Complete totally real submanifolds of a complex projective space

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1 Complete totally real submaifolds of a complex projective space Yoshio Matsuyama Abstract. The preset paper deals with the classificatio of a complete totally submaifold of a complex projective space by applyig Bocher formula. M.S.C. 00: 53C7, 53C40, 53C4. Key words: complete submaifold; totally real submaifold; complex projective space, Bocher formula; Gauss ad Weigarte formula. Itroductio The study of submaifolds of a Riemaia space form (i particular complex space form) has bee a area of iterest for may differetial geometers for may years. I [], Barros studied the properties of compact miimal submaifolds of the Euclidea sphere S ad obtaied a characterizatio of S. Moreover usig Obata s theorem [9], Okumura [0] proved that a ( )-dimesioal complete simply coected totally umbilical submaifold with o-zero costat mea curvature of a -dimesioal locally product Riemaia maifold is isometric to a sphere. I [6], Rio, Kupeli ad Ual characterized Euclidea sphere usig a stadard differetial equatio which is the aother versio of Obata s differetial equatio. O the other had, Djoric ad Okumura [5] discussed -dimesioal CR-submaifolds with ( ) as CR-dimesio i a complex projective space ad established a iequality betwee Ricci tesor, the scalar curvature ad the mea curvature. Later, Pak ad Kim [] studied CR-submaifolds with ( ) as CR-dimesio i a complex hyperbolic space. Recetly, we studied of the geometry of complete submaifolds of a Riemaia space form ad proved the follwoig [8]; Let M be a complete submaifold of a Riemaia space form M +p (c), (c 0) with the Ricci curvature bouded from below ad without boudary. If M admits a real valued o-costat fuctio f such that f + λf = 0 ad λ c, the M is either isometric to a sphere S for λ > 0 or isometric to a warped product of the Euclidea lie ad a complete Riemaia maifold whose warpig fuctio ψ satisfies the equatio d ψ dt + λ ψ = 0. Ad, let M be a complete -dimesioal CR-submaifold without boudary ad with the Ricci Differetial Geometry - Dyamical Systems, Vol.0, 08, pp c Balka Society of Geometers, Geometry Balka Press 08.

2 0 Yoshio Matsuyama curvature bouded from below ad CR-dimesio( ) i the complex space form M (+p) (4). If f : M R is ay smooth fuctio o M satisfyig the coditios f + λf = 0 ad λ, the M is isometric to oe of the followig: (a) coected compoet of the hyperbolic space, (b) warped product of the Euclidea lie ad a complete Riemaia maifold, where the warpig fuctio ψ satisfies the equatio d ψ dt + λ ψ = 0, (c) Euclidea sphere. The purpose of the paper is devoted to study the geometry of a totally real submaifolds of a complex projective space. The mai result of the paper is the followig: Theorem Let M be a complete totally real submaifold of a complex projective space M with the Ricci curvature bouded from below ad without boudary. If M admits a real valued o-costat fuctio f such that f + λf = 0 ad λ, the M is isometric to oe of the followig: (a) coected compoet of the hyperbolic space, (b) warped product of the Euclidea lie ad a complete Riemaia maifold, where the warpig fuctio ψ satisfies the equatio d ψ dt + λ ψ = 0, (c) Euclidea sphere. We remark i the future we wat to apply these way of this paper to CRsubmaifolds i quaterioic space forms which was defied by M. Barros, B-Y Che ad F. Urbao []. Prelimiaries Let M be the -dimesioal complex projective space with the Fubii-Study meric of costat holomorphic sectioal curvature 4 ad let M be a complete submaifold of M. Let us cosider a immersio ψ : M M ad let {e, e,..., e, Je,..., Je } be a adapted orthoormal frame of M such that {e, e,..., e } is a orthoormal frame to M ad {Je,..., Je } is a orthoormal frame of the ormal budle T M of M, where J is the complex sturucture of M. We deote by ad the Levi-Civita coectio o M ( ) ad M, respectively. The the Gauss ad Weigarte formulas are give by (.) (.) X Y = X Y + h(x, Y ), X Je i = A i X + XJe i, i =,,..., for ay vector X, Y taget to M [4], where A i is give by A Jei. Here deotes the ormal coectio iduced from i the ormal budle T M of M, ad h ad A α are the secod fudametal form ad the shape operator correspodig to Je i, respectively. Further, h ad A i are related as (.3) h(x, Y ) = g(a i X, Y )Je i. i=

3 Totally real submaifolds The we have the followig equatio g(h(e i, e j ), Je k ) = g(a i e j, e k ).. The mea curvature vector H is give by H = Gauss is give by (tra i )Je i. The equatio of R(X, Y, Z, W ) = g(y, Z)g(X, W ) g(y, W )g(x, Z) + g(jy, Z)g(JX, W ) g(jx, Z)g(JY, W ) The we have (.4) Ric(e i, e j ) = ( )g(e i, e j ) + i= +g(x, JY )g(jz, W ) + g(h(y, Z), h(x, W )) g(h(x, Z), h(y, W )). (tr A k )g(a k e i, e j ) k= g(h(e k, e i ), h(e j, e k )). The followig geeralized maximum priciple due to Omori [] ad Yau [3] will be used i order to prove our theorems. Theorem.. Let M be a complete Riemaia maifold whose Ricci curvature is bouded from below ad f C (M) a fuctio bouded from above o M. The, for ay ɛ > 0, there exists a poit p M such that (.5) k= f(p) supf ɛ, gradf < ɛ, f(p) < ɛ. For a fuctio f : M R, Bocher formula is give by [] f = Hess f + Ric( f, f) + g( f, ( f)) where Hess, Ric ad stad for the Hessia form, Ricci tesor ad the Laplacia, respectively, ad the square of the orm of a operator A is give by A = tr(aa ). 3 Applicatio of Bocher formula i space forms The results of the paper will be proved by appyig Bocher formula. To prove theorem, we eed the followig lemma which we will state ad prove first. Lemma 3. Let M be a submaifold without boudary of a complex projective space M, Let f : M R be ay fuctio o M ad λ be the first eigevalue of the Laplacia of M, i.e. f + λf = 0. The for ay t R we have Hess f = Hess f tfi (t + t λ )( f f ), where Hess f ad I deote the Hessia operator of f ad the idetity operator, respectively. The orm of ay operator A is Euclidea, i.e. A = tr(aa ). Proof. We have Hess f tfi = Hess f + t f I tfihessf.

4 Yoshio Matsuyama for ay t R. It is clear that I = tr(ii ) = ad IHess f = trhess f. Now Therefore which implies that f = g ij j i f = i i f = trhessf. Hess f tfi = Hess f + t f + tλf, (3.) Hess f tfi = Hess f + (t + t λ )λf. Also we kow that This gives f = f f + f. (3.) λf = f f. From equatios (3.) ad (3.) we get which implies that Hess f tfi = Hess f + (t + t λ )( f f ), (3.3) Hess f = Hess f tfi (t + t λ )( f f ). Proof of Theorem: Equatio (.4) yields Ric(f i e i, f j e j ) = ( )f i f j g(e i, e j ) +,k f i f j g(h(e i, e k ), h(e j, e k )), f i f j g(h(e i, e i ), h(e j, e j )) where f = i f i e i. This gives us Ric(f i e i, f j e j ) = ( ) f + f i f j g(h(e i, e i ), h(e j, e j )) f i f j g(h(e i, e k ), h(e j, e k )),k = ( ) f + + g(h( f, e i ), h( f, e i )) (3.4) = ( ) f + h( f, e i ).

5 Totally real submaifolds 3 It remids Bocher formula (.4) f = Hess f + Ric( f, f) + g( f, ( f)). Now pluggig the values of Hess f ad Ric( f, f) from equatios (3.3) ad (3.4) ito equatio (.4), we get f = Hess f tfi (t + t λ )( f f ) + ( ) f + h( f, e i ) λ f. Also accordig to the defiitio of the first eigevalue λ we must have λ [3], [9] ad the assumptio of f + λf = 0 ad hece f = Hess f tfi + t (t + λ ) f Ric( f, f) ( ) f +( ) f + h( f, e i ) ( ) λ f (t + t λ + λ ( ) λ ) f. If t = λ (3.5) It is easy to see that (3.6) the the R.H.S. of the above equatio reduces to f + λ f Hess f + λ fi 0. Hess f + λ fi 0. From the assumptio of the Ricci curvature bouded from below ad equatios (3.5), (3.6) we coclude that fi = 0, which implies that Hessf + λ fi = 0. The above result for λ 0 breaks up ito two possible isometries of M give by (i) M is isometric to a coected compoet of the hyperbolic space if ( f) p = 0 at some p M [6]. (ii) M is isometric to the warped product of the Euclidea lie ad a complete Riemaia maifold if f is o-vaishig, where warpig fuctio ψ o R satisfies the equatio [6] d ψ + λψ = 0, ψ > 0. dt

6 4 Yoshio Matsuyama Further if λ satisfies the iequality 0 < λ, the from equatio (3.5) we have (3.7) f + λ f fi 0. But we clearly have (3.8) fi 0. Combiig the assumptio of the Ricci curvature bouded from below ad the iequalities (3.7), (3.8), we obtai which gives fi = 0, Hessf + λ fi = 0 for 0 < λ. Hece M is isometric to a sphere [9]. This completes the proof of the theorem. Refereces [] A. Barros, B. Y. ad F. Urbao, Quaterio CR-submaifolds of quaterio maifolds, Kodai Math. J., 4 (98) [] A. Barros, Applicatios of Bocher formula to miimal submaifold of the sphere, J. of Geom. ad Phys., 44 (00), [3] M. Berger, P. Gauducho, E. Mazet, Le spectre d ue variété Riemaiaee, Lecture Notes i Math., 94, Spriger-Verlag, Berli,97. [4] B. Y. Che, Geometry of submaifolds, Marcel Dekker, Ic. New York, 973. [5] M. Djoric ad M. Okumura, Certai applicatio of a itegral formula to CRsubmaifold of complex projective space, Publ. Math. Debrece, 6/-(003), 3-5. [6] E. Garcia-Rio, D. N. Kupeli ad B. Ual, O a differetial equatio characterizig Euclidea sphere, J. of Diff. Eqs., 94(003), [7] Y. Norifumi ad Y. Matsuyama, O a Kaehler hypersurface with the cyclic Ricci semi-symmetric tesor, Acta Math. Si. (Egl. Ser.) 5 (009), [8] Y. Matsuyama, A. A. Shaikh, M. H. Shahid ad M. Jamali, Complete sumbmaifolds of space forms ad applicatio of Bocher formula, J. Adv. Math.Stud. 8 (05), [9] M. Obata, Certai coditios for a Riemaia maifold to be isometric with a sphere, J. Math. Soc. Japa, 4(96), [0] M. Okumura, Totally umbilical hypersurface of a locally product Riemaia maifold, Kodai Math. Sem. Rep, 9(967), [] H. Omori, Isometric immersios of Riemaia maifolds, J. Math. Soc. Japa 9 (967), 05-4.

7 Totally real submaifolds 5 [] J. S. Pak ad H. S. Kim, Applicatio of a itegral formula to CR-submaifolds of complex hyperbolic space, It. J. of Math. ad Math. Sci., 7(005), [3] S. T. Yau, Harmoic fuctios o complete Riemaia maifolds, Comm. Pure ad Appl. Math. 8(975), 0-8. Author s address: Y. Matsuyama Departmet of Mathematics, Chuo Uiversity, -3-7 Kasuga, Bukyo-Ku, Tokyo -855, Japa. matuyama@math.chuo-u.ac.jp