Abstract
In this work, we first derive a generalized Wintgen type inequality for a Lagrangian submanifold in a generalized complex space form. Further, we extend this inequality to the case of bi-slant submanifolds in generalized complex and generalized Sasakian space forms and derive some applications in various slant cases. Finally, we obtain obstructions to the existence of non-flat generalized complex space forms and non-flat generalized Sasakian space forms in terms of dimension of the vector space of solutions to the first fundamental equation on such spaces.
Highlights
The classical Wintgen inequality is a sharp geometric inequality established in [1], according to which the Gaussian curvature K of any surface N 2 in the Euclidean space E4, the normal curvature K ⊥, and the squared mean curvature kHk2 of N 2, satisfy kHk2 ≥ K + |K ⊥ |and the equality is attained only in the case when the ellipse of curvature of N 2 in E4 is a circle
The generalized Wintgen inequality was extended for several kinds of submanifolds in many ambient spaces, e.g., complex space forms [8], Sasakian space forms [9], quaternionic space forms [10], warped products [11], and Kenmotsu statistical manifolds [12]
Recall that the notion of generalized complex space form was introduced in differential geometry by Tricerri and Vanhecke [13], the authors proving that, if n ≥ 3, a 2n-dimensional generalized complex space form is either a real space form or a complex space form, a result partially extendable to four-dimensional manifolds
Summary
The classical Wintgen inequality is a sharp geometric inequality established in [1], according to which the Gaussian curvature K of any surface N 2 in the Euclidean space E4 , the normal curvature K ⊥ , and the squared mean curvature kHk2 of N 2 , satisfy kHk2 ≥ K + |K ⊥ |. The DDVV-conjecture generalizes the classical Wintgen inequality to the case of an isometric immersion f : Mn → N n+ p (c) from an n-dimensional Riemannian submanifold Mn into a real space form N n+ p (c) of dimension (n + p) and of constant sectional curvature c, stating that such an isometric immersion satisfies ρ + ρ⊥ ≤ kHk2 + c, where ρ is the normalized scalar curvature, while ρ⊥ denotes the normalized normal scalar curvature. The generalized Wintgen inequality was extended for several kinds of submanifolds in many ambient spaces, e.g., complex space forms [8], Sasakian space forms [9], quaternionic space forms [10], warped products [11], and Kenmotsu statistical manifolds [12]. Very recently, Bejan and Güler [28] obtained an unexpected link between the class of generalized Sasakian space-forms and the class of Kähler manifolds of quasi-constant holomorphic sectional curvature, providing conditions under which each of these structures induces the other one
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