$\mathrm{c}.\mathrm{l}$ & (Naoyuki Koike) (Science University of Tokyo) 1. $[\mathrm{l}],[\mathrm{p}],[\mathrm{s}1],[\mathrm{s}

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1 $\mathrm{c}.\mathrm{l}$ & (Naoyuki Koike) (Science University of Tokyo) 1. $[\mathrm{l}],[\mathrm{p}],[\mathrm{s}1],[\mathrm{s}2]$ 1960 ( ) 1980 Terng ([Te2]) 2 (F),(P) (F) (P) (F) $(\mathrm{p},)$ 2. parallel transport $[0, 1]\cross G$ $H^{0_{-}}$ $H^{0}([0,1], \text{ })$ ( ) $\phi$ $H^{0}([0,1], \text{ })$ $L^{2}$ Ad(G) $\mathrm{c}.\mathrm{l}$. Terng G. Thorbergsson([TeTh]) $\pi$ $N=G/K$ $G/K$ $H^{0}$ $\phi$ ([0, 1], )\rightarrow G parallel transport (parallel transport $\pi\circ\phi$ $\overline{\phi}$ 5 ) $N$ $M$ $\tilde{\phi}^{-1}(m)$ $H^{0}([0,1], \text{ })$ $M$ $\overline{\phi}^{-1}(m)$ ( (F) )

2 $\mathrm{c}.\mathrm{l}$ $\overline{hi}$ 163 $1\backslash$.Terng([KiTe]) 1993 C.King- $\leq$ \mu 1 $-/l_{2}, \leq\cdots<0<\cdots$ $\leq\lambda_{2}\leq\lambda_{1}$ $\lim_{sarrow 1-0}(\sum_{i=1}^{\infty}\lambda_{i}^{s}-\sum_{i=1}^{\infty}$ \mu ( $1\backslash$ ) ( ) D3A $i$ A $\lambda_{i}=\mu_{i}=0$ $\mathrm{t}\mathrm{r}_{\zeta}a_{v}$ $H_{\zeta}$ I $\mathrm{t}\mathrm{r}_{\zeta}a_{v}=<h_{\zeta},$ $(\forall v\in T^{[perp]}M)$ $v>$ $H_{\zeta}=0$ ( je ) $ H_{\zeta} $ $M$ $\mathrm{c}\mathrm{m}\mathrm{c}$ ( ) $M$ $M$ $\phi^{-1}(m)$ $M$ $M$ CMC $\phi^{-1}(m)$ $\phi$ CMC parallel transport ([K1]) $\{\lambda_{i} i=1,2, \cdots\}$ $ \lambda_{\mathrm{i}} > \lambda_{i+1} $ ( or ) $\lambda_{i}=-\lambda_{i+1}>0$ $\lambda_{i}$ $\sum_{i=1}^{\infty}$ m mi\lambda ( ) A Tr $i$ $\lambda_{i}=0$ ( ) $x= \lim_{karrow\infty}\sum_{\lambda\in\lambda_{k}}<x,$ $e_{\lambda}>e_{\lambda}(\{e_{\lambda}\}_{\lambda\in\lambda}$, $\Lambda_{k}=\{\lambda\in\Lambda <x, e_{\lambda}> >\frac{1}{k}.\})$ Tr $A_{v}=0$ ( ) Tr $H$ [KiTe] 1 $A_{v}=<H,$ $v>(\forall v\in T^{[perp]}M)$ $M$ CMC $ H $ $G/K$ curvature adapted $M$ $\overline{\phi}^{-1}(m)$ $M$ $G/K$ $ \backslash \mathrm{s}$ (0 ) $M$ $G/K$ $\overline{\phi}^{-1}(m)$ $M$ $\sum\infty$ (\lambda $-\mu_{i}$ ) ) $i=1$ E.Heintze-X.Liu-C Olmos([HLO]) ( \mu 1 $\leq-\mu_{2}\leq\cdots<0<\cdots\leq\lambda_{2}\leq\lambda_{1}$ ( ) A ( $D_{r}A_{v}\text{ }$ $i$ $\lambda_{i}=\mu_{i}=0$ $\mathrm{t}\mathrm{r}_{r}a_{v}$ $H_{r}$ $\mathrm{t}\mathrm{r}_{r}a_{v}=<h_{r},$ $v>(\forall v\in T^{[perp]}M)$ $H_{r}$

3 164 CMC $V$ $N$ $\psi$ $N$ $M$ $\psi^{-1}(m)$ $N$ $M$ CMC $\psi^{-1}(m)$ CMC ( ) $A$ $\mathrm{t}\mathrm{r}_{\zeta}a\neq \mathrm{t}\mathrm{r}a$ $\mathrm{t}\mathrm{r}_{\zeta}a\neq \mathrm{t}\mathrm{r}_{r}a$ $A_{v}\neq \mathrm{h}a_{v}$, $\mathrm{t}\mathrm{r}_{\zeta}a_{v}\neq \mathrm{t}\mathrm{r}_{r}a_{v}$ $\mathrm{c}.\mathrm{l}$. Terng G. Thorbergsson([TeTh]) equifocal ( 2 ) $\mathrm{a}([\mathrm{t}\mathrm{e}\mathrm{t}\mathrm{h}])$ $M$ $M$ equifocal $\tilde{\phi}^{-1}(m)$ equifocal $\mathrm{l}\mathrm{i}\mathrm{u}([\mathrm{h}\mathrm{l}2])$ E. Heintze X. ( ) 2 ( ) G. Thorbergsson 3 U. Christ([C]) 2 equifocal parallel transport Hcintze-Liu [TeTh] 1 Terng-Thorbergsson equifocal? ([K2]) $N=G/K$ parallel $[0, 1]\cross G$ $H^{0}$ ([0, 1], 9)( ) $G/K$ $\phi$ transport $\pi$ $\pi\circ\phi$ $\tilde{\phi}$ ( ) 1\

4 165 ( ) equifod ( ) [K2] $\mathrm{b}$ ([K2]) $M$ $\overline{\phi}^{-1}(m)$ $M$ equifocal $\overline{\phi}^{-1}(m)$ $N$ $N$ curvature adapted $M$ $M$ ( ) \mathrm{g}$ $\mathrm{c}$ $\not\in ([K2]) $M$ $G/K$ curvature adapted (i) $M$ $\overline{\phi}^{-1}(m)$ equifocal ( (ii) $M$ equifocal ( $gk$ ) ( $\pm\alpha(g_{*}^{-1}v)$ $\alpha$ $\alpha(g_{*}^{-1}v)\neq 0$ ) $\overline{\phi}^{-1}(m)$ ( ) extremality $G/K$ curvature adapted $M$ $\overline{\phi}^{-1}(m)$ ( )extremal equifocal ([K2] ) $V$ $V^{\mathrm{c}}$ $G/K$ $G^{\mathrm{c}}/K^{\mathrm{c}}$ $V$ $G/K$ ( ) $V^{\mathrm{c}},$ $G^{\mathrm{c}}/K^{\mathrm{c}}$ $M$ Mc( ) ( )

5 ( ) $M$ ([K2] ) $M^{\mathrm{c}}$ ( ) $M$ $M^{\mathrm{c}}$ $V$ $G/K$ $M$ $G^{\mathrm{c}}/K^{\mathrm{c}}$ equifocal $\phi^{\mathrm{c}}$ $G/K$ parallel transport Gc- $[0, 1]$ $\cross G^{\mathrm{c}}$ $H^{0}$ $H^{0}$ ([0, 1], c)( ([0, 1], ) $G^{\mathrm{c}}$ ) $\pi^{\mathrm{c}}$ $\phi^{\mathrm{c}}$ $G^{\mathrm{c}}arrow $\overline{\phi}^{\mathrm{c}}$ G^{\mathrm{c}}/K^{\mathrm{c}}$ $\mathrm{d}$ ([K3]) $M$ $G^{\mathrm{c}}/K^{\mathrm{c}}$ $G/K$ (i) $M$ $(\overline{\phi}^{\mathrm{c}})^{-1}(m)$ equifocal ( (ii) $M$ curvature adapted $M$ equifocal ( $gk^{\mathrm{c}}$ ) $\pm\alpha(g_{*}^{-1}v)(\alpha$ $\alpha(g_{*}^{-1}v)\neq 0$ $\tilde{\phi}^{\mathrm{c}-1}(m)$ ) $\mathrm{i}\sigma$ $\mathrm{e}([\mathrm{k}3])m\text{ }$ $G/K$ $M$ equifocal $\overline{\phi}^{-1}(m)$ equifocal $\mathrm{i}\mathrm{e}$ $\mathrm{f}([\mathrm{k}3])(m, J)\text{ ^{ } ^{ } $\{\lambda_{i} i\in I\}$ }$ $(V,\tilde{J})$ $E_{i}$ - $\lambda_{i}$ J- $v_{i}$ $\lambda_{i}$ J- $i$ $\lambda_{i}=0$ (i), (ii) (i) $(M, x)$ $\infty\bigcup_{i=1}(x+\lambda_{i}(x)^{-1}(1))$ (ii) $E_{i}$ $M$ ( ) $(V,\overline{J})$

6 $D_{zv}^{\mathrm{c}o}$, $\gamma_{v}$ $A_{v}^{\mathrm{c}}$ 167 REJECT}.\varphi$ $\ovalbox{\tt\small ( $M$ ) $E_{\ovalbox{\tt\small REJECT}}$ $ \lambda_{7}(v_{7}) $ ( $M$ ) 2. Equifocal equifocal $M$ $G/K$ $M$ curvature adapted $ \backslash \mathrm{s}v$ $R(\cdot, v)v$ $M$ A $R$ $G/K$ 1993 J. Berndt L. Vanhecke generic( ) curvature adapted curvature adapted ( ) curvature $\prime adapted $M$ equifocal equifocal (E-i) $\Lambda,I$ \mathrm{e}$ $M$ 2 (E-ii) $M$ $\tilde{v}_{x}(x\in M)$ $x$ ( $M$ ) 1995 ( $\mathrm{c}.\mathrm{l}$. Terng G. Thorbergsson([TeTh]) ( $M$ $x=gk$ \gamma $\gamma_{v} (\mathrm{o})=v$ $Y$ $Y(\mathrm{O})=X(\in T_{x}M),$ $Y (0)=-A_{v}X$ $Y(s)=(P_{\gamma_{v}1_{[0,s]}}\circ(D_{sv}^{co}-sD_{sv}^{si}\circ A_{v}))(X)$ $\overline{\nabla}$ $Y (0)=\overline{\nabla}_{v}Y,$ ( $G/K$ ) $P_{\gamma_{v}1_{[0,s]}}$ $\gamma_{v} [0,s]$ $D_{sv}^{co},$ $D_{sv}^{si}$ $T_{x}.M$ $D_{sv}^{\mathrm{c}o}=g_{*}\circ\cos(\sqrt{-1}\mathrm{a}\mathrm{d}(sg_{*}^{-1}v))\circ g_{*}^{-1}$ $D_{sv}^{si}=g_{*} \circ\frac{\sin(\sqrt{-1}\mathrm{a}\mathrm{d}(sg_{*}^{-1}v))-}{\sqrt{-1}\mathrm{a}\mathrm{d}(sg_{*}v)}\circ g_{*}^{-1}$ (ad ) $=\mathrm{l}\mathrm{i}\mathrm{e}g$ $\mathrm{k}\mathrm{e}\mathrm{r}(d_{sv}^{co}-sd_{sv}^{si}\circ A_{v})\neq\{0\}$ $\gamma_{v}$ $s$ $\mathrm{k}\mathrm{e}\mathrm{r}(d_{zv}^{co}-zd_{zv}^{si}\mathrm{o}a_{v}^{\mathrm{c}})\neq\{0\}$ $G/K$ $z$ ([K2]) $D_{zv}^{si}$ $(g_{*}\circ\cos(\sqrt{-1}\mathrm{a}\mathrm{d}(zg_{*}^{-1}v))\circ g_{*}^{-1}) _{T_{x}M}(T_{x}Marrow(T_{x}G/K)^{\mathrm{c}}),$ $(g_{*} \mathrm{o}\frac{\sin(\sqrt{-1}\mathrm{a}\mathrm{d}(zg_{*}^{-1}v))-}{\sqrt{-1}\mathrm{a}\mathrm{d}(zg_{*}v)}\mathrm{o}$ $g_{*}^{-1}) _{T_{x}M}$ $(T_{x}Marrow(T_{x}G/K)^{\mathrm{c}})$ $G/K$ $\mathfrak{p}=\mathrm{k}\mathrm{e}\mathrm{r}(\sigma+\mathrm{i}\mathrm{d}),$ $\mathrm{f}=$ ( $\sigma$ ) $G/K$

7 $\mathrm{k}\mathrm{e}\mathrm{r}(\sigma-\mathrm{i}\mathrm{d})$ $\tilde{v}_{x}$ 168 $\mathfrak{p}$ $T_{eK}G/K$ p=h+\mbox{\boldmath $\alpha$}\in \Sigma \triangle +p $\mathfrak{h}$ 2.1([K2]) $A_{v}X=\lambda X$ g*-1x\in p $X(\neq 0)$ $(\mathrm{i})\sim(\mathrm{i}\mathrm{i}\mathrm{i})$ (i) $ \lambda > \alpha(g_{*}^{-1}v) =0$ $\frac{1}{\lambda}$ $g_{*}^{-1}x\in \mathfrak{h}$ $\gamma_{v}$ (ii) $ \lambda > \alpha(g_{*}^{-1}v) >0$ \mathrm{a}\mathrm{r}\mathrm{c}\tanh\frac{\alpha(g_{*}^{-1}v)}{\lambda}+j\pi\sqrt{-1})(j\in \mathrm{z})$ $\gamma_{v}$ 7 $( -l) (iii) $ \lambda < \alpha(g_{*}^{-1}v) $ I-(arctanh\mbox{\boldmath $\alpha$}(g\lambda *-L)+(j+D\pi $\sqrt$ $(j\in \mathrm{z})$ $\gamma_{v}$ (E-i) (CE) equifocal (CE) $\mathit{1}1l$ $\tilde{v}_{x}(x\in M)$ $x$ ( $M$ ) 3. $M$ $V$ A $\{0\}\cup\{\lambda_{i} i=1,2, \cdots\}( \lambda_{i} > \lambda_{i+1} $ $(i=1,2, \cdots))$ ( $\lambda_{i}=-\lambda_{i+1}>0$ $\lambda_{i}$ $\lambda_{i}$ $i$ ) ( ) 0 ( $M$ $\tilde{v}_{x}$ ) $x\in M$ $i$ $x\in M$ $\tilde{v}_{x}$ $M$ $M$ $i$ 2 $M$ (I-i) $M$ (I-ii) $M$ $x\in M$ $M$ (I-ii) ( $\infty$

8 $\lambda_{i}$ $\text{ }\ovalbox{\tt\small REJECT}-\mathrm{f}\mathrm{f}\mathrm{i}^{\backslash }$ 169 (I-ii ) $M$ $A_{\overline{v}_{x}}(x\in M)$ $M$ $M$ 1 $x_{0}$ $T_{x_{0}}M=\overline{\bigoplus_{i\in I}E_{i}^{x_{0}}}$ $T_{x}^{[perp]}M)$ $A_{v}(v\in$ $T_{x_{0}}^{[perp]}M$ $\lambda_{i}^{x_{0}}(i\in I)$ $A_{v} _{E_{i}^{x_{0}}}=\lambda_{i}^{x_{0}}(v)\mathrm{i}\mathrm{d}(v\in T_{x_{0}}^{[perp]}M)$ $\lambda_{i}$ $\lambda_{i}(x_{0})=\lambda_{i}^{x_{0}}$ $T^{[perp]}M^{*}$ $M$ $x$ $T_{x}M$ $T_{x}M=\overline{\bigoplus_{i\in I}E_{i}^{x}}$ $T_{x}^{[perp]}M)$ $E_{i}(x)=E_{i}^{x}$ $\mathrm{i}\mathrm{d}$ $E_{i}^{x_{0}}$ $A_{v} _{E_{i}^{x}}=(\lambda_{i}(x))(v)\mathrm{i}\mathrm{d}(v\in$ $M$ $E_{i}$ $\lambda_{i}(i\in I)$ $M$ $E_{i}$ $M$ ( $x$ $T_{x}^{[perp]}M$ ) $\bigcup_{i\in I}(x+\lambda_{i}(x)^{-1}(1))$ $\lambda_{i}(x)^{-1}(1)$ $R_{i}^{x}(i\in I)$ $M\mathit{0}\supset \text{ }$ $\nearrow^{\text{ }}$ $\text{ }\mathrm{a}^{\mathrm{a}}$ ( $x\in M$ $\nearrow^{\text{ }}$ $U\triangleleft J\mathrm{s}\mathrm{f}\mathrm{f}\mathrm{i}^{\backslash f 77 }$) $R_{j}^{x}( \bigcup_{i\in I}\lambda_{i}(x)^{-1}(1))=\bigcup_{i\in I}\lambda_{i}(x)^{-1}(1)$ $(j\in I)$ 4. [K2] ( ) $V$ $<,$ $>$ $V$ $V$ $V=V_{-}\oplus V_{+}\text{ ^{}\backslash }\backslash <$, >lv-x $<,$ $> _{V\cross V}++$ $V$ $V$ $V=V_{-}\oplus V_{+}l-$ $V$ ( ) $<,$ $>_{V}\pm$ $<,$ $>_{V}=-\pm\pi_{V_{-}}^{*}<,$ $>+\pi_{v_{+}}^{*}<,$ $>$ $V$ $\pi v_{\pm}$ $V_{\pm}$ $V$ $V=V_{-}\oplus V_{+}\vee C_{\text{ }^{}\backslash }\backslash (V, <, >_{V_{\pm}})$ $<,$ $>_{V}\pm$ $V$ $(V, <, >)$ $M$ $(V, <, >_{V})$ $C^{k}$ $(k \geq 1)$ $M$ $x$ $T_{x}M$ $V$, $>_{V}$ $<,$ $>$ $M$ $(0, 2)$ $T^{*}M\otimes T^{*}M$ $C^{k}$ $M$ $x$ $<,$ $>_{x}$ $M$ $x$ $x$ $U$ $W_{+},$ $W_{-}$ 2 $C^{k}$ $(\Lambda/I, <, >)$ $C^{k}$ (PRH) $U$ $y$ $W_{\pm y}$ $(T_{y}M, <, >_{y})$ $(T_{?/}M, <, >_{y,w})\pm\nu$ $(V, <, >_{V})$

9 $\tilde{v}_{x}$ $\tilde{v}_{x}$ $A_{v}^{\mathrm{c}}$ $\text{ ^{}\backslash ^{\backslash }},\tau_{\text{ }J\mathrm{s}\text{ }^{}-}$ 170 $C^{\omega}$ $f$ $C^{k}$ $M$ $(V, <, >_{V})$ $C^{k}$ $(M, f^{*}<, >_{V})$ $C^{k-1}$ (ni, $f^{*}<,$ $>_{V}$ ) ( $M$ ) \text{ ^{}-}Tl2;\text{ ^{}\backslash }\mathrm{j}\underline{\wedge}\text{ }$ $f\#arrowarrow $_{arrow} $ $(V, <, >_{V})$ 0). $C^{k-1}$ $k=\infty$ codilllm $<\infty$ (F) $V=V_{-}\oplus V_{+}$ $M$ $\underline{(v,<,>_{v})\text{ }\mathit{0}\supset \text{ }}$ $(V, <, >v_{\pm})$ $M$ $f^{*}<,$ $>v_{\pm}$ $M$ $(V, <, >v)$ ( $\{0\}\cup\{\lambda_{i} i=1,2, \cdots\}( \lambda_{i} > \lambda_{i+1} $ $\lambda_{i}=-\lambda_{i+1}>0(i=1,2, \cdots))$ $A_{v}^{\mathrm{c}}$ $\lambda_{i}$ $i$ $\{0\}\cup\{\mu_{i} i=1,2, \cdots\}( \mu_{i} >$ $ \mu_{i+1} $ ( $ \mu_{i} = \mu_{i+1} $ $\arg\mu_{i}<\arg\mu_{i+1}$ $(i=1,2,$ ) $\cdots)$ $ _{\sqrt}\mathrm{a}$ $\mu_{i}$ $M$ $i$ (RI) $\Lambda I$ $M$ l $x(\in M)$ $M$ $M$ (CI) $M$ $M$ $x(\in M)$ $M$ (PCI) $M$ $T_{x}M^{\mathrm{c}}(x$ ) $T_{x}M^{\mathrm{c}}$ $T_{x}M^{\mathrm{c}}$ $\{e_{i}\}_{i=1}^{\infty}$ 1 (i) ( $i\in \mathrm{n}$ $ <f_{*}e_{i},$ $f_{*}e_{j}>_{v}^{h} =\delta_{\hat{i}j}(j=1,2, \cdots)$ $2\in \mathrm{n}$ 2 $>_{V}^{H}$ $<,$ $<,$ $>_{V}$ (ii) $\bigoplus_{i=1}^{\infty}\mathrm{s}\mathrm{p}\mathrm{a}11\{e_{i}\}=t_{x}m^{\mathrm{c}}$ $M$ $M$ 1 $x_{0}$ $A_{v}^{\mathrm{c}}(v\in T_{x}^{[perp]}M)$ $\mu_{i}^{x_{0}}(i\in I)$ $T_{x_{0}}M^{\mathrm{c}}=\overline{\bigoplus_{i\in I}E_{i}^{x_{0}}}$ $A_{v}^{\mathrm{c}} _{E_{i}^{x_{0}}}=\mu_{i}^{x_{0}}(v)\mathrm{i}\mathrm{d}(v\in T_{x_{0}}^{[perp]}M)$ $T_{x_{0}}^{[perp]}M^{\mathrm{c}}$ id

10 $E_{i}^{x_{0}}$ $(\begin{array}{lllll}g_{u}. g_{u}(0)=e k_{\mathrm{c}}^{\mathrm{y}}[\mathrm{o}^{\backslash ^{\backslash }} \epsilon_{\grave{\{}ffi 7_{\sim} Tg_{u*}^{-1}g_{u} =u}\backslash \mathrm{k}j\triangleright\wedge^{\backslash ^{\backslash }}J\triangleright \backslash J -ffl^{\backslash }H^{1}([0,1],G)\mathit{0}\supset\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}_{\backslash }\end{array})$ $\text{ }$ 171 $\mu_{i}$ $\mu_{i}(x_{0})=\mu_{i}^{x_{0}}$ (TlMc $M$ $T_{x}M^{\mathrm{c}}$ $x$ $T_{x}M^{\mathrm{c}}=\overline{\bigoplus_{i\in I}E_{i}^{x}}$ $Aarrow _{E_{i}^{x}}=$ $(\mu_{i}(x))(v)\mathrm{i}\mathrm{d}(v\in T_{x}^{[perp]}M^{\mathrm{c}})$ $E_{i}(x)=E_{i}^{x}$ $M$ $E_{i}$ $\mu_{i}(i\in I)$ $ki$ $E_{i}$ $\mu_{i}$ 5. Parallel transport $\mathrm{a}\mathrm{d}(g)$ $\phi$ - $[0, 1]$ $\cross G$ $H^{0}-$ $H^{0}$ ([0, 1], ) $\phi(u)=g_{u}(1)$ $(u\in H^{0}([0,1], \text{ }))$ $(*)$ $e$ $g_{u} $ $\mathrm{g}$ $(t\in[0,1])$ $H^{0}([0,1], \text{ })$ $g_{u*}^{-1}g_{u} $ ( g ( $(g_{u*}^{-1}g_{u} )(t)=l_{g_{u}(t)_{*}}^{-1}(g_{u} (t))$ $\mathit{0}\supset \mathrm{p}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{e}\mathrm{l}$ $\phi$ G transport $\underline{5.1([\mathrm{k}\mathrm{i}\mathrm{t}\mathrm{e}])}\phi$ $H^{1}([0,1], G)$ ( $H^{0}$ $[0,1]$, ) $g*u=\mathrm{a}\mathrm{d}(g)u-g g_{*}^{-1}$ $(g\in H^{1}([0,1], G), u\in H^{0}([0,1], \text{ }))$ $\Omega_{e}(G)=$ $\{g\in H^{1}([0,1], G) g(0)=g(1)=e\},$ $P(G, e\cross G)=\{g\in H^{1}([0,1], G) g(0)=e\}$ $\underline{5.2([\mathrm{t}\mathrm{e}\mathrm{t}\mathrm{h}])}(\mathrm{i})h^{1}([0,1], G)$ $H^{0}([0,1], \text{ })$ (ii) $P(G, e\cross G)$ $H^{0}([0,1], \text{ })$ $u\in H^{0}$ ( $[0,1]$, ) (iii) $\phi(g*u)=g(0)\phi(u)g(1)^{-1}(g\in H^{1}([0,1], G),$ $\phi$ (iv) $H^{0}([0,1], \text{ })arrow G$ $\Omega_{e}(G)$ ( ) $\phi(u)=x_{0}\phi(v)x_{1}^{-1}$ ( $u,$ $v\in H^{0}$ ([0, 1], ), $x_{0},$ $x_{1}\in G$ ) $u=g*$ $v,$ $g(0)=x_{0}$ $g(1)=x_{1}$ $g\in H^{1}([0,1], G)$ $u\in H^{0}$ ([0, 1], ) $g\in P(G, e\cross G)$ $u=g*\hat{0}$ $\hat{0}$ 0

11 \mathfrak{p}}$ ([Kl]) $T_{u}H^{0}([0,1], \text{ })$ $\phi_{*u}(v)=(\int_{0}^{1}\mathrm{a}\mathrm{d}(g^{-1})vdt)g(1)_{*}^{-1}$ \phi * $u=g*\hat{0}$ $\#^{-}\ddagger$ $\phi$ $u$ ( $H^{0}$ ([0, 1], ) ( ) $X\in$ $\phi_{*\hat{0}}(\hat{x})=x$ $\hat{x}$ $\phi$ ( $\hat{x}$ $X$ ( $\in$ $=T_{e}G$ $\hat{0}$ ) $\hat{x}$ $X$ $T_{u}H^{0}([0,1], \text{ })$ ) ([K2]) parallel transport Ad(G)- $<,$ $>$ $=\mathrm{f}$ $(<)> _{\mathrm{f}\cross \mathrm{f}}$ $> _{\mathfrak{p}\cross $\mathfrak{p}$, $<,$ ) $\text{ }+=\mathfrak{p}$, - $<,$ $>_{9\pm}=-<,$ $=\mathrm{f}$ $> _{9-\cross \mathrm{g}-}+<,$ $> _{9+^{\mathrm{X}}9+}$ $L^{2}$ $[0, 1]$ $\text{ },$ $\text{ }+$ - $H^{0}$ ([0, 1], ), $H_{\pm}^{0}$ $H^{0}([0,1], \text{ _{}+})$ $H^{0}([0,1], \text{ _{}-})$ $H^{0}([0,1], \text{ })$ $H_{\pm}^{0}$ ( $H^{0}([0,1], \text{ _{}\pm})$ $<,$ $>_{0}$ $u,$ $v>_{0}= \int_{0}^{1}<u(t),$ $v(t)>dt$ ([0, 1], ), $H^{0}$ $<,$ $>_{0}$ ) $(H^{0}([0,1], \text{ })$, $<,$ $>_{0,H_{\pm}^{0}})$ $(H^{0}([0,1], \text{ }), <, >_{0})$ $H^{0}$ $\phi$ parallel transport ([0, 1], )\rightarrow G $(*)$ $\phi$ 5.4([K2]) 5.2, [K3] equifocal $(M, <, >)$ ( ) $M$ 2 $(M, <, >, J)$ (i) $<JX,$ $JY>=-<X,$ $Y>(\forall X, Y\in TM)$ (ii) $\nabla J=0$ ( $\nabla$ $<,$ $>$. ) $(M, <, >, J)$ $(N, <, >,\tilde{j})$ $\dim_{\mathrm{r}}m=2n$ $\overline{j}\circ f_{*}=f_{*}\circ J$ $f$ $f$ $(M, <, >, J)$ $(N, <, >,\tilde{j})$ exp $(M, <, >, J)$ $\exp^{[perp]}(av+bjv)(a, b\in \mathrm{r})$

12 $\overline{j}$ \mathrm{c}}$ $\mathfrak{p},$ 173 $(M, <, >, ])$ $a+b\sqrt{-1}$ $h$ $(M, <, >, J)$ 2 $H$ $H_{x}= \frac{1}{2n}\sum_{i=1}^{2n}<e_{i},$ $e_{i}>h(e_{i}, e_{i})$ $\{e_{i}\}_{i=1}^{2n}$ ( $T_{x}M$ ) $h(x, Y)=<X,$ $Y>H-<JX,$ $h$ $Y>\overline{J}H(\forall X, Y\in TM)$ $(\mathrm{r}^{2m},$ $(M, <, >, J)$ $<$ $>,\overline{j})$, $(\mathrm{r}^{2m}, <, >)$ $(x_{1}, \cdots, x_{2m})$ $< \frac{\partial}{\partial x_{i}}$ $\frac{\partial}{\partial x_{j}}>=(-1)^{i+1}\delta_{ij}$ $(i=1, \cdots, 2m)$, $\overline{\mathcal{j}}(\frac{\partial}{\partial x_{2i-1}})=\frac{\partial}{\partial x_{2i}}$ $\overline{\mathcal{j}}(\frac{\partial}{\partial x_{2i}})=-\frac{\partial}{\partial x_{2i-1}}(i=1, \cdots, m)$ $\mathrm{r}^{2m}$ $z_{i}=x_{2i-1}+\sqrt{-1}x_{2i}(i=1, \cdots, m)$ $z_{1}^{2}+\cdots+z_{m}^{2}=\kappa^{2}(\kappa\in \mathrm{c})$ $\kappa$ $S_{\mathrm{c}}^{m-1}(\kappa)$ $\mathrm{c}^{m}$ $(\mathrm{r}^{2m}, <, >,\tilde{j})$ $\mathrm{i}\mathrm{e}6.1$ ([K3]) $G/K$ ( ) $K^{\mathrm{c}}$ $\mathrm{f}^{\mathrm{c}},$ $\mathfrak{p}^{\mathrm{c}},$ $=\mathrm{f}$ $G^{\mathrm{c}},$ $\mathrm{f},$ $\mathfrak{p}$, $G,$ $K$ $G/K$ Ad(G) $\mathrm{a}\mathrm{d}(g^{\mathrm{c}})$, $>$ $G^{\mathrm{c}}$-, $>$ $G^{\mathrm{c}}/K^{\mathrm{c}}$ $T_{eK^{\mathrm{c}}}(G^{\mathrm{c}}/K^{\mathrm{c}})$ $\mathfrak{p}^{\mathrm{c}}(=\mathfrak{p}+\sqrt{-1}\mathfrak{p})$ $> _{\mathfrak{p}\mathrm{x}\mathfrak{p}}$ $<,$ $\mathfrak{p}$ $<,$ $> _{\sqrt{-1}\mathfrak{p}\cross\sqrt{-1}\mathfrak{p}}-$ $\sqrt{-1}\mathfrak{p}$ $<,$ $>$ Gc- $G^{\mathrm{c}}/K^{\mathrm{c}}$ $J_{eK^{\mathrm{c}}}(X+\sqrt{-1}Y)=-Y+\sqrt{-1}X(X, Y\in \mathfrak{p})$ $(G^{\mathrm{c}}/K^{\mathrm{c}}, <, >,\overline{j})$ $G^{\mathrm{c}}/K^{\mathrm{c}}$ $G/K$ \emptyset $(M, <, >, J)$ equifocal (AKE) $(M, <, >, J)$ $\tilde{v}_{x}(x\in M)$ $x$ ( $M$ ) $G^{\mathrm{c}}/K^{\mathrm{c}}$ $G/K$ $\iota$ $\iota$ $\iota(gk)=gk^{\mathrm{c}}(gk\in G/K)$ $G/K$ $M$ $M$ $M^{\mathrm{c}}$ $G^{\mathrm{c}}/K^{\mathrm{c}}$ $M=G x_{0}(g \subset$ $G)$ $G^{\prime_{\mathrm{C}}}\iota(x_{0})$ $G^{\prime ( $G $ ) $G^{\mathrm{c}}/K^{\mathrm{c}}$

13 $\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}.\ovalbox{\tt\small REJECT}$ ( REJECT}\ovalbox{\tt\small REJECT}_{\backslash }4\mathrm{b}k\mathrm{D}^{\backslash \prime}\mp\lambda^{\backslash }\backslash _{\mathrm{o}}\backslash \simarrow(dk\doteqdot\backslash \mathit{1}^{\backslash }R\sigma)\ovalbox{\tt\small REJECT}\not\equiv\hslash\grave{\grave{>}}ffi^{\backslash }V)$ $\tilde{j}$ 174 $*\iota_{-}^{\tau}rx$ $7\ovalbox{\tt\small $\sim-\mathcal{x}\mathrm{b}km^{\mathrm{c}}k\ovalbox{\tt\small REJECT} \mathrm{b}_{\backslash }MU$ ) $\circ$ 62. (i) $M$ ( 2 $M^{\mathrm{c}}$ ) $\iota_{*}v$ ( ) $M^{\mathrm{c}}$ (ii) $M$ $M^{\mathrm{c}}$ equifocal equifocal $M^{\mathrm{c}}$ $G/K=H^{m}(-1)(=\{(x_{1}, \cdots, x_{m+1})\in \mathrm{r}_{1}^{m+1} -x_{1}^{2}+x_{2}^{2}+\cdots+x_{m+1}^{2}=$ $-1\},$ $M=S^{m-1}(1)(=\{(x_{1}, \cdots, x_{m+1}) \in H^{m}(-1) x_{1}=\sqrt{2}\}$ $G^{\mathrm{c}}/K^{\mathrm{c}}=$ $\{(z_{1}, \cdots, z_{m+1})\in \mathrm{c}^{m+1}=\mathrm{r}_{m+1}^{2m+2} -z_{1}^{2}+z_{2}^{2}+\cdots+z_{m+1}^{2}=-1\}(=h^{nl}(-1)^{\mathrm{c}})$, $M^{\mathrm{c}}=\{(z_{1}, \cdots, z_{m+1})\in \mathrm{c}^{m+1} z_{1}=\sqrt{2}, z_{2}^{2}+\cdots+z_{m+1}^{2}=1\}$ $G/K$ $G^{*}/K$ ( ) $G^{*\mathrm{c}}/K^{\mathrm{c}}$ $G^{\mathrm{c}}/K^{\mathrm{c}}$ $G^{*\mathrm{c}}/K^{\mathrm{c}}$ 7. [K3] $V$ $<,$ $>$ $V$ -id $J^{\tilde{2}}=$ $V$ $V$ $V V_{-}\oplus V_{+}$ $(V, <, >v_{\pm})$ $\tilde{j}v_{\pm}=v_{\mp}$ $(V, <, >,\tilde{j})$ $M$ $(V, <, >_{V})$ $M$ $(0, 2)$ $T^{*}M\otimes T^{*}M$ (C $M$ $x$, $>_{x}$ $M$ $(1, 1)$ $T^{*}M\otimes T\Lambda I$ (C $M$ $x$ $J_{x}$ $J_{x}^{2}=-\mathrm{i}\mathrm{d}$ $M$ $x$ $x$ $U$ 2 $(C^{\infty})$ $W_{+},$ $(M, <, >, J)$ $W_{-}$ (AKH) $U$ $(T_{l/}M, <, >_{y,w})\pm v$ $y$ $W_{\pm y}$ $(T_{y}M, <, >_{y})$ $(V, <, >_{V})$ $J_{y}W_{\pm y}=w_{\mp y}$ $f$ $M$ $(V, <, >_{V},\overline{J})$ $(C^{\infty})$ $(M, f^{*}<, >v)$ $\overline{j}(f_{*}t_{x}m)\subset f_{*}t_{x}m(x\in M)$

14 $\lambda_{i}$ $\tilde{v}_{x}$ 175 $\mathrm{c}\mathrm{o}\dim M<\infty$ $J\Leftrightarrow f_{*}\circ J=\overline{J}\circ $(M, <, >, ])$ ( $<,$ $>=f^{*}<,$ $>_{V}$, f_{*}$) $(V, <, >_{V},\overline{J})$ (AKF) $V=V_{-}\oplus V_{+}\vee C_{\text{ }^{}\backslash }\backslash (V, <, >_{V_{\pm}})$ $JV_{\pm}=V_{\mp}$ $ \backslash \mathrm{k}\mathrm{s}v$ $M$ A $f^{*}<,$ $>v_{\pm}$ $(V, <, >_{V},\overline{J})$ $(M, <, >, J)$ $(M, <, >, J)$ $A_{v}X=aX+bJX$ $X(\neq 0)$ $a+b\sqrt{-1}$ $X$ $a+b\sqrt{-1}$ 2 0 $\{\lambda_{i} i=1,2, \cdots\}$ ( $ \lambda_{i} > \lambda_{i+1} $ or $ \lambda_{i} = \lambda_{i+1} \ \arg\lambda_{i}<\arg\lambda_{i+1}$ ) $\lambda_{i}$ $ij$ $(M, <, >, J)$ (AKCI) $x\in M$ $M$ (AKPCI) $M$ A I $\Lambda I$ 1 $x_{0}$ $(M, <, >, J)$ $T_{x_{0}}M$ $A_{v}(v\in T_{x}^{[perp]}M)$ $T_{x_{0}}M=\overline{\bigoplus_{i\in I}E_{i}^{x_{0}}}$ $A_{v} _{E_{i}^{x_{0}}}={\rm Re}(\lambda_{i}^{x_{0}}(v))\mathrm{i}\mathrm{d}+{\rm Im}(\lambda_{i}^{x_{0}}(v))J(v\in T_{x_{0}}^{[perp]}M)$ $T_{x_{0}}M$ $\lambda_{i}^{x_{0}}(i\in I)$ $T_{x_{0}}M$ $\mathrm{i}\mathrm{d}$ $\lambda_{i}$ $E_{i}^{x_{0}}$ $T^{[perp]}M$ $T^{[perp]}M^{*}$ $M$ $x$ $T_{x}M$ $T_{x}M=\overline{\oplus E_{i}^{x}}$ $i\in I$ $\lambda_{i}(x_{0})=\lambda_{i}^{x_{0}}$ $A_{v} _{E_{i}^{x}}={\rm Re}((\lambda_{i}(x))(v))\mathrm{i}\mathrm{d}+$ Im((\lambda (x))(v))j $(v\in T_{x}^{[perp]}M)$ $\lambda_{i}(i\in I)$ $(M, <, >, J)$ $E_{i}(x)=E_{i}^{x}$ $M$ $l_{arrow} \text{ }\backslash \text{ }J$ $E_{i}$ \yen $\mathrm{f}\mathrm{f}\mathrm{i}^{\sigma} \neq_{j\mathrm{j}}^{t_{\underline{\backslash }}/\backslash \text{ }}$ $(M, <, >, J)$ $\mathrm{f}$ $E_{i}(i\in I)$

15 $\phi^{\mathrm{c}}$ \mathrm{c}}$ $V^{\mathrm{c}}$ $\text{ }$ $> _{\mathrm{g}_{+}^{\mathrm{c}}\cross \mathfrak{g}_{+}^{\mathrm{c}}}$ [K3] ( 5 ) parallel transport $\phi$ $H^{0}$ ( $[0,1]$, )\rightarrow G $V^{\mathrm{c}}$ $(V, <, >)$ $V$ ( $V^{\mathrm{c}}=V\oplus\sqrt{-1}V$) $> $ $<,$, $V^{\mathrm{c}}$ $>$ $\overline{j}$ $\tilde{j}x=\sqrt{-1}x(x\in V^{\mathrm{c}})$ $(V^{\mathrm{c}}, <, >,\tilde{j})$ $V_{\pm}$ $(V, <, >)$ $(V, <)>_{\pm})$ $V_{+}^{\mathrm{c}}=V_{+}+\sqrt{-1}V_{-},$ $V_{-}^{\mathrm{c}}=V_{-}+\sqrt{-1}V_{+}$ $(V^{\mathrm{c}}, <, > )$ $(V^{\mathrm{c}}, <, >_{V_{\pm}^{\mathrm{c}}} )$ $(V^{\mathrm{c}}, <, >,\tilde{j})$ $\overline{j}v_{\pm}^{\mathrm{c}}=v_{\mp}^{\mathrm{c}}$ $\underline{(v,<,,>)}$ \emptyset $(V, <, >)$ $M$ $M$ $(V^{\mathrm{c}}, <, >,\tilde{j})$ $M^{\mathrm{c}}$ $M=G u_{0}$ ( $G $ $(V,$ $<,$ $>)$ $u_{0}\in$ $V$ ) $G^{\prime \mathrm{c}}u_{0}$ $G^{\prime ( ) $(V^{\mathrm{c}}, <, >,\tilde{j})$ $M^{\mathrm{c}}$ $M$ 8.1. (i) $M$ $M^{\mathrm{c}}$ ( ) (ii) $M$ $M^{\mathrm{c}}$ $M^{\mathrm{c}}$ Ad(G) $<,$ $>$ $> _{\mathrm{f}\cross \mathrm{f}}$ ( $<,$ $> _{\mathfrak{p}\cross \mathfrak{p}}$, $<,$ ) c+ $=$ $\mathfrak{p}+\sqrt{-1}\mathrm{f},$ $\text{ _{}-}^{\mathrm{c}}=\mathrm{f}+\sqrt{-1}\mathfrak{p}$ $<,$ $>$ $> $, $<,$ $>_{\mathrm{g}_{\pm}^{\mathrm{c}}} =-<,$ $> _{9_{-^{\mathrm{X}}\mathfrak{g}_{-}^{\mathrm{c}}}^{\mathrm{c}}}+<,$ $L^{2}$ $H^{0}$ ([0, 1], $<$, > gc\pm [ $\mathrm{c}-$ $\text{ ^{}\mathrm{c}},$ $\text{ _{}+}^{\mathrm{c}}$ $[0, 1]$ $H^{0}([0,1], \text{ _{}+}^{\mathrm{c}}),$ $H^{0}([0,1], \text{ _{}-}^{\mathrm{c}})$ $H^{0}$ ), ( $[0,1]$, $H_{\pm}^{0,\mathrm{c}}$ $\mathrm{c}\pm$ ) $H^{0}([0,1], \text{ ^{}\mathrm{c}})$, $>_{0}$ $u,$ $v>_{0} = \int_{0}^{1}<u(t),$ $v(t)> dt$ $H_{\pm}^{0,\mathrm{c}}$ (7) ( ( $(H^{0}([0,1], \text{ ^{}\mathrm{c}}), <, >_{0} )$ $H^{0}$ $[0,1]$, c), $<,$ $>_{0,H_{\pm}^{0,\mathrm{c}}} )$ $(H^{0}([0,1], \text{ ^{}\mathrm{c}}), <, >_{0} )$ $H^{0}$ ([0, 1], c) $H^{0}$ ([0, 1], ( ([0, 1], c), $H^{0}$ $<,$ ) $)$ $\tilde{j}$ $\overline{j}u=\sqrt{-1}u(u\in$ $\tilde{j}$ $>_{0},\overline{J}$) ( ([0, 1], ), $H^{0}$ $<,$ $>_{0}$ ) ( 5 ) $>^{J}$ $\mathrm{a}\mathrm{d}(g^{\mathrm{c}})$ $<,$ $JX=\sqrt{-1}X(X\in \text{ ^{}\mathrm{c}})$ $G^{\mathrm{c}}$

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