Recent developments in Wintgen inequality and Wintgen ideal submanifolds

Abstract

P. Wintgen proved in [Sur l'inégalité de Chen-Willmore. C. R. Acad. Sci. Paris 288, 993-995 (1979)] that the Gauss curvature G and the normal curvature K D of a surface in the Euclidean 4-space E 4 satisfy G + |K D | ≤ H 2 , where H 2 is the squared mean curvature. A surface M 2 in E 4 is called a Wintgen ideal surface if it satisfies the equality case of the inequality identically. Wintgen ideal surfaces in E 4 form an important family of surfaces; namely, surfaces with circular ellipse of curvature. In 1999, P. J. De Smet, F. Dillen, L. Verstraelen and L. Vrancken gave a conjecture for Wintgen inequality on Riemannian submanifolds in real space forms, which was well-known as the DDVV conjecture. Later, the DDVV conjecture was proven by Z. Lu and by Ge and Z. Tang independently. In this paper, we provide a comprehensive survey on recent developments in Wintgen inequality and Wintgen ideal submanifolds.

Bounds for Statistical Curvatures of Submanifolds in Kenmotsu-like Statistical Manifolds

“…The solution of this conjecture was independently proven by Lu [9] and Ge and Tang [10] for general case. Since then, many remarkable articles were published and several inequalities of this type were obtained for other kinds of submanifolds in different ambient spaces (see [11][12][13][14][15]). The derivation of inequality in terms of Casorati curvatures for various submanifolds in various ambient spaces is focused on an optimization approach that establishes that the polynomial of quadratic type in the components of the second fundamental form is parabolic.…”

Casorati Inequalities for Statistical Submanifolds in Kenmotsu Statistical Manifolds of Constant ϕ-Sectional Curvature with Semi-Symmetric Metric Connection

“…The study of simple relationships between the main intrinsic and extrinsic invariants of submanifolds is a fundamental problem in submanifold theory [ 1 ]. Recent research shows a growing trend in approaching this fascinating problem through an approach that proves some types of geometric inequalities (see, e.g., [ 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 ]).…”

“…The topic of -Casorati curvatures will appeal to more geometers focused on finding new solutions of the above problem. In this respect, some recent developments are devoted to inequalities on various submanifolds of a statistical manifold , notion defined by Amari [ 18 ] in 1985 in the realm of information geometry [ 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 ]. In this setting, the Fisher information metric is one of the most important metrics that can be considered on statistical models [ 19 ].…”

Generalized Wintgen inequality for BI-SLANT submanifolds in conformal Sasakian space form with quarter-symmetric connection

“…In fact, many remarkable articles were published in the recent years and several inequalities of this type have been obtained for other classes of submanifolds in several ambient spaces for example, for statistical submanifolds in statistical manifolds of constant curvature [9]; for Legendrian submanifolds in Sasakian space forms [10]; for submanifolds in statistical warped product manifolds [11]; for quaternionic CR-submanifolds in quaternionic space forms [12]; for submanifolds in generalized (κ, μ)-space forms [13]; for totally real submanifolds in LCS-manifolds [14] and so on. For more details, see [15].…”