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(1)Geometry of orbits of the isotropy group in a compact Riemannian symmetric space. ,. by. s. Shigeo AKIBA* (Received May 6, 1986). gO. Introduction. We study in this paper orbits of the isotropy group of a compact Riemannian symmetric space. These orbits are closed submanifolds and foliate the sYmmetric. space. The orbit of an antipodal point of the origin is known to be a totally geodesic submanifold. In the case of compact symmetric space of rank one, see [2], Prop. 10.4., p. 330, and in the general compact case, see [1], lemma 2.1., p.. 406. 0rbits of the other points are proved to be pseudo-umbilical with respect. to a normal vector ditected to the "radius" in the case that the symmetric space is rank one and has unique positive restricted root. Another purpose of this paper is to explore a Lie algebraic version of the covariant differentiation. and the second fundamental form of the orbit of the isotropy group. g1. Lie algebraic versien of the covariant differentiation.. Let M=U/K be a compact Riemannian symmetric space, where U is the identity component of its isometry group and K be the isotropy subgroup of U. at oEML Let u and f be the Lie algebra of U and K respectively, then there exists a subspace p of u such that tt==E+p be a Cartan decomposition.. We fix a ULinvariant Riemannian metric induced by the restriction of -B to pxp, where B is the Killing-Cartan form of u. In calculations later, let <,> denote both the minus of the Killing-Cartan form on u and the metric tonsor on. M Let z denote the natural projection U->M) and by (dre)e we identify the tangent space of M at o with p. Let T denote the action of U on M; i.e., T(g)z(g') ==. T(gg'), for g,g'E U. Let P be an arbitrary point of 11Z let M=K(P) be the KLorbit of P, and let. ffEp be such that P=Exp ffZ Then M is a closed submanifold of ML * Department of Mathematics, Faculty of Education, Yokohama National Universlty..

(2) 12 S. AKIBA. Let a* denote a maximal abelian subspace of p containing H and b a max-. imal abelian subalgebra of u containing a*. Put a==bnt. Let ScK be the subgroup leaving the point P fixed and let i denote its Lie algebra. A oneparameter group of isometry T(exp tZ), for ZEIu and tEiR, generates a curve cz (may be singular) on M starting at P and a vector field Z on M We call Z a vector field on M generated by ZEu.. '. Then. cz (t) =T(exp H)T(exp te-ad"Z) (1) z (p) = d cz (t) dt t=o = dT (exp H) ,o (dT),(e-ad HZ). =dT(expH)o(sinh(-adH)e+cosh(-adH)X), (2) where 8 [X] respectively] is the E [p, respectively]-component of Z. Since U acts transitively on M) each tangent vector of M at P is in a form. Z7(P) for some ZGu. For eEf, c4 is a curve in ML LEMMA 1. Let 8 and rp be elements of E, then (i) Vector .field rp' generated by rp along ce(t) is written. rp'(c(t))==dT(expH)c,(t)odT(expte-adIZg),o(dT),(e-adHe-taderp),. where ci(t) =T(exp te-ad He). (ii) in Particular, rector .17eld 6' generated by e along c6(t) is written 6' (c (t)) = dT (exp H) c,(t)odT (exp te-ad He) .o (du),(e-ad He) .. PRooF. Generating curve of rp' is. rt(s)=T(exps)7)T(expt8)P. ". Calculate 7t(O), we obtain (i). Put rp=6 in (i), (ii) follows immediately.. Put t==O in Lemma 1, we have Li. LEMMA 2. Any tangent vector of M at P is written rp' ip) =rp' (c (O)) =dr (exp H),(sinh(-ad H) rp) , for some rpEIE.. LEMMA 3. I7br A(!la*, A'(P) is a normal vector to M at P.. PRooF. From Lemma 2, it suffices to prove <rpt (p), A, (P)>=O. for any rpEIE. Since cosh(-ad H)A=A and since dT(exp H)o is an isometry, we. have.

(3) Geometry of orbits of the isotropy group in a compact Riemannian symmetric space 13 <rp' (p), A' ip)>=<sinh(-ad H) rp, A>= <rp, sinh (adEl) A>= O.. '. Let V denote the Riemannian connection of the compact symmetric space M] V that of submanifold M induced by V, and cr the second fundamental forM of A(Z Covariant derivative of vector fields tangent to M is represented in Lie algebraic version as follows:. b. PRoposlTIoN 1. Let g' and rp' are tangent vector helds to M generated by e and rp, respectiz"ely. 772en the covariant derivative of rp' in the direction of 6'(P) at P zs zvrztten. (Ve,rp') p= - aT (exp H),([cosh (ad H) rp, sinh (ad H) e]) .. PROoF. Since T(exp(-H)) is an isometry, it leaves V invariant, transforms a. tangent vector 8' (P) at P into sinh(-ad U)6Ep=Mb, and a tangent vector field rp'(c(t)) along c(t) into a tangent vector field dT(exp(-H))c(t)rp'(c(t)) along ci(t)･. Since. dT. (exp se-ad Hrp) ci (t) = dT (exp (- H))e(t)rp' (c (t)),. ds .s=o. dT(exp(-H))rp'(c(t)) is a vector field along a curve ci(t) generated by e-adHrp. Thus, by Corollary 1.3. of [3], p. 188, and since the torsion T vanishes identically,. we have (Ve'rp')p=dT(expH)o(VdT(exp(-H))etdT(exp(-H))c(t)rp'(c(t)))t=o. = dT (exp H)o(- Ae-adHn sinh(-ad H) 8) = dT (exp H).([[e-adffny]{, sinh(-ad H) 6]). = dT (exp H).([cosh(-ad H) rp, sinh(-ad H) g) ,. where AiY=-fryX-T(X) Y) and [ ]f means the E-compont of an element of :. u. This completes the proof. Since Veop' is the tangent component of Vs,rp' and cr(6',v') the normal component, we have, from Proposition 1: PROpOSITION 2. Let 6', rp' and 4' be arbitrary tangent vectors of M at P and. let A'lv) be a normal vector to M at p generated by AEa*, then. (i) A'lv)-comPonent of the secondfatndamentalform of M in M zs <cr (g', v')., A' (P) >== -<sinh (ad H) e, cosh (ad H) [A, rp]> ,. and (ii) the 4'-comPonent of the covariant derivative v6op' in M is <V6,rp', 4'>p=<[cosh (ad H) rp, sinh (ad H) 6], sinh (ad H) C>. = <cosh (ad ll) rp, [sinh (ad H) 6, sinh (ad H) C]> .. g2. Representations in terms of restricted roots..

(4) 14 S. AKIBA. Let u* and b be as in g1. Then 6C is a Cartan subalgebra of uC. Let go. denote the dual Lie algebra of u in uC. Then go=E+vi=Ip is a Cartan decomposition of go. An involutive automorphism 0 of uC is defined to be +1 on EC and '. -1 on pC. Its restrictions to u and go are denoted by the same symbol 0. Put ao==bC+vi ::'lp, then a=bCnfand ao=V=la*. Let A be the set of all nonzero. roots of uC with respect to bC. Each crEa is real on bR:=ao+vi=Ia. Choose a compatible ordering in bC with respect to ao. Let a' be the set of positive roots with respect to the ordering. Let X be the set of the restriction to ao of. roots crEa which do not vanish identically on ao and let :' be the set of positive restricted roots.. Then uC[go, respectively] is decomposed into the direct sum of root spaces with respect to bC[ao, respectively]:. uC= bC+Zu.[go=go,+Z go, R, respectively].. aEd Ze2. Put u2=(go,2+go,2-R)nE+Vi=1(go,R+go,-2)np for apositive restricted root 2. Since go,2+go,-2 is invariant by the involutive automorphism 0 (see [2], p. 263), dim u2=2m2:=2dim go,2 and we obtain decompositions of u, C and p into direct sum of subspaces:. u=6+ Z uz, E=a+ Z h, p=a*+ Z pA. 2EX+ 2E2+ 2EE+･. where f2=uanE=(go,2+go,-2)nE and pA=u2np=V](go,2+go,-A)np Let X be an arbitrary element of p2, then X=V-1(X2+X-2) for some XhE go,2 and X.2(Eigo,-2. For any AEa*, since v/=IAEiao,. [A, x] ., [vi=1A, X>+XL2] =1(V=1A) (.[Y>-L[YH2).. Since [A, X]EE, X2-XLRc!C. Conversely, we observe. -. [A, X2-X-2]. =-[vi=1[v! IlrmIA, X2-XH2]. ...V.12(V=IA)(X2-X-2) = -1(viqA) X. Thus we have LEMMA 4. thr an arbilrary elemenl X of p2, there exists an element YEEi such that. [A,X]=2(V'::'-IA)Y and [A, Y]=-2(VX=ua1-A)X for any AEa*. H2nce p2+f2 is an eigen sPace of (adA)2 with eigen value - (2(v/=-IA))2 and adA exchanges p2 and E2. Since [go,2, go,pt]cgo,2+pt (== (O), if R+pt is not a restricted root), we have. 4･.

(5) Geometry of orbits of the isotropy group in a compact Riemannian symmetric space 15. LEMMA 5. Let 2 and pt be Positive restricted roots, then. [PR, Pp,]CER+F,+El2-p,l, [PR, Et,] C P2 + tt + Pi2- tt i, [f2, fpt] C f2 + pt + fi2- pt i,. where, ijC RgEpt, lZ-pt1 means Positive one of 2-Ft and lt-2, and of R== lt, E12-ptt=. a and pi2-pti=a*.. By the invariance of the Killing-Cartan form, we have. LEMMA 6. 71he direct decomposition. u=a+ Z ER+a*+2 pR 2El2+ 2E2+ is orthogonal.. PRooF. Since the Killing-Cartan form is invariant by automorphisms of u, especially bY the involutive automorphism 0, u==f+p is an orthogonal decomposition. So it suMces to prove that the decompositions. f=a+:Ei and p=a*+Zp2 are orthogonal.. Let Z>EE2+p2 and 4EE,+p, then <(ad A) 2Z>, Zb> =-<[A, ZR], [A, 4]>=<Zh, <ad A) 21,>. for any AEIa*. From Lemma 4, we have -R(V -IA) 2<a, 4>= - ge (V -IA) 2<a, ZL>. If R±pt, then 2(N/-IA) ?ept(vi-A) for some AEa*. Thus <Zh, 4>=O if Zipt. Similary we can prove the orthogonality of a and ER [a* and pA, respectively].. Now we shall develop formulas in Proposition 2 according to the decomposiF. tion E==a+ZREx+ E2.. l. PRoposlTIoN 3. Let 6, rp and C(EE) be decomPosed as. e=eo+Z&,rp=rpo+Em and C=Co+Z42.. REE+ ZE2E+ REE+. I7br tangent vectors 6', ny' and C' generated by 6, rp and 4(ii!E and a normal vector. A' generated by AGa*, we have sin (2R (vi - IH)) <a (6', rp'), A'>=- Z <[H) gd,[A, za]>,. x. 2z(vgH). and <v6･ny',4'>==,,e..Si"(2,(V(vtZl,IHB/in(fth(V.=)iH)).

(6) 16 S. AKIBA. × <cos((2+ pt) (H))m+,+cos((R- pt) (H))rp T2-,i, [[H] &], [H) 4pt]]>. PRooF. By means of Lemma 6, we have sinh(-ad H)s =.?¥ .Si:Ev2 (lmli-Hi)H)[H e2]. tL. and. cosh(-ad H)e= z cos(2(VtrlH))e.. -1 '. , EX+. Now the proof is an immediate consequense of Proposition 2, Lemma 5 and 6.. As stated in g1, each tangent vector of M at p is generated by some ZEu. Let g denote a map which assigns to ZEiu the generated vector Z' (p) Eit. Then q is a linear map of u onto iil,. Restriction of g to E [p, respectively] is written. - dT (exp H),osinh (ad H) [- dT (exp H)oocosh (ad H), respectively] . We define subspaces of u: '. Ez =￡ {ER; 2(vi -IH) EIi 7. Z} ,. ET= Z (fR; 2(V=IH) qi 7vz}, PN == Z {P2; Z(vi -IH) EzZ},. pz=Z{p2;Z(V=IH)= n+- ff,(i) nEZ)' and. PT=Z PR; 2(v/-1H) {g Gi-Z}. Then tt is decomposed into a direct sum of subspaces:. u= ET+Ez+a+a*+piv+pz+PT, and we have. ' PROPOSITIoN 4. Let ua denote thetangent sPace to M at P, and let Ml,±. t7. denote the normal sPace to M at P. 772en. (i) q maPs fT+pT onto Mlo and maPs CT isomorPhically onto ca. (ii) g maPs a*+pN isomorPhically onto M),±. (iii) 71e2e leernel of the linear q is. tz+a+Pz+b, whereb is a linear span of {2(vi=H)sin(Z(V-IHDX-cos(1(Vt-'tH))[H] X]; XEp2} with 2 such that 2(v'=IH)e(z/2)Z. PROOF. Since T(expte)P, 6EE, is a curve on M starting at P, q(e) is tangent. to Mat P. Since Kacts transitively on M] q mapsEonto Mp. Observing g(6) =dT (exp H),(sinh (ad H) 6). 1' '.

(7) Geometry of orbits of the isotropy group in a compact Riemannian symmetric space 17 = dT (exp H) .(z Sil "((ill)(-".!i-. 'Hi)H)[H; e2]),. ,. g(e)=O if and only if 2(V---IH)EilTZ. Thus g maps fT isomorphically onto ca.. For AEa*, we have alreadY shown that g(A) is normal to M (Lemma 3). Let XEp be such that <q(X),g(E)>=O for any eEE. Since ,. sin(2(V=1'H))crmos(R./Y-IH)). <g(X),g(e)>=Z2- 2(7t-H) <[ILXi],&> and since <,> is nondegenerate on E2xra, we have. sin(2(V=H))cos(1(V=IH)) [E X,]= O,. ny -･2(V='IH). for each ZEg'. Thus, if2(vi=rH)G(z/2)Z, we have [IZ XR]=O. By Lemma 6, it follows that XA=O. It is obvious thatq(X2)=O, if R(vi-IH)=(n+(1/2))n,. nEZ. Thus we have (ii). Let XGpT. It is easily seen that <g(X), g(Y)>=O for any Y(Ia*+pN. So g(X)GMp and we have (i). Let ZEu be such that g(Z)=O. If we write. Z== Xb+ Z X2+e,+Z &,. a2. then q(Z) Xb + 4 cos (Z (V L7IH)) X> + ¥ tj-"2(t-Vtti }- g))[IZ 6R]･. So we have Xb=O and ' cos(2(v=1' H))x+ Si:2/ttttit#))[I4 62]=O. for any REIS". Thus (iii) follows.. This completes the proof.. PRoposlTIoN 5. Let 7iZf be a compact symmetric spaceofranle oneandsumpose that there evists zaniqace positive restricted root R.. 71hen orbit sPace K(P) is Pseudo-umbilical with respect to the normal direction H'(p) at p.. 2111 ]ietrther, Z(-H) eTZ then H' (p) is the anique nGrmal direction and K(P) is an umbilical hmpersu2zf17ce of th; in particular, ijC R(vi=IH)=(n+(1/2))z for some nEZ, K(P) is a totally geodesic hmpersurftice of ]iiZL. PRooF. Observing that <e' (P), rp' (P) > = <sinh (ad H) g, sinh (ad H) rp>.

(8) S. AKIBA. 18. = sin2(2 (V --IH))<6, rp> ,. the assertions are immediate consequenses of Proposition 3 and 4. Bibliography [1]. ,. B. Y. Chen and T. Nagano, Totally geodesic submanifolds of symmetric spaces II, Duke Math. J., 45 (1978), 405-425.. [2]. S. Helgason, Differtial geometry, Lie groups, and symmetric spaces, (1978), Academic. [3]. S. Kobayashi and K. Nomizu, Foundations of differential geometry, Vol. II (1969), Inter-. sl. Press. ･ sclence.. 4.

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