The Involutions of Compact Symmetric Spaces III

Abstract

TANAKA at every point $x$ such that (1) the singleton $\{x\}$ is open in $F(s_{x}, M),$ (2) $s_{x}$ is involutive and (3) $M$ admits an affine connection $\nabla$ which every symmetry $s_{x}$ leaves invariant.4.2.A homomorphism (of a symmetric space) $f:M\rightarrow N$ is a smooth map of ' symmetric space $M$ into another which commutes with the symmetries; one $ha_{\backslash }^{I}$ $s_{f\langle x)}\circ f=f\circ s_{x}$ for every point $x$ of $M$ .Thus ageodesic is ahomomorphism $c:R\rightarrow A4$ of the real line.We have a category of the symmetric spaces and the $homomorphism\{$ between them.A subspace $M\subset N$ is a symmetric space of which the inclusion is ' homomorphism.The automorphism group of $M$ is denoted by $I(M)$ and its connectee component $I(M)_{\langle 1)}$ through the unit element 1 is denoted by $G$ , or $G_{M}$ or $G(M)$; generally we denote by $X_{(p)}$ the connected component of $X$ through the point $p$ .We may follow the practice of using $G$ to denote its covering group when it is convenient, $especiall\backslash $ . in describing specific spaces such as the complex Grassmann manifold $G_{r}(C^{n})=SU(n)$ $S(U(r)\times U(n-r))$; i.e. $G$ may not be effective on $M$ .If $M$ is connected as we usuall.assume, $G$ is transitive on $M;M=G/K,$ $K$ the stabilizer at a point $0$ , and $K$ is open $i1$ $F(ad(s_{o}), G)$ .When $M$ happens to be a Lie group, we call a gr-homomorphism $j$ homomorphism of $M$ as a Lie group.Every local isomorphism class of $\infty nnectedsimpl($ spaces $M$ contains a unique space $M^{\%}$ of which every $M$ in the class is a covering space We call $M^{\%}$ the bottom space (the adjoint space in [H]); $M^{\%}=c\%/F$($ad(s_{o}),$ G). $($ write $M^{\sim}$ for the universal covering space of $M$ .We say $M$ is locally isomorphic witl $N$ if $M^{\sim}$ is isomorphic with $N^{\sim};$ the symbol is $M=.N$.4.3.We assume compactness of $M$ , unless otherwise mentioned.Thus th $($ uniqueness of $\nabla$ implies that $\nabla$ is the Levi-Civita connection of a Riemannian metri $($ which $I(M)$ leaves invariant (provided $M$ is simple).Another consequence is that $Mi$ a local product of a torus and simple spaces, if $M$ is connected.We will freely use known facts in Bourbaki [B] and Helgason [H].We now recall and explain definitions of basic terms.4.4.We call every component of F$(s_{o}, M)apolarofoinManddenoteitbyM^{\cdot}$ or $M^{+}(p)$ if it contains a point$p$.We also call it apole when it is a singleton $M^{+}(p)=\{p\}$ .4. $4a$ .Every homomorphism carries a polar into another.4. $4b$ .If $M$ has a pole $p$ of $0$ , then $M$ admits a double covering homomorphisn onto another space $M/\{0, p\}$ which carries $\{0, p\}$ onto a single point.4.5.Given a subspace $N$ , we call a subspace $N^{\perp}the$ c-orthogonal space to $N$ a $p$ if the tangent space $T_{p}N^{\perp}$ is the orthogonal complement of $T_{p}N$ in $T_{p}M$.If a subspac $N$ equals $F(\tau, M)_{(0)}$ for some involution $\tau$ , then $N^{\perp}:$ $=F(\tau\circ s_{o}, M)_{\langle 0)}$ is c-orthogonal $t|$ $N$ at $0$ .One notes that the intersection $N\cap N^{\perp}$ is then contained in $F(s_{o}, M)$ and henc in $F(s_{o}, N)$; in case $F(\tau, M)$ is connected one has the equality $N\cap N^{\perp}=F(s_{o}, N^{\perp})$ .4.6.The c-orthogonal space $M^{-}(p)$ to a polar $M^{+}(p)$ at $p$ is called the meridia