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B.-Y. Chen inequalities for slant submanifolds in quaternionic space forms

One of the most powerful tools to find relationships between intrinsic invariants and extrinsic invariants of a submanifold is provided by Chen’s invariants. In this paper some B.-Y. Chen inequalities for slant submanifolds in quaternionic space forms are established. | Turk J Math 34 (2010) , 115 – 128. ¨ ITAK ˙ c TUB doi:10.3906/mat-0807-14 B.-Y. Chen inequalities for slant submanifolds in quaternionic space forms Gabriel Eduard Vˆılcu Abstract In this paper some B.-Y. Chen inequalities for slant submanifolds in quaternionic space forms are established. Key Words: Chen’s invariant, squared mean curvature, quaternionic space form, slant submanifold. 1. Introduction One of the most powerful tools to find relationships between intrinsic invariants and extrinsic invariants of a submanifold is provided by Chen’s invariants. This theory was initiated in [9] where B.-Y. Chen established a sharp inequality for a submanifold in a real space form using the scalar curvature and the sectional curvature (both being intrinsic invariants) and squared mean curvature (the main extrinsic invariant). On the other hand, the slant submanifolds of complex manifolds were defined in [8] and Chen-like inequalities for slant submanifolds in complex space forms and in generalized complex space forms were obtained in [27] and [23]. The slant submanifolds of contact manifolds were introduced in [24], and Chen-like inequalities for slant submanifolds of Sasakian space forms were obtained in [14]. The study of slant submanifolds in S -manifolds and B.-Y. Chen inequalities in S -space forms has been realized in [6] and [7]. Other Chen-like inequalities in different settings and submanifolds satisfying Chen’s equality can be found in [1], [2], [3], [5], [12], [13], [15], [17], [18], [19], [20], [28], [29], [30], [32], [34]. Some B.-Y. Chen inequalities for totally real submanifolds in quaternionic space forms are established in [33]. Recently, S ¸ ahin [31] introduced the slant submanifolds of quaternionic K¨ ahler manifolds, as a natural generalization of both quaternionic and totally real submanifolds. Motivated by the above considerations, we’ll be studying Chen-like inequalities in the context of slant submanifolds in quaternionic space forms. The paper