and the path-component of the identity in .
$\mathcal{H}(M)$$\mathcal{G}_{0}$$\mathcal{G}$$\mathcal{G}_{1}$$\mathcal{G}$is a good Radon measures on $M$ . The subgroups
are defined as follows:

Definition 1.3. Suppose$?t(M;\mu- \mathrm{r}\mathrm{e}\mathrm{g})\subset \mathcal{H}(M)$(1)$\mathcal{H}(M;\mu)\subset$

is -preserving if $\mu(h(B))=\mu(B)$ for any $B\in B(M)$ .
consisting of -preserving homeomordenotes the subgfoup of$h\in \mathcal{H}(M)$$\mathcal{H}(M;\mu)$$\mu$

phisms of $M$ .

$\mu$$\mathcal{H}(M)$

$\mu$

11
(2)

$h\in \mathcal{H}(M)$is -biregular if$\mu$$\mathcal{H}(M;\mu- \mathrm{r}\mathrm{e}\mathrm{g})$“$\mu(h(B))=0$ iff $\mu(B)=0$ for any $B\in B(M)$ ”.

denotes the subgroup of$\mathcal{H}(M)$consisting of -biregular homeomor$\mu$

phisms of $M$ .

The topological groupwhere$h_{*}\mu\in \mathcal{M}_{g}^{\partial}(M)$$\mathcal{H}(M)$acts continuously on the spaceis defined by $(h_{*}\mu)(B)=\mu(h^{-1}(B))$

by $h\cdot\mu=h_{*}\mu$ ,$\mathcal{M}_{\mathit{9}}^{\partial}(M)_{w}$$(B\in \mathcal{B}(M))$. The subgroupcoincides with the stabilizer of under this action.
We also use the following terminologies.

$\mathcal{H}(M;\mu)$$\mu$

Definition 1.4. Suppose $X$ is a space and
(1)

$A$

is a subspace of $X$ .

is a SDR (strong deformation retract) of $X$ if there exists a homotopy

and $\varphi_{t}|_{A}=id_{A}(0\leq t\leq 1)$ .
is HD (homotopy dense) in $X$ if there exists a homotopy

such that
(2)

$A$

$A$

$\varphi_{0}=id_{X}$$\varphi_{0}=id_{X},$$\varphi_{t}$: $Xarrow X$$\varphi_{1}(X)=A$$\varphi_{t}$: $Xarrow X$ such thatand $\varphi_{t}(X)\subset A(0<t\leq 1)$ .

In both cases the inclusion map $A\subset X$ is a homotopy equivalence with a homotopy

inverse$\varphi_{1}$: $Xarrow A$ .
2. COMPACT CASE–

FATHI’S RESULTS

Suppose $M$ is a compact connected -manifold. The von Neumann-Oxtoby-Ulam theo$n$

rem [10] asserts that the above action is essentially transitive.
Theorem 2.1. (von Neumann-Oxtoby-Ulam) Suppose $M$ is compact andwith $\nu(M)=\mu(M)$ . Then there exists
such that $h_{*}\mu=\nu$ .

$\mu,$$\nu\in \mathcal{M}_{\mathit{9}}^{\partial}(M)$$h\in \mathcal{H}_{\partial}(M)_{0}$A parametrized version of this theorem was obtained by A. Fathi [6]. Let
We need to restrict ourselves to the following subspace ofDefinition 2.1.
denotes the subset ofwhich has the same total mass and the same null sets as$\mathcal{M}_{g}^{\partial}(M;\mu- \mathrm{r}\mathrm{e}\mathrm{g})$The action of
the subspace$\mathcal{H}(M)$on$\mathcal{M}_{g}^{\partial}(M)$$\mathcal{M}_{g}^{\partial}(M;\mu- \mathrm{r}\mathrm{e}\mathrm{g})_{w}$

$\pi$

:

$\mathcal{M}_{g}^{\partial}(M)$$\mathrm{A}\mathrm{t}_{g}^{\partial}(M)$$\mu$consisting of$\nu\in \mathcal{M}_{g}^{\partial}(M)$.

. We obtain the orbit map

:

.

.

restricts to the action of the subgroup$\mathcal{H}(M;\mu- \mathrm{r}\mathrm{e}\mathrm{g})arrow \mathcal{M}_{g}^{\partial}(M;\mu- \mathrm{r}\mathrm{e}\mathrm{g})_{w}$$\mu\in \mathcal{M}_{g}^{\partial}(M)$$\pi(h)=h_{*}\mu$.

$\mathcal{H}(M;\mu- \mathrm{r}\mathrm{e}\mathrm{g})$on12Theorem 2.2. (A. Fathi [6], 1980)(1) The orbit map(2)

Suppose $M$ is a compact connected n-manifold.

admits a section$\pi$$\sigma$:$\mathcal{M}_{g}^{\partial}(M;\mu- reg)_{w}arrow \mathcal{H}_{\partial}(M;\mu- reg)_{1}\subset \mathcal{H}(M;\mu- reg)$$\mathcal{H}(M;\mu- reg)\cong \mathcal{H}(M;\mu)\mathrm{x}\mathcal{M}_{g}^{\partial}(M;\mu- reg)_{w}$(3)

Weak $HD$$SDR$$\mathcal{H}(M;\mu)$$\mathcal{H}(M, \mu- reg)$$\subset$$\mathcal{H}(M)$$\subset$$\mathrm{O}\cap D$(4) $n=2$Corollary 2.1. (Yagasaki [13].)

$ANR$

$ANR$

$ANR$

$\mathcal{H}(M;\mu)$

is an$\ell_{2}$-manifold.

Corollary 2.1 easily follows from the next topological characterization of$\ell_{2}$-manifold.

Theorem 2.3. (T. Dobrowolski-H. Torutczyk [5])A topologi $cal$ group is a -manifold iff is a separable, non locally compact, completely
$G$

$G$

$p_{2}$

metrizable $ANR$ .

3. NON-COMPACT$\mathrm{c}_{\mathrm{A}\mathrm{S}\mathrm{E}}-\mathrm{R}$. BERLANGA’S RESULTS

Suppose $M$ is a noncompact connected -manifold possibly with boundary. First we$n$introduce some notations on the ends of M.

Definition 3.1.

of $M$ is a function which assigns to each compact subset $K$ of $M$ a.
connected component $e(K)$ of $M-K$ such that $e(K_{1})\supset e(K_{2})$ if(2) $E(M)$ denotes the space of ends of $M$ .
$\overline{M}=M\cup E(M)$ denotes the end compactification of $M$ .
(3) The topology ofis described by the following conditions:(i) $M$ is an open subspace of .
(ii) Fundamental open neighborhoods of $e\in E(M)$ is given by$N(e, K)=e(K)\cup\{e’\in E(M)|d(K)=e(K)\}$( $K\subset M$ : compact)(1) An end$e$$K_{1}\subset K_{2}$$\overline{M}$$\overline{M}$is a compact metrizable space and $E(M)$ is a O-dim compact subset of.
Let$\overline{M}$$\mu\in \mathcal{M}_{g}^{\partial}(M)$$\overline{M}$

.

.

13
Definition 3.2.
(1) $e\in E(M)$ is -finite if $\mu(e(K))<\infty$ for some compact subset $K$ of $M$ (i.e.,
$\mu$

$e$

has

a neighborhood with finite -mass).
$\mu$(2) $E_{f}(M;\mu)$ denotes the subspace of -finite ends of $M$ .
$\mu$

The von Neumann-Oxtoby-Ulam theorem is extended to the non-compact case in thefollowing form.
Theorem 3.1. (R. Berlanga [1], 1983)
Suppose$\mu,$$\nu\in \mathcal{M}_{g}^{\partial}(M)$

has same total mass and same finite ends. Then there enists

with $h_{*}\mu=\nu$ .

$h\in \mathcal{H}_{\partial}(M)_{1}$A parametrized version of this theorem is obtained recently by R. Berlanga [3]. Simple
examples show that the weak topology

$w$

on

$\mathcal{M}_{\mathit{9}}^{\partial}(M;\mu- \mathrm{r}\mathrm{e}\mathrm{g})$is not enough to extend thesection theorem (Theorem 2.2 (1)) to the noncompact case. R. Berlanga introduces a little
stronger topology called the finite-end weak topology, which turns out to be the correct
topology for this purpose.
Definition 3.3. (Finite-end weak topology)(1)$\mathcal{M}_{g}^{\partial}(M;\mu- \mathrm{e}\mathrm{n}\mathrm{d}- \mathrm{r}\mathrm{e}\mathrm{g})$Letdenotes the subset of$\mu\in \mathcal{M}_{g}^{\partial}(M)$$\nu\in \mathcal{M}_{g}^{\partial}(M)$.

which has the same total

mass, same null sets and same finite ends as .
$\mu$(2) Consider the inclusions.

$M\subset\iota M\cup E_{f}(M;\mu)\subset\overline{M}$The map induces the natural map$\iota$$\iota_{*}$:$\mathcal{M}_{g}^{\partial}(M;\mu- \mathrm{e}\mathrm{n}\mathrm{d}- \mathrm{r}\mathrm{e}\mathrm{g})arrow \mathcal{M}_{g}^{\partial}(M\cup E_{f}(M;\mu))_{w}$(3) The finite-end weak topology $ew$ onsuch that
The space$\iota_{*}$$\nu\mapsto\overline{\nu}=\iota_{*}\nu$$\mathcal{M}_{g}^{\partial}(M;\mu- \mathrm{e}\mathrm{n}\mathrm{d}- \mathrm{r}\mathrm{e}\mathrm{g})$is the weakest topology

is continuous.

$\mathcal{M}_{g}^{\partial}(M;\mu- \mathrm{e}\mathrm{n}\mathrm{d}- \mathrm{r}\mathrm{e}\mathrm{g})_{ew}$Definition 3.4.

:$\mathcal{H}(M;\mu- \mathrm{e}\mathrm{n}\mathrm{d}- \mathrm{r}\mathrm{e}\mathrm{g})$admits the contraction $\varphi_{t}(\nu)=(1-t)\nu+t\mu(0\leq t\leq 1)$ .
denotes the subgroup of$\mathcal{H}(M)$consisting of$h\in \mathcal{H}(M)$which preserves -null sets and -finite ends of $M$ .
$\mu$

The group$\mathcal{H}(M;\mu- \mathrm{e}\mathrm{n}\mathrm{d}- \mathrm{r}\mathrm{e}\mathrm{g})$$\mu$acts continuously on$\mathcal{M}_{g}^{\partial}(M;\mu- \mathrm{e}\mathrm{n}\mathrm{d}- \mathrm{r}\mathrm{e}\mathrm{g})_{ew}$by $h\cdot\nu=h_{*}\nu$

and we obtain the orbit map$\pi:\mathcal{H}(M;\mu- \mathrm{e}\mathrm{n}\mathrm{d}- \mathrm{r}\mathrm{e}\mathrm{g})arrow \mathcal{M}_{g}^{\partial}(M;\mu- \mathrm{e}\mathrm{n}\mathrm{d}- \mathrm{r}\mathrm{e}\mathrm{g})_{\mathrm{e}w}$: $\pi(h)=h_{*}\mu$ .

14Theorem 3.2. (R. Berlanga [3], 2003)(1) The orbit map$\sigma$

(2)

:

$\pi$

has a section$\mathcal{M}_{g}^{\partial}(M;\mu- end- reg)_{ew}arrow \mathcal{H}_{\theta}(M;\mu- end- reg)_{1}\subset \mathcal{H}(M;\mu- end- reg)$.

$\mathcal{H}(M;\mu- end- reg)\cong \mathcal{H}(M;\mu)\cross \mathcal{M}_{g}^{\partial}(M;\mu- end- reg)_{ew}$

(3)

$SDR$

$7t(M;\mu)$$\subset$$\mathcal{H}(M, \mu- end- reg)$

The relation between the two groups$\subset \mathcal{H}(M)$$\mathcal{H}(M, \mu- \mathrm{e}\mathrm{n}\mathrm{d}- \mathrm{r}\mathrm{e}\mathrm{g})\subset \mathcal{H}(M)$is not known for $n\geq 3$ .

In $n=2$ we can apply our results on homeomorphism groups of noncompact 2-manifolds$[11, 12]$ to obtain the following conclusions.

$\ell_{2}$

-MFD

ANR

ANR

HD

The main statement$\mathcal{H}(M,\mu- \mathrm{e}\mathrm{n}\mathrm{d}- \mathrm{r}\mathrm{e}\mathrm{g})_{0}$ing arguments. When $M$ is a PL -manifold,$n$consisting of

can be derived by the follow-$\mathcal{H}(M)_{0}$$\subset$$\mathcal{H}^{\mathrm{P}\mathrm{L}}(M)$denotes the subgroup of$\mathcal{H}(M)$-homeomorphisms of $M$ .

$\mathrm{P}\mathrm{L}$(1) Suppose $M$ is a noncompact connected 2-manifold. Then(i) $M$ admits a PL-structure.
(ii)$\mathcal{H}^{\mathrm{P}\mathrm{L}}(M)_{0}$is HD in$\mathcal{H}(M)_{0}$for any(2) Suppose $M$ is a PL -manifold and$n$be$\mathrm{i}\mathrm{s}o$toped to a new$\mathrm{P}\mathrm{L}$$\mathrm{P}\mathrm{L}$-structure on $M[12]$ , cf. [7].

$\mu\in \mathcal{M}_{g}^{\partial}(M)$-structure so that

. Then the

Suppose $M$ is a noncompact connected -manifold and.
4.1. Topological Vector SpaceFirst we define a topological vector space-structure on $M$ can.

$\mathcal{H}^{\mathrm{P}\mathrm{L}}(M)\subset \mathcal{H}(M;\mu- \mathrm{r}\mathrm{e}\mathrm{g})[15]$4. MASS FLOW TOWARD ENDS ON NON-COMPACT$n$$\mathrm{P}\mathrm{L}$$n$-MANIFOLDS$\mu\in \mathcal{M}_{g}^{\partial}(M)$.

$V_{\mu}(M)$ends by -preserving homeomorphisms.
$\mu$$V_{\mu}(M)$, which parametrizies mass flows toward15
Definition 4.1.

{ $B\in B(M)|$ Fr: Compact}(1)$\mathcal{B}_{c}(M)=$(2)$W(M)$ denotes the space of all functions $a:B_{c}(M)arrow \mathbb{R}$$B$.

(i) $W(M)$ is a real vector space under the addition and the scalar product of realvalued functions.
(ii) $W(M)$ is equipped with the product topology,i.e., the topology induced by the projections$\pi_{C}:W(M)arrow \mathbb{R}$(3)$V(M)=\{a :: $\pi_{C}(a)=a(C)$$(C\in B_{\mathrm{c}}(M))$.

B_{c}(M)arrow \mathbb{R}|(*)_{1}, (*)_{2}, (*)_{3}\}$$(*)_{1}C,$ $D\in B_{c}(M),$$Cl(C-D),$ $Cl(D-C)$ : compact$(*)_{2}C,$ $D\in \mathcal{B}_{\mathrm{c}}(M),$$C\cap D=\emptyset$$a(C)=a(D)$$\Rightarrow$$a(C\cup D)=a(C)+a(D)$$\Rightarrow$$(*)_{3}a(M)=0$$V_{\mu}(M)=\{a\in V(M)|(*)_{4}\}$$(*)_{4}C\in B_{c}(M),$$V(M)$

and$V_{\mu}(M)$$\mu(C)<\infty$$\Rightarrow$$a(C)=0$

are linear subspaces of $W(M)$ , which are equipped with thesubspace topology.

4.2. Mass flow homomorphism toward ends
Next we define a continuous group homomorphism
measures a mass moved toward ends by eachhas a unique extension.

$J:\mathcal{H}_{E}(M, \mu)arrow V_{\mu}(M)$$J$:$\mathcal{H}_{E}(M, \mu)arrow V_{\mu}(M)$$h\in \mathcal{H}_{E}(M, \mu)$$h\in \mathcal{H}(M)$.
, which. Let $E=E(M)$ . Each$\overline{h}\in \mathcal{H}(\overline{M})$

Definition 4.2.
(1)

(2)

$\mathcal{H}_{E}(M, \mu)=\{h\in \mathcal{H}(M, \mu)|\overline{h}|_{E}=id_{E}\}$(a subgroup of$\mathcal{H}(M,$ $\mu)$)$J:H_{E}(M, \mu)\ni h-J_{h}\in V_{\mu}(M)$$J_{h}(C)=\mu(C-h(C))-\mu(h(C)-C)$

The group$\mathcal{H}_{E}(M, \mu)$$J:\mathcal{H}_{E}(M, \mu)arrow V_{\mu}(M)$acts continously on$V_{\mu}(M)$$(C\in B_{\mathrm{c}}(M))$

by $h\cdot a=J_{h}+a$ and the homomorphism

coincides with the orbit map at$\mathrm{O}\in V_{\mu}(M)$.

16
Theorem 4.1. (Yagasaki [14])(1) The map

with(2) (i)$\mathrm{K}\mathrm{e}\mathrm{r}J$$J$

admits a section$s(\mathrm{O})=id_{M}$$s$:$V_{\mu}(M)arrow \mathcal{H}_{\partial}(M, \mu)_{1}\subset \mathcal{H}_{E}(M, \mu)$(i.e., $Js=id$ ).
(ii)$\mathcal{H}_{E}(M;\mu)\cong \mathrm{K}\mathrm{e}\mathrm{r}J\cross V_{\mu}(M)$contains the subgroup$\mathcal{H}^{c}(M;\mu)$$\mathrm{K}\mathrm{e}\mathrm{r}J\subset \mathcal{H}_{E}(M;\mu)$: a $SDR$of -preserving homeomorphisms with compact$\mu$

support. Our next aim is the study of relation between these groups.

5. SPACES OF VOLUME FORMS ANDGROUPS OF VOLUME-PRESERVING DIFFEOMORPHISMS

Suppose $M$ is a connected oriented$C^{\infty}n$-manifold without boundary.

Deflnition 5.1.
(1) $D^{+}(M)$ deotes the group of orientation-preserving diffeomorphisms of $M$ with thecompact-open$C^{\infty}$-topology.

(2) For a positive volume form

(3)$\omega$

on $M$ ,$D(M;\omega)$

denotes the subgroup of -preserving diffeomorphisms of $M$ .

$\mathcal{V}^{+}(M)_{w}$denotes the space of positive volume forms on $M$ equipped with the weak$C^{\infty}$For$\omega$

topology.
$m\in(\mathrm{O}, \infty],$$\mathcal{V}^{+}(M, m)_{w}=\{\mu\in \mathcal{V}^{+}(M)|\mu(M)=m\}$(the weak$C^{\infty}$

topology).

Eachdetermines a unique good Radon measure on $M$ , which is denotedby the same symbol . This defines an inclusion.
$\mu\in \mathcal{V}^{+}(M)$$\mathcal{V}^{+}(M)\subset \mathcal{M}_{g}^{\partial}(M)$$\mu$

The topological group $D^{+}(M)$ acts continuously on
and$h_{*}\mu(=(h^{-1})^{*}\mu)$ .
The subgroup $D(M;\omega)$ coincides with the stabilizer of$\mathcal{V}^{+}(M)_{w}$$\mathcal{V}^{+}(M, m)_{w}$$\omega$by $h\cdot\mu=$under thisaction.

5.1. Compact case.
Suppose $M$ is a compact connected oriented$C^{\infty}n$-manifold without boundary. Moser’s

theorem [9] implies the transitivity of this action and its parametrized version.

17
Theorem 5.1. Suppose $M$ is a compact connected oriented(1) (Transitivity) For any , \iota ノ$\mu$$\in \mathcal{V}^{+}(M, m)$(2) (Parametrized version) Let ru$C^{\infty}$n-manifold.

there exists $h\in D(M)_{1}$ such that $h_{*}\mu=\nu$ .
. Then the orbit map: $D^{+}(M)arrow$$\in \mathcal{V}^{+}(M;m)$$\pi$, admits a section a : $\mathcal{V}^{+}(M;m)_{w}arrow D(M)_{1}\subset D^{+}(M)$ .

$\mathcal{V}^{+}(M;m)_{w\rangle}\pi(h)=h_{*}\omega$

5.2. Non-compact case.
Suppose $M$ is a non-compact connected$C^{\infty}n$-manifold without boundary. Recall that

$E=E(M)$ is the space of ends of $M$ and $\overline{M}=M\cup E(M)$ is the end compactification. For
denotes the subspace of $E(M)$ consisting of -finite ends of $M$ .
of $M$ . Each $h\in D(M)$ has a unique extension$\overline{h}\in \mathcal{H}(\overline{M})$$\mu\in \mathcal{V}^{+}(M),$$E_{f}(M, \mu)$$\mu$

Definition 5.2. Suppose $F\subset E(M)$ is an open subset.
(a subgroup of $D^{+}(M)$ )(1) $D^{+}(M;F)=\{h\in D^{+}(M)|\overline{h}(F)=F\}$(2) $\mathcal{V}^{+}(M;F)=\{\mu\in \mathcal{V}^{+}(M)|E_{f}(M, \mu)=F\}$$\mathcal{V}^{+}(M;m, F)=\mathcal{V}^{+}(M;m)\cap \mathcal{V}^{+}(M;F)$$\mathcal{M}_{g}^{\partial}(M;F)=\{\mu\in \mathcal{M}_{g}^{\partial}(M)|E_{f}(M, \mu)=F\}$(3) (Finite-end weak topology)$\iota$The inclusion $M\subset M\cup F(\subset\overline{M})$ induces the injection$\iota_{*}$$\iota_{\#}$:$\mathcal{V}^{+}(M;m, F)$$\subset$$\mathcal{M}_{g}^{\partial}(M;F)$$\nu$$\mapsto$$\nu$

The finite-end weak topology $ew$ on
the maps$\iota_{\#}$—$\mathcal{M}_{g}(M\cup F)_{w}$$rightarrow$$\mathcal{V}^{+}(M;m, F)$.

$\overline{\nu}=\iota_{*}\nu$

is the weakest topology such thatand $id:\mathcal{V}^{+}(M;m, F)arrow \mathcal{V}^{+}(M;m, F)_{w}$ are continuous.

The group $D^{+}(M;F)$ acts continuously on$\mathcal{V}^{+}(M;m, F)_{ew}$by $h\cdot\mu=h_{*}\mu$ and thestabilizer ofcoincides with the subgroup $D(M;\omega)$ . Transitivity of thisaction was verified by R. E. Greene-K. Shiohama [8].
$\omega\in \mathcal{V}^{+}(M;m, F)_{w}$Theorem 5.2. (R. E. Greene-K. Shiohama [8])For any$\mu,$$\nu\in \mathcal{V}^{+}(M;m, F)$

there exists $h\in D(M)_{1}$ such that $h_{*}\mu=\nu$ .

-modification of R. Berlanga’s argument [3] leads to the parametrized version ofthis theorem.
A$C^{\infty}$

18
Theorem 5.3. (Yagasaki [15])are maps such:is a paracompact Hausdorff space andthat $\mu_{p}(M)=\nu_{p}(M)(p\in P)$ . Then there exists a map $h:Parrow D(M)_{1}$ such that, then $h_{p}=id_{M}$ .
and (ii) if $p\in P$ and(i)

Suppose$P$$Parrow \mathcal{V}^{+}(M;F)_{ew}$$\nu$$\mu,$$h_{p_{*}}\mu_{\mathrm{p}}=\nu_{p}(p\in P)$

Corollary 5.1. Let$\omega\in \mathcal{V}^{+}(M;m, F)$(1) The orbit map$\sigma$$\mu_{p}=\nu_{p}$

$\pi$

.

: $D^{+}(M;F)arrow \mathcal{V}^{+}(M;m, F)_{ew},$ $\pi(h)=h_{*}\omega$ , admits a section: $\mathcal{V}^{+}(M;m, F)_{ew}arrow D(M)_{1}\subset D^{+}(M;F)$ .
(ii) $D(M;\omega)\subset D^{+}(M;F)$ : a $SDR$(2) (i) $D^{+}(M;F)\cong \mathcal{V}^{+}(M’, m, F)_{\epsilon w}\cross D(M;\omega)$n-manifolds.
5.3. Mass flow toward ends on non-compact-manifold without boundary andSuppose $M$ is a non-compact connected
and a continuous group homomor. The topological vector space $V(M),$$C^{\infty}$$C^{\infty}n$$\omega\in$$V_{\omega}(M)$$\mathcal{V}^{+}(M)$.

$\mathrm{F}\mathrm{o}\mathrm{r}h\in D_{E}(M;\omega)$$\mathrm{p}\mathrm{h}\mathrm{i}\mathrm{s}\mathrm{m}J^{\omega}:D_{E}(M,\omega)arrow V_{\omega}(M)\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{e}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{d}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{n}\S 4.1\mathrm{a}\mathrm{n}\mathrm{d}\S 4.2$

$J_{h}^{\omega}:B_{c}(M)arrow \mathbb{R}$:$J_{h}^{\omega}(C)=\omega(C-h(C))-\omega(h(C)-C)(C\in B_{c}(M))$

The group $D_{E}(M,\omega)$ acts continuously on

The map$J^{\omega}$by$V_{\omega}(M)$$(h\in D_{E}(M, \omega),$ $a\in V_{\{v}(M))$$h\cdot a=J_{h}^{(d}+a$.

.

: $D_{E}(M,\omega)arrow V_{\omega}(M)$ coincides with the orbit map at $0\in V_{d}‘(M)$ .

Definition 5.3. For two maps$\mu,$$\nu$:

for any $p\in P$ there exists a neighborhoodonsuch that

$U$

of in

$M-K\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{y}q\in U$$\mu_{q}=\nu_{q}$we write as$Parrow \mathcal{V}^{+}(M)$

$p$

$P$

$\mu\sim \mathrm{c}\nu$

and a compact subset $K\subset M$.

is a paracompact Hausdorff space andTheorem 5.4. Suppose, $(\mu-\nu)(M)=0$ and: $Parrow V(M)$ are maps such that$P$$\mu,$$\nu$:$Parrow \mathcal{V}^{+}(M)_{w}$,$(p\in P)$.

$a_{p}\in V_{\mu \mathrm{p}}(M)$$\mu\sim\nu$

$a$

if

$c$

Then there exists a map $h:Parrow D(M)_{1}$ such that(i)$h_{\mathrm{P}*}\mu_{\mathrm{p}}=\nu_{\mathrm{p}}(p\in P)$

Corollary 5.2. Let

if $p\in P$ andand (ii)$\omega\in \mathcal{V}^{+}(M)$$\mu_{p}=\nu_{p}$$J_{h_{\mathrm{p}}}^{\mu_{\mathrm{p}}}=a_{\mathrm{p}}$.

.

: $D_{E}(M,\omega)arrow V_{\omega}(M)$ admits a section$D_{E}(M,\omega)(J^{\omega}s=id_{V_{\omega}(M)})$ with.

(1) The map, then$J^{\omega}$$s$:$V_{\omega}(M)arrow D(M, \omega)_{1}\subset$$s(\mathrm{O})=id_{M}$(2) (i)$D_{E}(M;\omega)\cong \mathrm{K}\mathrm{e}\mathrm{r}J^{v}‘\cross V_{\omega}(M)$(ii)$\mathrm{K}\mathrm{e}\mathrm{r}J^{\omega}\subset D_{E}(M;\omega)$

Our next aim is to study the relation between two groups: a $SDR$$D^{c}(M;\omega)\subset \mathrm{K}\mathrm{e}\mathrm{r}J^{\omega}$.

19
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$\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{h}.\mathrm{G}\mathrm{T}/0507328)$$\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{h}.\mathrm{G}\mathrm{T}/0512231$

Tatsuhiko Yagasaki
Division of Mathematics, Department of Comprehensive Science,
Kyoto Institute of Technology,
Matsugasaki, Sakyoku, Kyoto 606-8585, Japan-mail address: yagasak i Qkit. ac. jp$E$