On the equivalence problems associated with a certain class of homogeneous spaces

Abstract

type $(L)$ under a weaker hypothesis on $L$ (Theorems 9.1 and 9.2), which just generalizes the notion of the normal projective connection.In \S 1, we construct the homogeneous space $M_{L}=G/G^{\prime}$ and study the fundamental properties of it.In \S 2, we define the notion of an irreducible l-system (Definition 2.1).It is shown that an arbitrary l-system $L$ is decom- posed into a product of irreducible l-systems (Propositions 2.1 and 2.2) and that the set of all irreducible l-systems is devided into two classes called of type $(R)$ and of type $(C)$ (Definition 2.2). \S 3 is devoted to the study of the duality which lies among l-systems.This leads to the investigation of the maximal compact subgroups of the group $G$ .In \S 4, it is shown that the set of all isomorphism classes of irreducible l-systems of type $(C)$ is in a one to one correspondence with the set of all isomorphism classes of compact irre- ducible hermitian symmetric spaces.\S 5, \S 6 and \S 7 are preliminary to the subsequent two sections.In \S 8, it is proved that to every G-structure $\tilde{P}$ on a manifold $M$ there is associated a tensor field $T$ of type $\left(\begin{array}{l}1\\2\end{array}\right)$ on $M$ called the torsion tensor field of $\tilde{P}$ .This notion turns out to be a generalization of the notion of the Nijenhuis tensor field of an almost complex structure.\S 9 is concerned with the construction of the normal connection of type \ l a n g l e $L$ ), which will be carried out following the construction of the normal projec- tive connection given in [9].Hypotheses on $L$ for the existence of the normal connection of type $(L)$ will be stated in terms of an endomorphism $\Phi_{L}$ and an integer $\delta(L)$ (as for the cohomological interpretations of these hypotheses, see Appendix).The endomorphism $\Phi_{L}$ has an intimate relationship to the quad- ratic form which appears in the study of discrete subgroups of a Lie group, for example, in [11].The integer $\delta(L)$ is the dimension of the first derived space associated with the representation $(\tilde{G}, \mathfrak{m})$ .Finally in \S 10, we study the endomorphism $\Phi_{L}$ and the integer $\delta(L)$ by using the results in \S 2, \S 3 and \S 4 and show that hypotheses on $L$ for the existence of the normal connection of type $(L)$ are generally satisfied.Owing to the results in \S 2, the problems will be reduced to the case where $L$ is an irreducible l-system of type $(C)$ . Preliminary remarkThroughout this paper, we always assume the differentiability of class $C^{\infty}$ , and by a manifold we shall mean a manifold satisfying the second counta- bility axiom.$R$ (resp.$C$ ) will denote the field of real numbers (resp. of complex numbers).$ ada\xi\equiv l(a)\xi$ mod $fi^{\prime}$ for all $a\in G^{\prime}$ and $\xi\in \mathfrak{n}\tau$ .We shall denote by the same letter 1 the correspond- ing homomorphism of $Q^{\prime}$ into gl(m), $i$ .$e.$ , [X, $\xi$ ] $\equiv l(X)\xi mod \mathfrak{g}^{\prime}$ for all $X\in \mathfrak{g}^{\prime}$ and $\xi\in \mathfrak{m}$ .We see easily that 1(X) coincides with the g-com- ponent of $X$ in the decomposition $Q^{\prime}=\mathfrak{m}^{*}+\tilde{\mathfrak{g}}$ .Now it is clear that the restric- tion of 1 $(: G^{\prime}\rightarrow GL(\iota \mathfrak{m}))$ to $\tilde{G}$ is the identity transformation of $\tilde{G}$ .We have $ad(\exp\omega)\xi=\xi+ [\omega, \xi]+\frac{1}{2} [\omega, [\omega, \xi]]$ for all $\omega\in \mathfrak{m}^{*}$ and $\xi\in \mathfrak{m}$ , which implies that the group $\exp \mathfrak{m}^{*}$ is contained in the kernel of 1.PROPOSITION 1.5.$l(G^{\prime})=\tilde{G}$ .PROOF.As we have seen above, the restriction of 1 to $\tilde{G}$ is the identity transformation of $\tilde{G}$ .Hence $l(\tilde{G})=\tilde{G}$ .Therefore we have only to prove that $l(G^{\prime})\subset\tilde{G}$ .Take any $a$ in $G^{\prime}$ .We have $[adaA, \xi]=ada[A, ada^{-1}\xi]$ $\equiv ada(Al(a)^{-1}\xi)\equiv l(a)Al(a)^{-1}\xi mod \mathfrak{g}^{\prime}$ for all $\xi\in \mathfrak{m}$ and $A\in\tilde{\mathfrak{g}}$ .This implies that $l(a)Al(a)^{-1}$ coincides with the g-component of $adaA$ (in the decomposition $Q^{\prime}=\mathfrak{m}^{*}+\tilde{\mathfrak{g}})$ for all $A\in\tilde{\mathfrak{g}}$ .Therefore, $l(a)\in\tilde{G}$ by Prop.1.3.LEMMA 1.6.Let $a\in G^{\prime}$ be in the kernel $ofl$ . (1)$ ada\omega=\omega$ for all $\omega\in \mathfrak{m}^{*}$ .(2) If we put $\omega=adaE_{L}-E_{L}$ , then we have $\omega\in \mathfrak{m}^{*}$ and $ada\xi-\xi\equiv[\omega, \xi]mod \mathfrak{m}^{*}$ for all $\xi\in \mathfrak{m}$ .PROOF.(1) By Lemma 1.1, we have $\langle\xi, ada\omega\rangle=\varphi(ada^{-1}\xi, \omega)=\langle\xi, \omega\rangle$ for all $\xi\in \mathfrak{m}$ , whence $ ada\omega=\omega$ . (2) It follows from the proof of Prop. 1.5 that $adaA\equiv Amod \mathfrak{m}^{*}$ for all $A\in\tilde{\mathfrak{g}}$ .Hence $\omega\in \mathfrak{n}\iota^{*}$ .Since $E_{L}$ is in the center of $\tilde{\mathfrak{g}}$ , we get $[E_{L}, ada^{-1}\xi]\equiv\xi mod \mathfrak{m}^{*}$ .Therefore we have $[\omega, \xi]$ $=ada[E_{L}, ada^{-1}\xi]-\xi\equiv ada\xi-\xi mod \mathfrak{m}^{*}$ .PROPOSITION 1.6.The kernel of $l$ is equal to $\exp \mathfrak{m}^{*}$ .PROOF.As we have seen above, the group $\exp \mathfrak{m}^{*}$ is contained in the kernel of $l$ .Take any $a\in G^{\prime}$ in the kernel of 1.By Lemma 1.6, we can find an $\omega\in \mathfrak{m}^{*}$ such that $ada\xi-\xi\equiv[\omega, \xi]mod \mathfrak{m}^{*}$ for all $\xi\in \mathfrak{m}$ .If we put $b=(\exp(-\omega))a$ , then we have easily $adb\xi\equiv\xi mod \mathfrak{m}^{*}$ for all $\xi\in \mathfrak{m}$ .Let us prove that $ adb\xi=\xi$ for all $\xi em$ .We set $\gamma(\xi)=adb\xi-\xi$ , which is in $\mathfrak{m}^{*}$ .We have $adb[\xi, \xi^{\prime}]=[\xi, \gamma(\xi^{\prime})]-[\xi^{\prime}, \gamma(\xi)]=0$ .Hence we get $\langle\xi, \gamma(\xi^{\prime})\rangle$ $=\langle\xi^{\prime}, \gamma(\xi)\rangle$ by Prop.1.4.On the other hand, we have $adb^{-1}\xi^{\prime}=\xi^{\prime}-\gamma(\xi^{\prime})$ by Lemma 1.6, (1).Hence we have $\langle\xi^{r}, \gamma(\xi)\rangle=\varphi(\xi^{\prime}, adb\xi-\xi)=\varphi(\xi^{\prime}, adb\xi)$ $=\varphi(adb^{-1}\xi^{\prime}, \xi)=\varphi(\xi^{\prime}-\gamma(\xi^{\prime}), \xi)=-\langle\xi, \gamma(\xi^{\prime})\rangle$ .Consequently, we have $\gamma=0,$ $i$ .$e.$ , $(R)$ (resp. of type $(C)$) if the representation $(\tilde{\mathfrak{g}}, \mathfrak{m})$ is of first class (resp. of second class).In general, let $(\mathfrak{g}, V)$ be a real representation, $i$ .$e.,$ $V$ is a real vector space and $\mathfrak{g}$ is a subalgebra of $\mathfrak{g}I(V)$ .$V^{c}$ denoting the complexification of $V,$ $\mathfrak{g}$ may be regarded as a (real) subalgebra of the complex Lie algebra of all complex endomorphisms of $V^{c}$ .Assuming that the representation $(\mathfrak{g}, V)$ is irreducible, we say that $(\mathfrak{g}, V)$ is of first class (resp. of second class) if the complex repre- sentation $(\mathfrak{g}, V^{c})$ is irreducible (resp.reducible) [3].Let $L=(\mathfrak{g}, \mathfrak{m}, \mathfrak{n}\iota^{*},\tilde{\mathfrak{g}})$ be an l-system.We shall denote by $\tilde{C}$ (resp.i) the center of the group $\tilde{G}$ (resp.the Lie algebra g).By Prop. 1.3, we have $\sim c=C(\tilde{\mathfrak{g}}, \mathfrak{g}I(\mathfrak{m}))$and $\tilde{C}\subset C(\tilde{\mathfrak{g}}, GL(\mathfrak{m}))\subset\tilde{G}$ .This being said, we have PROPOSITION 2.3.The notation being as above, we assume that $L$ is irredu- cible.(1) If $L$ is of type $(R)$ , then 7 consists of all the elements $\lambda E_{L}$ , where $\lambda\in R$ .(2) If $L$ is of type $(C)$ , then there is an element $I_{L}$ of $\sim_{C}$ such that $I_{L}^{2}=-E_{L}$ and such $that\sim c$ consists of all the elements $\lambda E_{L}+\mu I_{L}$ , where $\lambda,$ $\mu\in R$ .Moreover, $I_{L}$ is unique up to the factor $-1$ .(3) $\tilde{C}=C(\tilde{\mathfrak{g}}, GL(\mathfrak{m}))=\sim c\cap GL(\mathfrak{m})$ .We shall now show that an irreducible l-system of type $(C)$ is really " complex".