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1216 surface modification of acrylic fibers by cutinase and esterase. Biotechnology Journal. 7:842-879. Meghwanshi, GK., Agarwal, L., Dutt, K. and Saxena, RK. 2006. Characterization of 1,3-regiospecific lipase from new Pseudomonas and Bacillus isolates. Journal of Molecular Catalysis. B: Enzyme 40:127-131. Mohan, ST., Palavesan, A. and Immanvel, G. 2008. Isolation and characterization of lipase-producing Bacillus strains from oil mill waste. African Journal of Biotechnology. 7:2728-2735. O'Neill, A., Araújo, R., Casal, M., Guebitz, G. and Cavaco Paulo, A. 2007. Effect of the agitation on the adsorption and hydrolytic efficiency of cutinase on polyethylene terephthalate fibers. Enzyme and Microbial Technology. 40:1801-1805. Rohit, S., Yusuf, C. and Ullamchand, B. 2001. Production, purification, characterization and application of lipases. Biotechnology Advances. 19:627-662. Shalaby, SE., El-Balacosy NG. and Abo El Ola, SM. 2007. Alkaline treatment of polyethylene glycol modified poly(ethyleneterephthalate) fabrics. Journal of Textile Association. May-June 31-38. Sharma, R., Chisti, Y. and Banerjee, UC. 2001. Production, purification, characterization, and applications of lipase. Biotechnology Advances. 19:627- 662. Silva, C. and Cavaco-Paulo, A. 2004. Monitoring biotransformations in polyamide fibers. Biocatalysis and Biotransformation. 22:357-360. Silva, C., Araujo, R., Casal, M., Gubitz, GM. and Cavaco- Paulo, A. 2007. Influence of mechanical agitation on cutinases and protease activity towards polyamide substrates. Enzyme and Microbial Technology. 40:1678- 1685. Souissi, N., Bougatat, A., Triki-ellouz, Y. and Nasri, M. 2009. production of lipase and biomass by Staphylococcus simulans grown on sardinella (Sardinella aurita) hydrolysates and peptone. African Journal of Microbiology. 8:451-457. Stukus, PE. 1997. Investigating microbiology: A laboratory manual for general microbiology. Saunders College Publishing, New York. pp. 143-209. Vertommen, MAME., Nierstrasz, VA., van der Veer, M. and Warmoeskerken, MMCG. 2005. Enzymatic surface modification of poly(ethyleneterephthalate). Journal of Biotechnology. 120:376-386. Received: April 3, 2010; Accepted: May 26, 2010 Canadian Journal of Pure and Applied Sciences
SENRA Academic Publishers, Burnaby, British Columbia Vol. 4, No. 2, pp. 1217-1219, June 2010 ISSN: 1715-9997 ON SUBMANIFOLDS OF INDEFINITE COMPLEX SPACE FORM Augustus Nzomo Wali Department of Applied Mathematics Kigali Institute of Science and Technology, B. P. 3900, Kigali, Rwanda ABSTRACT n Sun (1994) showed that if M is a maximal spacelike submanifold of M n () c then either M is totally geodesic ( n≥2, c≥ 0) or c 0≤ S ≤− n( n−1 ) ,( n≥ 2, c< 0) . The purpose of this paper is to study the geometry of an n- 4 dimensional compact totally real maximal spacelike submanifold M immersed in an indefinite complex space form n+ p M p () c . In this manuscript 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> 1or (1 − n)( n+ 2 p) S ≤ c for c< 0, n> 1and thus generalized Sun (1994) 4 result. Keywords: Totally real submanifold, complex space form, totally geodesic. INTRODUCTION Among all submanifolds of a Kaehler manifold there are two classes; the class of totally real submanifolds and the class of holomorphic submanifolds. A submanifold of a Kaehler manifold is called totally real (resp. holomorphic) if each tangent space of the submanifold is mapped into the normal space (resp. itself) by the almost complex structure of the Kaehler manifold, Chen and Ogiue (1974). A Kaehler manifold of constant holomorphic sectional curvature is called a complex space form, Wali (2005). Let M be an n-dimensional totally real maximal spacelike submanifold isometrically immersed in a 2(n+p)- n+ p dimensional indefinite complex space form M p () c of holomorphic sectional curvature c and index 2p. We call M a spacelike submanifold if the induced metric on M from that of the ambient space is positive definite, Ishihara (1988). Let J be the almost complex structure of n+ p M p () c . An n-dimensional Riemannian manifold M n+ p isometrically immersed in M p () c is called a totally n+ p real submanifold of M p () c if each tangent space of M is mapped into the normal space by the almost complex structure, Yano and Kon (1976). n+ p Let h be the second fundamental form of M in M () c *Corresponding author email: tabnzo@yahoo.co.in p and denote by S the square of the length of the second fundamental form h. Sun (1994), proved that if M is a maximal spacelike n submanifold of M n () c then either M is totally geodesic ( n≥2, c≥ 0) or c 0≤S ≤− n( n−1 ) ,( n≥ 2, c< 0) . The 4 purpose of this paper is to study an n-dimensional compact totally real maximal spacelike submanifold M immersed n+ p M () c . in an indefinite complex space form p Our main result is: Theorem: Let M be an n-dimensional compact totally n+ p real maximal spacelike submanifold of M p () c . Then either 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 LOCAL FORMULAS n+ p Let M p () c be an indefinite complex space form of holomorphic sectional curvature c, dimension 2(n+p), p ≠ 0 and index 2p. Let M be an n-dimensional totally real maximal spacelike submanifold isometrically n+ p immersed in M p () c . We choose a local field of orthonormal frames
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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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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
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Page 50 and 51: 1176 note that though the Mn(II) co
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Page 54 and 55: 1180 prepare the rats feed. The rat
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Page 58 and 59: 1184 Table 8. Histopathology result
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Page 78 and 79: 1206 Degtyarev, AI, Smirnova, TG. a
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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
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Page 94 and 95: eadings (Q), density (), viscosity
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Page 100 and 101: 1130 Evaluation by EvalAccess 2.0 a
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