**1. Introduction**

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<sup>⊥</sup>, and also the squared mean curvature -H-2 of N 2, satisfy

$$\|\mathcal{H}\|^2 \ge \mathcal{K} + |\mathcal{K}^\perp|$$

and the equality is attained only in the case when the ellipse of curvature of N 2 in E<sup>4</sup> is a circle. Later, this inequality was extended independently by Rouxel [2] and Gaudalupe and Rodriguez [3] for surfaces of arbitrary codimension *m* in real space forms N *<sup>m</sup>*+<sup>2</sup>(*c*) with constant sectional curvature *c* as

$$\|\mathcal{H}\|^2 + c \ge \mathcal{K} + |\mathcal{K}^\perp|.$$

The generalized Wintgen inequality, also known as the DDVV-inequality or the DDVV-conjecture, is a natural extension of the above inequalities that was conjectured in 1999 by De Smet, Dillen, Verstraelen and Vrancken [4] and settled in the general case independently by Ge and Tang [5] and Lu [6]. The

*Mathematics* **2019**, *7*, 1151; doi:10.3390/math7121151 DDVV-conjecture generalizes the classical Wintgen inequality to the case of an isometric immersion *f* : *Mn* → *Nn*+*p*(*c*) from an *n*-dimensional Riemannian submanifold *Mn* into a real space form *Nn*+*p*(*c*) of dimension (*n* + *p*) and of constant sectional curvature *c*, stating that such an isometric immersion satisfies

$$
\rho + \rho^{\perp} \le \|\mathcal{H}\|^2 + c\_\prime
$$

where *ρ* is the normalized scalar curvature, while *ρ*⊥ denotes the normalized normal scalar curvature. Notice that there are many examples of submanifolds satisfying the equality case of the above inequality and these submanifolds are known as Wintgen ideal submanifolds [7].

Recently, 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]. In the first part of the present paper, we obtain generalized Wintgen-type inequalities for different types of submanifolds in generalized complex space forms and also in generalized Sasakian space forms, generalizing the main results in [8,9], and also discuss some applications. The last part of the paper is devoted to the investigation of the Hessian equation on both generalized complex space forms and generalized Sasakian space-forms. In particular, some obstructions to the existence of these spaces are established. 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 2*n*-dimensional generalized complex space form is either a real space form or a complex space form, a result partially extendable to four-dimensional manifolds. However, the existence of proper generalized complex space form in dimension 4 was obtained by Olszak [14], using some conformal deformations of four-dimensional flat Bochner–Kähler manifolds of non-constant scalar curvature. It is important to note that the generalized complex space forms are a particular kind of almost Hermitian manifolds with constant holomorphic sectional curvature and constant type in the sense of Gray [15].

On the other hand, Alegre, Blair and Carriazo [16] generalized the notions of Sasakian space form, Kenmotsu space form and cosymplectic space form, by introducing the concept of generalized Sasakian space form. Notice that several examples of non-trivial generalized Sasakian space-forms are given in [16] using different geometric constructions, such as Riemannian submersions, warped products, and *D*-conformal deformations. Afterwards, many interesting results have been proved in these ambient spaces (see, e.g., [17–27]). We only recall that, 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.
