SOME INTEGRAL FORMULAS FOR CLOSED MINIMALLY IMMERSED HYPERSURFACE IN THE UNIT SPHERE S n+1

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1 TWS J. Pure App. ath. V.1 N pp SOE INTEGAL FOULAS FO CLOSED INIALLY IESED HYPESUFACE IN THE UNIT SPHEE S +1 IHIBAN KÜLAHCI 1 AHUT EGÜT 1 Abstract. I this paper we obtai some itegra formuas for miima hypersurfaces i the uit sphere S +1. Keywords: Cifford Torus; miima hypersurfaces; ricci curvature; scaar curvature. AS Subject Cassificatio: 53C65 1. Itroductio The Cifford torus is the oy miima surface i S 3 with costat cotact age. The study of miima surfaces has payed a formative roe i the deveopmet of mathematics over the ast two ceturies. Today miima surfaces appear i various guises i diverse areas of mathematics physics chemistry ad computer graphics but have aso bee used i differetia geometry to study basic properties of immersed surfaces i cotact maifods [6]. ay wors have bee doe reated to itegra formuas by may mathematicias (see [1] [3] [4] ad [7]. For exampe Cao [1] obtaied itegra formuas for miima space-ie hypersurfaces i (+1-dimesioa idefiite space form. I additio Ximi [7] gave simiar itegra formuas for miima space-ie hypersurfaces i (+p-dimesioa idefiite space form. Later sigificat wors i this directio have bee obtaied by Küahcı Ergüt ad Betaş [34]. I this paper we coduct a study about miima hypersurfaces i the uit sphere S +1. However to the best of our owedge these itegra formuas have ot bee preseted for cosed miimay immersed hypersurface i the uit sphere S +1. Thus the study is proposed to serve such a eed.. Preimiaries Let be a -dimesioa hypersurface i a uit sphere S +1. We choose a oca orthoorma frame fied i e 1... e +1 } i S +1 so that restricted to e 1... e are taget to. Let w 1... w +1 deote the dua co-frame fied i S +1. The i It foows from Carta s Lemma that w +1 = 0. w +1i = j h ij w j h ij = h ji. (1 The secod fudameta form h ad the mea curvature H of are defied by h = ij h ij w i w j e +1 ad H = i h ii. ( 1 Departmet of athematics Firat UiversityTurey e-mai: mihribauahci@gmai.com e-mai: mergut@firat.edu.tr auscript received 4 December

2 8 TWS J. PUE APPL. ATH. V.1 N We reca that is by defiitio a miima hypersurface if the mea curvature of is ideticay zero. The coectio form w ij is characterized by the structure equatios dw i + w ij w j = 0 w ij + w ji = 0 j dw ij + w i w j = Ω ij (3 Ω ij = 1 ij w w where Ω ij (resp. ij deotes the curvature form (resp. the compoets of the curvature tesor of. The Gauss equatio is give by ij = (δ i δ j δ i δ j + (h i h j h i h j. (4 The covariat derivative h of the secod fudameta form h of with compoets h ij is give by h ij w = dh ij + h j w i + h i w j. The the exterior derivative of (1 together with the structure equatios yied the foowig Codazzi equatio h ij = h ij = h ji. (5 Simiary we have the covariat derivative h of h with compoets h ij as foows h ij w = dh ij + h j w i + h i w j + h ij w ad it is easy to get the foowig icci formua h ij h ij = m h im mj + m h mj mi. (6 From ow o we assume that is miima. Deote by S = ij h ij the square of egth of h. The compoets of the icci curvature ad the scaar curvature are give respectivey by ij = ( 1δ ij h i h j (7 = ( 1 S. (8 It foows from (8 that S is costat if ad oy if is costat. For ay fixed poit p i we ca choose a oca orthoorma frame fied e 1... e such that h ij = λ i δ ij. (9 Let S = h ij. The foowig formuas ca be obtaied by a direct computatio ij The Gauss-Kroecer curvature K of is defied by h ij = ( Sh ij (10 1 S = h ij S(S. (11 ij K = det(h ij. (1 Let be a -dimesioa cosed miimay immersed hypersurface i the uit sphere S +1. Assume i additio that has costat scaar curvature or costat Gauss-Kroecer curvature. I this paper we aouce that if has ( 1 pricipa curvatures with the same sig ( everywhere the is isometric to a iemaia product S 1 1 iemaia product aso correspod to Cifford Torus [5]. S 1 ( 1. This

3 . KULAHCI. EGUT : SOE INTEGAL FOULAS FO CLOSED From ow o we assume that is a iemaia product. ( Theorem.1. Let be a -dimesioa compact iemaia product S 1 1 the S 1 ( 1 mij + } mj + dv 0 (13 1 where mij is the square egth of the sectioa curvature mj is the square egth of the icci curvature tesor is the scaar curvature dv is the voume eemet ( of. ( Theorem.. Let be a -dimesioa compact iemaia product S 1 1 S 1 1 the 1 mij + } mj dv ( 1 S } V o( (14 where mij is the square egth of the sectioa curvature mj is the square egth of the icci curvature tesor S is the square of egth of secod fudameta form dv is the voume eemet of. ( ( Theorem.3. Let be a -dimesioa compact iemaia product S 1 1 S 1 1 the 1 mij (3 S + 1 } S dv ( + 3 1V o( (15 where mij is the square egth of the sectioa curvature S is the square of egth of secod fudameta form dv is the voume eemet of. 3. Proof of theorems Proof of Theorem.1. From the defiitio of Lapacia we have h ij = h im mj + h m mij. (16 From (4(16 ad taig ito cosideratio is miima we get hij h ij = h ij h m mij + h ij h im mj hij h ij = 1 (hij h m h mj h i mij + (h ij h im h ii h jm ( mj. (17 If the equaity (4 is used i the first term at the right side of the equaity (17 we fid 1 (hm h ij h mj h i = 1 [ (δ mδ ij δ mj δ i mij ]. (18 If the equaity (4 is used i the secod term at the right side of the equaity (17 we have (him h ji h ii h jm = (δ im δ ji δ ii δ jm + ijim. (19 If (18 ad (19 are writte i (17 we obtai hij h ij = 1 [ (δm δ ij δ mj δ i mij ] mij + or + [(δ im δ ji δ ii δ jm + ijim ] ( mj hij h ij = 1 mij + mj + 1 [ (δm δ ij δ mj δ i ] mij + (0 + [(δ im δ ji δ ii δ jm ] ( mj.

4 84 TWS J. PUE APPL. ATH. V.1 N After some cacuatio ast two terms at the right side of (0 are obtaied as the foowig: 1 [ (δm δ ij δ mj δ i ] mij + [(δ im δ ji δ ii δ jm ] ( mj =. (1 If (1 is writte i (0 we fid hij h ij = 1 mij + mj +. ( Sice h ij h ij } dv 0 [] we have the foowig: 1 mij + } mj + dv 0. This competes proof of theorem.1. Proof of Theorem.. If (8 is cosidered i (13 the proof of the theorem. is trivia. I order to prove theorem.3 we eed the foowig emma. Lemma. Let a 1... a be rea umbers the (ai 1 ( ai (3 where the equaity sig hods whe ad oy whe a 1 =... = a. Proof of Theorem.3. If (9 is cosidered i (7 we have mj = ( 1δ mj λ jδ mj. (4 If (3 is used i (4 we fid mj = ( 1 ( 1S + λ 4 j ( 1 ( 1S + 1 ( λ j mj ( 1 ( 1S + 1 S. (5 If (5 ad (8 are used i (13 we obtai 1 mij + ( S(3 + 1 } S dv 0 or This competes the proof. 1 mij (3 S + 1 } S dv ( + 3 1V o(. efereces [1] Cao X (1999 Itegra iequaities for maxima spaceie hypersurfaces i the idefiite space form Differetia Geometry ad its Appicatios North-Hoad 10 pp [] Cher S S Carmo.do ad Kobayashi S(1968 iima submaifods of a sphere with secod fudameta form of costat egth i: Fuctioa Aaysis ad eated Fieds Proc.Cof. Chicago (Spriger New Yor 1970 pp [3] Küahcı Ergüt ad Betaş (004 Some itegra formuas o compact iemaia submaifods i m F.Ü. Fe ve ühedisi Biimeri Dergisi 16(3 pp [4] Küahcı Ergüt ad Betaş (004 A itegra formua for compact submaifods i m Iteratioa Joura of Pure ad Appied athematics 16(1 pp [5] Luiz A Sousa J (001 igidity theorems of Cifford Torus A. Acad. Bras. Ciec. 73(3 pp [6] otes ad Verderesi J A(007 iima surfaces i S 3 with costat cotact age arxiv: v1[math.dg] ay.

5 . KULAHCI. EGUT : SOE INTEGAL FOULAS FO CLOSED [7] Ximi L (001 Itegra iequaities for maxima spaceie hypersurfaces i the idefiite space form Baa Joura of Geometry ad Its Appicatios 6(1 pp ihriba Küahcï - received the B.Sc.degree from Firat Uiversity Eazig Turey i 000..S. ad Ph.D. degrees i geometry from Firat Uiversity Eazig Turey i respectivey. She is assistat professor doctor at the athematics Departmet Firat Uiversity. Her mai research iterests are iemaia maifods semi- iemaia maifods itegra formuas o maifods ad curves. ahmut Ergüt - received the B.Sc.degree from Ege Uiversity Izmir Turey i S. ad Ph.D. degrees i geometry from Firat Uiversity Eazig Turey i respectivey. He received associate professorship i Sice 001 he has bee a professor at the athematics Departmet Firat Uiversity. Whie wide ragig i iterest his research is focused argey o the rued surfaces Loretzia geometry iemaia maifods itegra geometry.