On normal CR-submanifolds of S-manifolds

José Cabrerizo; Luis Fernández; Manuel Fernández

Colloquium Mathematicae (1993)

  • Volume: 64, Issue: 2, page 203-214
  • ISSN: 0010-1354

Abstract

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Many authors have studied the geometry of submanifolds of Kaehlerian and Sasakian manifolds. On the other hand, David E. Blair has initiated the study of S-manifolds, which reduce, in particular cases, to Sasakian manifolds ([1, 2]). I. Mihai ([8]) and L. Ornea ([9]) have investigated CR-submanifolds of S-manifolds. The purpose of the present paper is to study a special kind of such submanifolds, namely the normal CR-submanifolds. In Sections 1 and 2, we review basic formulas and definitions for submanifolds in Riemannian manifolds and in S-manifolds, respectively, which we shall use later. In Section 3, we introduce normal CR-submanifolds of S-manifolds and we study some properties of their geometry. Finally, in Section 4, we consider those submanifolds in the case of the ambient S-manifold being an S-space form.

How to cite

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Cabrerizo, José, Fernández, Luis, and Fernández, Manuel. "On normal CR-submanifolds of S-manifolds." Colloquium Mathematicae 64.2 (1993): 203-214. <http://eudml.org/doc/210185>.

@article{Cabrerizo1993,
abstract = {Many authors have studied the geometry of submanifolds of Kaehlerian and Sasakian manifolds. On the other hand, David E. Blair has initiated the study of S-manifolds, which reduce, in particular cases, to Sasakian manifolds ([1, 2]). I. Mihai ([8]) and L. Ornea ([9]) have investigated CR-submanifolds of S-manifolds. The purpose of the present paper is to study a special kind of such submanifolds, namely the normal CR-submanifolds. In Sections 1 and 2, we review basic formulas and definitions for submanifolds in Riemannian manifolds and in S-manifolds, respectively, which we shall use later. In Section 3, we introduce normal CR-submanifolds of S-manifolds and we study some properties of their geometry. Finally, in Section 4, we consider those submanifolds in the case of the ambient S-manifold being an S-space form.},
author = {Cabrerizo, José, Fernández, Luis, Fernández, Manuel},
journal = {Colloquium Mathematicae},
keywords = {S-manifolds; normal CR-submanifolds; S-space forms; normal submanifold; CR-submanifold; -manifold; -space forms; CR- product},
language = {eng},
number = {2},
pages = {203-214},
title = {On normal CR-submanifolds of S-manifolds},
url = {http://eudml.org/doc/210185},
volume = {64},
year = {1993},
}

TY - JOUR
AU - Cabrerizo, José
AU - Fernández, Luis
AU - Fernández, Manuel
TI - On normal CR-submanifolds of S-manifolds
JO - Colloquium Mathematicae
PY - 1993
VL - 64
IS - 2
SP - 203
EP - 214
AB - Many authors have studied the geometry of submanifolds of Kaehlerian and Sasakian manifolds. On the other hand, David E. Blair has initiated the study of S-manifolds, which reduce, in particular cases, to Sasakian manifolds ([1, 2]). I. Mihai ([8]) and L. Ornea ([9]) have investigated CR-submanifolds of S-manifolds. The purpose of the present paper is to study a special kind of such submanifolds, namely the normal CR-submanifolds. In Sections 1 and 2, we review basic formulas and definitions for submanifolds in Riemannian manifolds and in S-manifolds, respectively, which we shall use later. In Section 3, we introduce normal CR-submanifolds of S-manifolds and we study some properties of their geometry. Finally, in Section 4, we consider those submanifolds in the case of the ambient S-manifold being an S-space form.
LA - eng
KW - S-manifolds; normal CR-submanifolds; S-space forms; normal submanifold; CR-submanifold; -manifold; -space forms; CR- product
UR - http://eudml.org/doc/210185
ER -

References

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  1. [1] D. E. Blair, Geometry of manifolds with structural group U(n)×O(s), J. Differential Geom. 4 (1970), 155-167. Zbl0202.20903
  2. [2] D. E. Blair, On a generalization of the Hopf fibration, Ann. Ştiinţ. Univ. 'Al. I. Cuza' Iaşi 17 (1) (1971), 171-177. Zbl0244.53038
  3. [3] D. E. Blair, G. D. Ludden and K. Yano, Differential geometric structures on principal toroidal bundles, Trans. Amer. Math. Soc. 181 (1973), 175-184. Zbl0276.53026
  4. [4] J. L. Cabrerizo, L. M. Fernández and M. Fernández, A classification of totally f-umbilical submanifolds of an S-manifold, Soochow J. Math. 18 (2) (1992), 211-221. Zbl0753.53031
  5. [5] L. M. Fernández, CR-products of S-manifolds, Portugal. Mat. 47 (2) (1990), 167-181. Zbl0716.53051
  6. [6] I. Hasegawa, Y. Okuyama and T. Abe, On p-th Sasakian manifolds, J. Hokkaido Univ. Ed. Sect. II A 37 (1) (1986), 1-16. 
  7. [7] M. Kobayashi and S. Tsuchiya, Invariant submanifolds of an f-manifold with complemented frames, Kodai Math. Sem. Rep. 24 (1972), 430-450. Zbl0246.53038
  8. [8] I. Mihai, CR-subvarietăţi ale unei f-varietăţi cu repere complementare, Stud. Cerc. Mat. 35 (2) (1983), 127-136. 
  9. [9] L. Ornea, Subvarietăţi Cauchy-Riemann generice în S-varietăţi, ibid. 36 (5) (1984), 435-443. 
  10. [10] K. Yano, On a structure defined by a tensor fie1ld f of type (1,1) satisfying f 3 + f = 0 , Tensor 14 (1963), 99-109. Zbl0122.40705

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