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1218 { e1,..., en; en+ 1,..., en+ p; e1* = Je1 ,..., en* = Jen; e( 1* ) = Jen 1,..., n + e( n p) = Je + + * n+ p} n+ p in M () c p Canadian Journal of Pure and Applied Sciences such that restricted to M, the vectors { 1 } ,..., e e n are tangent to M and the rest are normal to n+ p M. With respect to this frame field of M p () c , let 1 n n+ 1 n+ p 1* n* ( n+ ) ( n+ p) ω ,..., ω ; ω ,..., ω ; ω ,..., ω ; ω ,..., ω be the field of dual frames. 1* * Unless otherwise stated, we shall make use of the following convention on the ranges of indices: 1 ≤ ABCD , , , ≤ n+ p; 1 ≤i, j, k, l, m≤ n; n+ 1 ≤α, βγ , ≤ n+ p; and when a letter appears in any term as a subscript or a superscript, it is understood that this letter is summed over its range. Besides ε = g e, e = g Je, Je = 1, when 1 ≤i≤ n ( ) ( ) ( ) ( ) i i i i i ε = g e , e = g Je , Je =− 1, when α α α α α n+ 1≤α≤ n+ p . n+ p Then the structure equations of M () c are; p A B B A A A B i i* dω +∑ε ω ∧ ω = 0 , ω + ω = 0 , ω = ω , B B j j* i* j* ω j = ωi , A dωB + A C 1 εω C C ∧ ωB = 2 C D εε C DRABCDω∧ω ∑ ∑ , C CD c R = εε δ δ − δ δ + J J − J J + J J 4 ( 2 ) ABCD C D AC BD AD BC AC BD AD BC AB CD where R ABCD denote the components of the curvature n+ p tensor R on M p () c . Restricting these forms to M we have; α ω = 0, α α i α α i i j h = h , dω =−∑ω ∧ ω , ω = ∑ h ω , i ij i ij ji i j ωj + ωi = 0 , i i k 1 k l dωj =−∑ωk ∧ ωj + ∑ Rijklω ∧ω , 2 kl Rijkl = Rijkl − α α α α ( hikhjl−hilhjk), ∑ ∑ α α α dω =− ω ∧ω , β β β α α γ 1 i j dωβ =−∑ ωγ ∧ ωβ + Rαβijω ∧ω , 2 γ j α β α β ( ) Rαβ ij = ∑ k hh ik jl −hh il jk (2.1) From the condition on the dimensions of M and n+ p M () c it follows that e1* ,..., e n* is a frame for p ⊥ T ( M) . Noticing this, we see that c α α α α Rijkl = ( δikδ jl −δilδ jk ) −∑ ( hh ik jl −hh il jk ) (2.2) 4 α 1 α We call H = ∑ trh the mean curvature of M and n α S h α = ∑ the square of the length of the second ijα ( ) 2 ij fundamental form. If H is identically zero then M is said to be maximal. M is totally geodesic if h=0. From (2.2) we have the Ricci tensor R ij given by ( n −1) α α Rij = ∑Rikjk = cδij + ∑ hik hkj (2.3) k 4 αk Thus the Ricci curvature R is c R = Rii = ( n− 1) + S (2.4) 4 From (2.3) the scalar curvature is given by n( n−1) ρ = ∑ R jj = c+ S (2.5) j 4 Let hijk α denote the covariant derivative of ij hα . Then we define hijk α by ∑h ω = dh + ∑h ω + ∑h ω + ∑ h ω (2.6) α k α α k α k β β ijk ij kj i ik j ij α k k k β α and hijk α hikj = . Taking the exterior derivative of (2.6) we define the second covariant derivative of ij hα by h ω = dh + h ω + h ω + h ω + h ω (2.7) ∑ ∑ ∑ ∑ ∑ α l α α l α l α l β β ijkl ijk ljk i ilk j ijl k ijk α l l l l β Using (2.7) we obtain the Ricci formula ∑ ∑ ∑ (2.8) h − h = h R + h R + h R α α α α β ijkl ijlk mj mikl im mjkl ij βαkl m m β The Laplacian ij hα ∆ of the second fundamental form ij hα ∆ h =∑ h . is defined as α ij α ijkk k Therefore, ∆ h c = ( n− 1) 4 h + h h h + h h h ∑ ∑ ∑ (2.9) α α α β β α β β ij ij mi mk kj km mk ij βmk βmk ∑ ∑ + hh h −2hhh β α β β α β ki jm mk ki mk mj βmk βmk
1 2 1 ∆ = + − + 2 4 From α 2 α 2 α α ∆ ∑( hij) = ∑( hijk ) + ∑ hij ∆h we obtain, ij αij αijk αij α 2 ∑( hij ) α 2 ∑( hijk ) c ( n α 2 1) ∑( hij) α α β β ∑ hijhklhlk hij αijαijk αij αβijkl α β β α α β β α ∑ ( hh li lj hh li lj )( hh ki kj hh ki kj ) + − − αβijkl PROOF OF THE THEOREM (2.10) Let M be an n-dimensional compact totally real maximal spacelike submanifold isometrically immersed in n+ p M () c . For each α let Hα denote the symmetric p matrix ( hij ) α and let S h h α β αβ ij ij ij = ∑ . Then the (n+2p)×(n+2p)-matrix ( Sαβ ) is symmetric and can be assumed to be diagonal for a suitable choice of e ,..., e . S = S 2 = trH and n+ 1 n+ p Setting α αα α S = ∑ Sα, equation (2.10) can be rewritten as α 1 α 2 c 2 2 ∆ S = ∑( hijk ) + ( n− 1) S+ ∑Sα + ∑tr( HαHβ −HβHα) 2 αijk 4 α αβ (3.1) α 2 c 2 1 2 = ∑( hijk ) + ( n− 1) S+ ∑Sα+ S 4 n+ 2p αijk α 1 + − + − n+ 2 p∑ ∑ ( Sα Sβ) tr( HαHβ HβHα) α> β αβ 2 2 ⎛c1 ⎞ 1 = + − + + − αijk ⎝4 n+ 2p ⎠ n+ 2p α> β α ∑( hijk ) ⎜ ( n 1) S⎟S ∑( Sα Sβ ) ∑ 2 2 ( ) 2 α β β α + tr H H −HH αβ From (3.1) we see that 1 α 2 ⎛c1 ⎞ ∫ ∆Sdv ≥ ( hijk ) dv ( n 1) S Sdv M 2 ∫ ∑ + M ∫ M⎜ − + ⎟ αijk ⎝4 n+ 2p ⎠ where dv is the volume element of M. By the well known theorem of Hopf (1950), ∆ S = 0 . Therefore, α 2 ⎛c1⎞ 0≥ ∫ ∑( hijk ) dv+ ( n 1) S Sdv M ∫M⎜ − + ⎟ αijk ⎝4 n+ 2p ⎠ which implies that Wali 1219 ⎛c1⎞ ( n 1) S Sdv 0 M ⎜ − + ⎟ ≤ (3.2) ∫ ⎝4 n+ 2p ⎠ Thus either S=0 implying M is totally geodesic or ( 1− n)( n+ 2p) S ≤ c. This shows that M is totally 4 geodesic for c≥ 0, n> 1 or ( 1− n)( n+ 2p) 0 ≤ S ≤ c 4 for c< 0, n> 1. This proves our theorem. CONCLUSION In this paper we studied the geometry of an n-dimensional compact totally real maximal spacelike submanifold M n+ p immersed in an indefinite complex space form M p () c by computing the square of the length of the second fundamental form. In conclusion, we have shown that either the square of the length of second fundamental form S=0, implying M is totally geodesic for c≥ 0, n> 1 or (1 − n)( n+ 2 p) S ≤ c for c< 0, n> 1. 4 This generalizes the result by Sun (1994). REFERENCES Chen, BY. and Ogiue, K. 1974. On totally real submanifolds. Transactions of the American Mathematical Society. 193:257-266. Hopf, E. 1950. A theorem on the accessibility of boundary parts of an open point set. Proceedings of the American Mathematical Society. 1:76-79. Ishihara, T. 1988. Maximal spacelike submanifolds of a Pseudoriemannian space of constant curvature. Michigan Mathematical Journal. 35:345-352. Sun, H. 1994. Totally real maximal spacelike submanifolds in indefinite complex space form. Journal of Northeastern <strong>University</strong>. 15:547-550. Wali, AN. 2005. On bounds of holomorphic sectional curvature. East African Journal of Physical Sciences. 6(1):49-53. Yano, K. and Kon, M. 1976. Totally real submanifolds of complex space forms II. Kodai Mathematical Seminar Reports. 27:385-399. Received: Nov 13, 2010; Revised: Feb 15, 2010; May 14, 2010
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EDITOR MZ Khan, SENRA Academic Publ
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SENRA Academic Publishers, Burnaby,
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Table1. Evolution of glucosamine an
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Dunsford, BR., Knabe, DA. and Hacns
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1134 Actinomyces, Micrococcus and S
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1136 shown in figure 1. Among wild-
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1138 (75/10) and tobramycin (10). T
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SENRA Academic Publishers, Burnaby,
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Sodium dodecyl sulfate-polyacrylami
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2). RCG1 was purified as a single 1
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(100 µg/ml), which showed highest
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Rabinovich, GA. and Gruppi, A. 2005
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1152 2009). The main vegetated area
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1154 vehicle disturbs the animals a
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1156 Population and Status of Small
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1158 6.95%, Cape Hare 3.64%, Mouse
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1160 S. No. Scientific name Common
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1162 Grimmett, R., Inskipp, C. and
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1164 housed in clean cages, and pla
Page 40 and 41: 1166 kidneys of all the experimenta
Page 42 and 43: 1168 Prockop, DJ. 1964. Isotopic st
Page 44 and 45: 1170 copper(II) nitrate and zinc(II
Page 46 and 47: 1172 Infrared spectra The relevant
Page 48 and 49: 1174 step was the loss of two molec
Page 50 and 51: 1176 note that though the Mn(II) co
Page 52 and 53: 1178 dimethyl-2H-indazol-6-yl)methy
Page 54 and 55: 1180 prepare the rats feed. The rat
Page 56 and 57: 1182 The result of the mean interna
Page 58 and 59: 1184 Table 8. Histopathology result
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Page 62 and 63: RMSE and MBE are statistical instru
Page 64 and 65: Figure 2 is similar to Figure 1, fo
Page 66 and 67: dH/dt nT/min (Obs) AE nT (Obs) Dst
Page 68 and 69: Tables 2 (a-f) contain summaries of
Page 70 and 71: Pulkkinen, A., Viljanen, A., Pajunp
Page 72 and 73: 1200 that a significant fraction of
Page 74 and 75: 1202 Nino and La Nina periods are a
Page 76 and 77: 1204 periods (1981, 1984, 1991, 199
Page 78 and 79: 1206 Degtyarev, AI, Smirnova, TG. a
Page 80 and 81: 1208 The purpose of the present stu
Page 82 and 83: 1210 Quantitative screening for lip
Page 84 and 85: 1212 A bsorbance 0.70 0.65 0.60 0.5
Page 86 and 87: 1214 Effect of supplementation of C
Page 88 and 89: 1216 surface modification of acryli
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Page 96 and 97: flow of crude oil and air in horizo
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Page 100 and 101: 1130 Evaluation by EvalAccess 2.0 a
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