A structure on an almost contact metric manifold is defined as a generalization of well-known cases: Sasakian, quasi-Sasakian, Kenmotsu and cosymplectic. This was suggested by a local formula of Eum [9]. Then we consider a semi-invariant $\xi^{\bot}$-submanifold of a manifold endowed with such a structure and two topics are studied: the integrability of distributions defined by this submanifold and characterizations for the totally umbilical case. In particular we recover results of Kenmotsu [11], Eum [9,10] and Papaghiuc [16].
References
A. Bejancu, CR-submanifolds of a Kähler manifold I., Proc. Amer. Math. Soc., 69 (1978), 135-142. MR467630 0368.53040 A. Bejancu, CR-submanifolds of a Kähler manifold I., Proc. Amer. Math. Soc., 69 (1978), 135-142. MR467630 0368.53040
A. Bejancu and N. Papaghiuc, Semi-invariant submanifolds of a Sasakian manifold, An. Ştiinţ. Univ. 'Al. I. Cuza', Iaşi, Secţ I a Math., 27(1) (1981), 163-170. 0463.53030 A. Bejancu and N. Papaghiuc, Semi-invariant submanifolds of a Sasakian manifold, An. Ştiinţ. Univ. 'Al. I. Cuza', Iaşi, Secţ I a Math., 27(1) (1981), 163-170. 0463.53030
A. Bejancu, Geometry of CR-submanifolds, Mathematics and its Applications, D. Reidel Publishing Co., Dordrecht, 1986. 0605.53001 A. Bejancu, Geometry of CR-submanifolds, Mathematics and its Applications, D. Reidel Publishing Co., Dordrecht, 1986. 0605.53001
B. Y. Chen, Riemannian submanifolds, in: Handbook of differential geometry, Vol. I, eds. F. Dillen and L. Verstraelen, North-Holland, Amsterdam, 2000, pp. 187-418. B. Y. Chen, Riemannian submanifolds, in: Handbook of differential geometry, Vol. I, eds. F. Dillen and L. Verstraelen, North-Holland, Amsterdam, 2000, pp. 187-418.
B. Y. Chen, $\delta $-invariants, inequalities of submanifolds and their applications, in: Topics in differential geometry, Ed. Acad. Române, Bucharest, 2008, pp. 29-155. B. Y. Chen, $\delta $-invariants, inequalities of submanifolds and their applications, in: Topics in differential geometry, Ed. Acad. Române, Bucharest, 2008, pp. 29-155.
D. Chinea and C. Gonzales, A classification of almost contact metric manifolds, Ann. Mat. Pura Appl., 156 (4) (1990), 15-36. 0711.53028 10.1007/BF01766972 D. Chinea and C. Gonzales, A classification of almost contact metric manifolds, Ann. Mat. Pura Appl., 156 (4) (1990), 15-36. 0711.53028 10.1007/BF01766972
S. S. Eum, A Kaehlerian hypersurface with parallel Ricci tensor in an almost contact metric space of constant $C$-holomorphic sectional curvature, Tensor, 21 (1970), 315-318. 0182.24803 S. S. Eum, A Kaehlerian hypersurface with parallel Ricci tensor in an almost contact metric space of constant $C$-holomorphic sectional curvature, Tensor, 21 (1970), 315-318. 0182.24803
K. Kenmotsu, A class of almost contact Riemannian manifolds, Tôhoku Math. J., (2), 24 (1972), 93-103. 0245.53040 10.2748/tmj/1178241594 K. Kenmotsu, A class of almost contact Riemannian manifolds, Tôhoku Math. J., (2), 24 (1972), 93-103. 0245.53040 10.2748/tmj/1178241594
V. Mangione, On submanifolds of a cosymplectic space form, Bull. Math. Soc. Sci. Math. Roumanie, 47(1-2) (2004), 85-95. 1114.53050 V. Mangione, On submanifolds of a cosymplectic space form, Bull. Math. Soc. Sci. Math. Roumanie, 47(1-2) (2004), 85-95. 1114.53050
J. C. Marrero, The local structure of trans-Sasakian manifolds, Ann. Mat. Pura Appl., 162 (4) (1992), 77-86. 0772.53036 10.1007/BF01760000 J. C. Marrero, The local structure of trans-Sasakian manifolds, Ann. Mat. Pura Appl., 162 (4) (1992), 77-86. 0772.53036 10.1007/BF01760000
M. I. Munteanu, Warped Product Contact $CR$-Submanifolds of Sasakian Space Forms, Publ. Math. Debrecen, 66(1-2) (2005), 75-120. 1063.53052 M. I. Munteanu, Warped Product Contact $CR$-Submanifolds of Sasakian Space Forms, Publ. Math. Debrecen, 66(1-2) (2005), 75-120. 1063.53052
N. Papaghiuc, Semi-invariant submanifolds in a Kenmotsu manifold, Rend. Mat.$($7$)$, 3(4) (1983), 607-622. 0541.53010 N. Papaghiuc, Semi-invariant submanifolds in a Kenmotsu manifold, Rend. Mat.$($7$)$, 3(4) (1983), 607-622. 0541.53010
N. Papaghiuc, On the geometry of leaves on a semi-invariant $\xi^\perp$-submanifold in a Kenmotsu manifold, An. Stiint. Univ. Al. I. Cuza Iasi Sect. I a Mat., 38(1) (1992), 111-119. 0805.53050 N. Papaghiuc, On the geometry of leaves on a semi-invariant $\xi^\perp$-submanifold in a Kenmotsu manifold, An. Stiint. Univ. Al. I. Cuza Iasi Sect. I a Mat., 38(1) (1992), 111-119. 0805.53050
Mohd. Shoeb, Mohd. Hasan Shahid and A. Sharfuddin, On submanifolds of a cosymplectic manifold, Soochow J. Math., 27(2) (2001), 161-174. 1009.53033 Mohd. Shoeb, Mohd. Hasan Shahid and A. Sharfuddin, On submanifolds of a cosymplectic manifold, Soochow J. Math., 27(2) (2001), 161-174. 1009.53033
K. Yano and M. Kon, Structures on Manifolds, Series in Pure Mathematics, World Scientific Publishing Co., Singapore, 1984. 0557.53001 K. Yano and M. Kon, Structures on Manifolds, Series in Pure Mathematics, World Scientific Publishing Co., Singapore, 1984. 0557.53001