L
ON INVARIANT SUBMANIFOLDS IN SASAKIAN
MANIFOLDS OF CONSTANTφ一SECTIONAL
CURVATURE＊BY
MASA田Ro KON
Introduction． In［6］the author studied the Simons’type formula for the second 負m（lamenta1 forrn of an invariant submanifold in ．a Sasakian manifold and proved the pinching theorem with resp㏄t to the length of the seco皿d fundamental fbrm ぐrheorem A）． 1ぷt M（k）be a Sasakjan manifbld of constantφ一s㏄tional curvature 2andハイbe its illvariant submanifbld． In§2 we study an invariant submanifbld M in．ルt（2）whose second fundamental form A satis丘es ll刈2＝（n十2）（k十3）／3， which was studied by Tanno［12］fbr an invariant submanifold in a sphere． Throughout 血this note， we〃24夕πo’aぷsu〃昭that the a〃zbient spaceルt iぷregular， i．e． the exist− ence of a fibra’輌o〃． In§3 we sha皿treat an垣variant submanifbld M of constant φ一s㏄tional curvature k 血1 M（k） and prove that M jb tota皿y geodesic or （k十3） ≧2（k十3），when元〉−3（Theorem 2）． Moreover if the second fundamental form of M ifη一parallel， thenハイis totally geodesic or（k十3）＝2（k十3）（Theorem 3）． Our notion ofη一paraUe1 sgcQnd fundamental form is essential ’for an invariant submanifold in a Sasakian manifbld because if the s㏄ond fUndamental fbrm is para皿el， then M iS totally geodesic［3，4］． In§4 we shaU prove the pinching theorem fbr the Ri㏄i curvature of an invariant submanifbld， which was proved by Harada［1］fbr an i皿一 vadant submanifbld in simply connected elliptic Sasakian space fbrms． Our theorem is the extension of． the theorem in［5］．1皿the last section an invariant submanifold with certain frame wM be treated．．ThiS was studied by口dden−Ogiue［7］fbr com− ・plex submanifblds hl Kaehler manifblds． 1．In▼ariant submanifolds in Sasaklan nianifolds． In this section we state some basic formUlas for an invariant submanfold in a Sasakian manifold and some results for latter use． Let M2（n＋♪）＋1（φ，ξ，η，8）be a Sasakian manifold of dimension 2（〃十P）十1． ture tensors of M satisfv f（）rany v㏄tor fields X and We denote the metric tensor field by＜，＞instead of g． The struc一 φ2−−1＋η⑧ξ， φξrσ；・η（φX）−0， η（ξ）−1， 〈φx，φγ〉−2（x，γ〉一η（x）v（γ）， η（x）一〈x，ξ〉 γon M． We denote by 7 the covariant’ differeutiation ＊ R㏄eived September 4，1974．［1］
2
M．KON
mMand by R the Riemannian curvature tensor of M． Then we can see
ゴ （7zφ）γ一η（γ）x−＜x， y＞ξ一」∼（x，ξ）】r， 7xξ一φx． DEFINITIoN． A submanifbld M of、荷of dimension 2〃十1 is said to be invariant ifξis tangent to ハイeverywhere onハイand φ・￥is tangent to M fbr any tangentV㏄torXtoM．
It iS well known that any invariant submanifold M with i lduced structure tensors， which w皿be denoted by the salne（φ，ξ，η，8）as in M， is also a Sasakian man− ifbld and minimal in M，（cf．［11］，［13］）． Let 7 be the covariant differentiation in M detern血ed by the induced metric on M． Then the Gauss−Weingarten formulas are given by 7xY＝7xY十β（−， Y）， x，γ∈駕（M）， 7xN＝−AN（X）十1）x1V， X∈駕（M）， 1V∈駕（M）⊥ Wherg D iS the linear conn㏄tion in the normal bundle T（ルの⊥and the second fUIIda− mental form of M satis丘es＜AN（X）， y＞＝＜B（X，η，∼V＞． The second fUnda− mental form A of M has the following pro’垂?窒狽奄?刀F φAN＝ノtφN＝−ANφ， ∠IN（ξ）＝・0． For the secorid fundamental fbrm A， we define its covar並mt derivative，7xA， by setti119 17x（A）N（η一7x（！IN（η）−ADxN（η一AtV−（7xγ）． If we assume that M iS of constantφ一s㏄tional curvature元， then we have （i・・5R（X， Y）Z−÷⇔（＜y，Z＞X−〈X，・Z＞η＋÷（2−1）（・（X）・（Z）・ 一η（γ）η（Z）X＋＜−，Z＞η（γ）ξ一＜】∼Z＞η（X）ξ 十＜φ｝∼Z＞φX十〈φZ，γ〉φγ一2＜φX，Y＞φZ）． Let．R be the Riemannian curvature tensor ofハ㌘． Then fbr any X，｝∼Z∈駕（M），we have
（1．2） R（−， ］1）Z＝」R（X， ］r）Z一ノIB（「・z）（X）十ノ4β｛x・z）（Y）， ”We denote by S the Rioci tensor of M． Then we have （1・・）・（＆ n−｝（・（E＋・）＋（元一1））〈X，γ〉丁（・＋1）（E−1）・（X）・（γ） 一〈A＊（x），Y・〉， 2P where the operator．4＊is de血1ed by．4＊一Σ（．4・）2 fbr a丘ame Vl，…， V2p for Tm（M）↓． a＝1 アrom this the scaler curvature．Kof M is given by （1．4） K＝n2（k十3）十〃（k十1）一・1レllj2 where ilAll denotes the length Of the s㏄ond fundamental fbmL If the R icci tensorSofMisofthef（）m
s（x，r）一α＜x，r＞＋々α）η（γ） whereαandゐ．are constants． then M is ca皿ed anη一Einstein manifold and whenクー0，』4誌an恥stein mni句ld・
Now we define the notion ofη一para皿el Ricci tensor and T−para皿el second funda− mental f（）rm．ON INVARIANT SUBMANIFOωS IN SASAKIAN MANIFOLI）S
3 DEFINITIoN． Let S be the Ri㏄i tensor o白Sasakian manifbld、M’． If 7x（S）（φ｝1， φZ）＝Of・・any・㏄t・・丘・1d・X・r・nd Z・n M・th・n w・SPy th・t th・．Ri・・i・t・P・q・ ぷof M isη一paralle1． ’PEFIMTIoN， Let M be all invariant submanifdd of a Sasakian．manifold Mf The second fundamental fbrmβis said to beη一para皿el if 7φx（B）（φY，φZ）＝O fbr anyvector飼ds X， rand Z on M．
・ If a Sasak海n manifbld」M’isη一Einste血1， then its Ri㏄i tensor is obviouslyη一par− allel． Wh㎝ぷ鉋η一paralle1， we．can prove easily that the scalar curvature ．K of M is constant and・the square of the］tength of the Ricci operatorρ鉋also cgnstant． We can， see that the s㏄㎝d fundamental fbm B isη一paraHel if．and・only if ＜7A・7・4＞＝311Al12．・ If the aml）ient space万is of conf㎞ntφ一s㏄tional．，Curvatute k，the Ricci tensorぷof M isη一parallel when、Bisη一para且el， apd hencC the scalar curvatule K of M is constant． Next．we shall state some reWlts which w皿be used垣this note．・F加t we・define the fb皿dwing tWo operatofs’翌?奄Uh are defined by Simons’［10］． ．’1 Letλイ2《蕗＋P｝＋1 be a Sasakian manifold andハイ2潟＋1 be an ilvariant sub血a五ifbld of ．M． We de五ne the symmetric， positive semi−definite operators Z and 4 by lsetting ∼ 2カ ・4＝tA。A and ，4＝ΣadAq．adAa． 1 P・ ． a＝1 1n［5，6］the author−proved the fdllowing ” 一＿LEMMA A・L・’M2n＋1 b・卿加・a・iant・ubmani2 old・！・、Sq・aki・4 manif・ld
ハf2（・＋カ）・・．：Then we加vε ’ ”’． 1 ’．’・内 ．・’ ÷11All・≦＜4・A・・A＞≦llAll・・己S［1・ll・≦〈A・i A5≦丁｜1・11・・ど鑑蒜蒜輌 1‘一θ晒醐∫⑭s’θ励”d°吟ヴ
LEMMA B・Lθ’M2剛＋1 be a Saぷakia〃〃anif・〃・∫c・ηぷtah’φ一sec伽al・c〃rv鋤re 肋〃d」lf2カ＋1．ゐθ㎝伽・廠〃’・ubmanif・ld・∫M乃窃ψρ・θ・・〃d吻damenial f・・mメぷa’輌ぷfies ・’
一く脳〉二・｜IAII・一〈 NA。A， A〉†〈4砲〉−ii （n＋・）ぽ）llメll・． LEMMA C・Let．M2（外＋ρ）＋1 bθaSaぷakia〃』manifo〃oゾcoηぷ’o〃’φ一sectional curvatqrekand M2”＋1舵an加〃iant submanif・〃・f M with T−pa・allel Ricei励w．ぴ
QA』Aaρ（a−1，…，ρ），ご乃ε〃． M∫ぷ，μ〃η一鋤ぷ’ein・姫nifQ．厄 一，． LEMMA D・L『’M2（π＋ρ）＋1 beαSasaki碗n ．an ifol4 ．Qf conぷtan’φ一ぷeetional c〃rvature k＞−3a〃d M2カ＋1 be・a・eo〃卿c’伽arian’ぷtibmaniftild of M with cons’砺ぷcalar eur． vat〃re・lf eAa一メ℃（4−1，・一， P）， then M・iぷαηη一Einstein manifold． ・ THEOREM A． Le’」豆2（銘士，｝＋1 bεaSasakian manifo）d”6ゾcoπぷταη’φ一seetio〃bl eur． γ微昧．励．M2や加αqempqc．t伽ρ・勧’submanif・14 ・f．M．．：．．． Then ，ei．ther M．輌ぷ’
4 M．KON
・−totally， geodeぷic or ll、4112＝（n十2X元十3）ノ3，0r atぷo〃le point〃z∈ルt， ・・ ’ … ll∠ll2（〃1）〉（n十2）（k十3）／3． THEOREM B． Let 荷2（錫＋P｝＋1 beα＆isakian〃ianifold o∫co〃ぷtantφ一sec’ional cur− vat〃re k a〃d M・・＋・be an invarian’η一E加ぷteinぷub〃ianif・ld・∫M． Then either M istotally 8θα脚c oア11All2≧〃（〃＋3）． ・’ 告
2．、’ln▼a，iant submani肋lds with l凶1・一（・＋2X万＋3）／3． In th誌・㏄ti・n． 浴E’唐?≠t・ttidy・⊇va血・t・ubma・if・ld M・…ili・．S・磁・m・・if・ld M2剛＋1・’with
constantφ一sectional curvatureえwhose s㏄ond｛undamental fbm ．4 sat麺es
l凶1・一（・＋2）（元＋3）／3，whi・h was s加di・d by士・P・・［12コwh・・’』m蹴卿r・ξ・ ・疵・己H・・ew・n㏄d・・t th・assumpti・・th・t th・．・mb輌・nt・p・C・iS・eg・la・ 8asak垣n man冠bld． … lTHEOREM 1． Let∼区2（n＋ρ）＋l be a Saぷakian〃lanifo〃o∫coπぷtan’φ一sec’ional c〃アー vat〃re 元〉−3 a〃d ．M2n＋1 bθan invariant sub〃ianifold o∫ハf． if I lメll2＝（π十2）（k十3）／3， ’物M∫ぷαηη一肋ぷ’ε加manif・ld・∫伽θπ∫’・η3and haぷ’乃￠ぷc吻r c〃γγαWεK−（k＋1）・『 ． ． ． ．
PRooF． By the assumption the．．square of the longth of the s㏄ond fundamentalform iS constant and we have−＜72A， A＞＝＜7A，7A＞． Therefore Lemma B
implies 〈・2，・∠〉一・IIAII・一〈A・♂A〉＋＜4・A， A＞一丁（・＋・）（k＋・）ll・1・・ °・th…hi・r h叫中・auth・・p・・v・d th・t＜7A・．7A＞≧31凶ll・．［．in［6コ・1Which sh°ws that（・．1） −．・≦〈屈λ〉＋＜銀，A＞一丁（〃＋・）（馴All・・ ．．
Si・ce・A i・ynm・t・i・， P・・itiv・・e面一d・血ite a・dφZ一砲，・・mg・・uit・bl・飴m・・A jS・・p…t・d bパ・亘・tt’X f・rm μ1 0 ・ ． い．．．∵．．Z斗・だ．’・∴．「『μ・・t一μ・．μ≧0・・ ．・・ 0 ’・、 ／．・ μ2♪ ． 、、’ 1
』FroM、this we obtain 、 ． 〈 NA・A，A〉・皇くぷ吃）・IAa＞一皇・・砲）・rVa＞・・．・tt・・
Q．・E．
ﾟ§。ε（2PΣμ、）iL・Sl。。μ、一＜A．i A＞，”・・
、 ． 「 、 ． ， ． σ＝1 、4＝1 σ≒6 ド、 二 and hen㏄・we get．the following ． J 乙』．．．． ’．．．、、 ．・（・②．・．・・、．《砧・かTll・1卜・t？x・・μ・・．
・Ah輌can』s已＜4。A，4＞≦1凶ドby i6㎜a A．．Thus（2．1）and（2．2）㎞ply
ON INVARIANT SUBMANIFOLI）S IN SASAKrAN MANIFOLDS
5 ・≦丁（・11A｜1・一（・＋・）僻・））ll・ll・一臓・・μ・・ ．： If llAll・一（。＋2）（2＋3）／3， th㎝￡μ。μ、−0． A。d w。 h。ve a1、。31剛・一＜7A，・7A＞． ’ aキめ ’ qnd the Ricci tensor S of M isη一parallgl． Moreover we may assume thatμ彦＝0 （a＝2，…，p）， and hence．4α＝0（a・・ 2，…，ク）． Therefbre we have．4＊＝2（ノ11）2． On the other hand，（1．3）㎞pHes． “・i−i（n（k＋3）＋（2−．1））X一丁（・＋1）（2−1）・（X）ξ一・（・・）・X・ From this we can prove that CAa− AaO（a＝1，…， p）． From this and Lemma C， ．M is anη一Einstein manHbld． If M is anη一Einstein manifbld， we get l凶12≧n（ん十3） by Theorem B， and hence we have inequality （n十2）（元十3）／3≧η（疋十3）， ・ which implies that〃＝1and dim M＝3． From this and（1．4）we obtain K＝（ん十3）． This completes our assertion． 3・ Invariant submanifOl己s wit血constantφ一sectional Furvature・ In this section we assumeてhatハイ2（カ＋♪）＋1 be a Sasa㎞n mallifbld of constantφ一sectional curvature． 泥and M2n†1 be an invariant submanifold of、M of constantφ一sectional curvature k： N ア Then Lemma A shows that＜4・A， A＞＝llAl14！〃． Next we shaU皿1culate＜A。A， A＞． N From（1．2）and the de血nitioll of．4， we obta血 ‘こ パ＋1 暑、〈AV（Ei）・R（Ei・，x）「−R（E，・x）・「〉 ．．’．． n＋1 ＝一Σ（〈AV（長），メB（x・「）（Ei）〉一〈AV（Ei），．4耶i・7）α）〉） ‘＝正 ● 一一＜AA（の（x）， r＞・． ．『 S垣ce、M and 」∪「are of constant φ一s㏄tional curvature k and k resp㏄tively，（1．1）・ implies N A・A＝（k−k）A． From this we obtain the fbllowi皿g ダ ロ ＜A・A，A＞一（k−k）1レtli2． By usillg the equation IlAII2＝n（〃十1）（元一k）， we have ． く属⇔。（．it，）ll小 Therefore Lemma A shows that if p＜n（n十1）／2， then M is totaHy gCodes掩祖π． And we obtdin， by Lemma B， ・≦＜・A・・7A＞一・ll・ll・一・ぴ＋1）（〃＋・）（E一り（丁（E＋・）一（・＋・））・ From this we have the fbllgwing ． 、．． THEOREM 2． Lθ’M2（錫＋♪｝＋1 be a Sasakian〃ianifold o∫eonstan’φ・ぷeetional eur・「 vature k andハ42n＋1 be an invariantぷubmanifold of M of eonstantφ一ぷebtional eur− vature k． ぴ瓦〉−3， then either M元ぷtotally geodeぷic， i． e．， k＝k」or （k十3）≧2（k十3）．6
M，KON
THEOREM 3〔五ε’M2｛”＋カ）＋1みeαSasakia〃〃2anifo〃o∫’coπぷ’antφ一ぷectional c〃r・ vature k and M2斑be an in’・variantぷ励ηα屍ノb〃of M of conぷ吻’φ一ぷectional C〃r一γ4航・左．〃’he・Secδnd伽dame％1允・m’A・∫M∫ぷη一畑α〃どL’乃・π泥＝瓦励」lf
is’oずally geodesicカ2π or（k＋3）＝2（ノヒ十3）， the latter case bris’ ing o〃［夕whehえ〉−3． 4．Ricci curvature of in▼ariant submanifolds． In this s㏄tion we prove the fbl− 10willg theoreロ1 which Was proved by Harada［1］fbr invarlait submanifblds i11 ぷ」〃⑳co雄cτεば鋤τ匡c＆isakianΨαcθ動〃and this theorem iS the’extension of the theorem in ［5］． THEoREM 4． Lθ’M2（n＋P｝＋1ゐθaSaぷakian栩anifo〃oアco〃ぷtantφ・ぷeetio〃al Cur・’ ．γature k＞−3 and M2n＋1 be a eo〃rpact invariantぷub〃mnifold o∫ルt， If every Ricci euryatu．re o∫Mis greateγthan n（k十3）／2−2， then M is totaltンgeodesic． PRooF． By（1．3）we have ＜4。！1，ノ1＞＝2乙（∠1＊）2 2ク2ヵ＋1 ＝（海（E十3）＋（X−1））1レII12−2ΣΣぷ（A・（Ei）， A・（Ei））≦（え＋3）1レlll2． 一 α＝1‘＝1 ． On the other hand， we have always＜4。A， A＞≧lIAlI4／n by Lemma A． Thus we obtain ． ．（4．1） r （π（元十3）−1レ引12）1レill2≧0． ’And the Simons’type formula implies ．
1．（＜7∠1，7／1＞−31レIll2）＝”！M（〈砧メ〉＋＜4・A・A＞一丁（〃＋・）（k＋・）1凶1・） − ≦Tl。（1凶12一刀（2＋3））llA‖・・ From this and（4．1）， we have the equat拓n （n（ん＋3）−1レtll2）llAll2−0． But we can prove easily that ll、4112＜n（ん十3）by using（1．3）． Therefbre we have lレ4U2＝O and henceハイis totaHy geodesic． ’ Next we ass㎜e that眺an恥stem manibld， i，e．，，8（X，γ）＝2π＜X， Y＞． And if 2n＞〃（元十3）／2−2， lhen M is tota皿y geodesic by the above theorem． Thus weobtain
THEoREM 5． Letハ42（抄＋ρ1＋1 beαSaぷakian manifo∼d o∫constant｛かsectional eur−． vatu「e えi ahd M2π＋1 ゐε an inva「iant Einstein ぷ〃6”2απ」ゾb”o／M＝ 1ア元＜1＋4／π， ’乃θη Misτoτ醐y geodesic and hence k＝1． COROLLARy 1． Let M2に＋ρ）＋1 be a Saぷakian〃2απ〃b∼d∂f conぷtantφ一ぷeetional cur鴎 vature k＞−3 a〃dルt3 beαco〃脚cτ3一碗〃iensional i’nvariant sub栩anifo∼ば《ゾM．．ぴ the seetional curyature ｛ゾMis greater than（2十3）／2−2 everγ｝vhere onル1，’乃θμM’∫totally geodesic． ． ・ ・ ． ・．． ．
ノ 、
ON INVARIANT SUBMANIFOIDS IN SASAKIAN MANIFOLDS
7
、 5．In▼ariant submanifblds wit血￠ertain frame・Iet M2（n＋♪）＋1 be a Sasakian man− ifbld of constantφ一sectional curvature k and M2n＋1 be an invariant submanif（）1d of M． In thjs s㏄tion we shall study the following condition（＊）w垣ch was studiedby Ludden−Og垣e［7］fbr complex submanifbldざ垣Kaehler manifolds．
（＊）At each pomt m of M， there exists an orthonomlal frame Ei，…， En，φE，，…． φE．，ξsuch that B（Ei， Ej）−O fbr 1≦i・4：ノ≦n． THEOREM 6． Lθ’ルf’2〔労＋P）＋1 be a Sasakian〃iani．fold o∫constantφ一ぷectional cur−vature E＞−3 and M2n＋1加aeo〃rpact invarian’、submanifo〃of M／with constant
ぷcalar eurvature． ∬ルt has the properりノ（＊）， thenルt∫ぷanη’E加∫’ε元〃〃2anifold． PRooF． The condition（＊）is equivalent to the fbllowing． （＊）’at each point．m bfルちthe佗e巫ts an orthonormal frame Ei，…， E．，φ昼． …，φE』，ξwith respect to which the matrices of the・4“’s are of the fbm1 、Aa＝
λla ． ・0
0
● z蕗” μ1α ・ ・ ・ ．0
0
0
Ptna μ1α ．0
・ 一λ1a ・ ・ ．0
0
0
Pt＃a0
コ ーλヵa0
0
0
● From this we can see that（Aa）2 is simultaneously diagonalizable as the following （Aa）2一 Tia ・ ● 万α Tia ● ● ● T．a0
， where Tia＝（λ‘り2十（μの2（匡＝1，…，〃）． From this we obtah1．（Aa）2Ab＝4b（ノ4α）2 fbr any a，b＝1，’■・・，ρby using the condition（＊）’． Then the Ri㏄i operator（∼of M sat＿ 姪fiesρ．4a＝．4℃（a＝1，…，」ρ）． Thus we have our assertion by using Lem輌na D． By the similar method and Lemma C， we have THEOREM 7． Let M2（n＋P｝＋1 be a Sasakiαn manifo〃ρ∫conぷ’απ’φ一ぷec’ional c〃r一 ツature lξ and M2π＋1加 an カ1ツariantぷ〃b〃ianifold（ゾ」M｝vith OP−para〃どl Ricci励SOア． ∬ハf．has’heρrρρθ’砂（＊），’乃θπM’iぷanη’Einstein〃lanifo〃． THEOREM 8． L訂M2（”＋♪）＋1（it＞1）ゐθaSasakian〃anifo〃oゾω〃ぷtan’φ一ぷec’io〃α∫ curyature瓦a〃d M2n＋1加an inヅarian’ぷubmanifo〃げMof COπぷ」fan’φ」ぷectio〃al cur− wat〃re k．・ぴMhas’乃θproperリノ（＊）， the〃バグ匡ぷtotaUンgeodeSic．8
1M． KON
PRooF． Theφ一s㏄tional curvature H of M is given by H（X）＝元一2113（X，X）ll2． By the assumption we get llβ（x， x）H・一丁（E−k）・ 、 Usi皿g the property（＊）， we can see 2P Σ〈β（Ei， Ei），β（Ei， Ei）〉＝Σ〈（Aつ2（Ej）， E」〉一丘一ん． ‘＝1 a＝1 Thelefbre we obtain l l、4112＝2η（k−k）． On the other hand， we see lレ倒12r〃（〃十1）（k−k）， and hence we get k＝k， i．e．，ノlf is totally geodesic． 一 REMARK． In this note we proved some theorems when the ambient space M has constantφ一sectional curvature k＞−3． In the fbllowing we prove a theorem in which the amblbnt space M has constantφ一s㏄tional curvature k≦−3． “ THEOREM． Let M2（π＋♪｝＋1 be a Sasakian〃m〃ifold of conぷtant ￠一ぷeeガo〃al curγature 瓦and M2n＋1●θan invarian’ぷubmanifold of M with OP ・parallel seeond funda〃iental ／b朋．if k≦−3，’乃θηMiぷ’o’α1砂9即伽ξc inλf． if k＞−3， then theぷea∼br・cur− vature K of Mぷa’isfies K≧η2（h十ρ＋1）（E＋3）／（η＋2P）−2〃and if equality holds， then Miぷanη一Ei，〃ぷ’痂％〃ifold． ’ PRooF． From Lemma A and Lemrna B， we get the following ・一＜7A・・7A＞一・IIAII・≧（W＋IP”Al）・一（〃＋21尭＋3））jlAll・・ Hence ifえ≦−3， then M is tota皿y geodesic五1 M． Let元〉−3． Then we obtahl l剛2≦pn（π＋2）（k＋3）／（π＋2P）． From this and（1．4）， we can see K≧n2（〃十p十1）（k十3）／（n十2p）−2〃． If the equality holds， then l凶12＝蜘十2）（疋十3）／（n十2p）and hence，’by Lemma A and Lemma B， （丁（n＋・）（綱一＄IIAII・）11All・≧＜4・A・・A＞÷剛・・ 丁（・＋・）（E＋・）−tll・1［・一÷1剛・・ which imply＜4。A， A＞＝IIAI14／n． Therefore by Lemrna A， M is an ll−E血ste血 manifbld．REFERENCES
］−﹂ −凸ウ一 ﹁﹂［ ［3］ ［4］ M．Harada：On Sasakian submanifolds， T6hoku MathJ．，25（1973），103−109． K．Kenmotsu：Invariant submanifolds in a Sasakian manifold， T6ho㎞Ma hJ．，21 （196g）， 495−500． M．Ko皿：Invariant submanifolds in norma1 contact metric manifolds， K6dai Math． Sem． R．ep．，25（1937），330−336． ． tt． M．Kon：Anote on血variant submanifolds in a K−contact Riema皿亘m manifold， Tensor，27（1973），158−−160．［5］ −﹂−■一 107’
