Wintgen inequality for statistical surfaces

Abstract

The Wintgen inequality (1979) is a sharp geometric inequality for surfaces in the 4-dimensional Euclidean space involving the Gauss curvature (intrinsic invariant) and the normal curvature and squared mean curvature (extrinsic invariants), respectively.In the present paper we obtain a Wintgen inequality for statistical surfaces. For surfaces M 2 of the Euclidean space E 3 , the Euler inequality K ≤ H 2 is fulfilled, where K is the (intrinsic) Gauss curvature of M 2 and H 2 is the (extrinsic) squared mean curvature of M 2 . Furthermore, K = H 2 everywhere on M 2 if and only if M 2 is totally umbilical, or still, by a theorem of Meusnier, if and only if M 2 is (a part of) a plane E 2 or, it is (a part of) a round sphere S 2 in E 3 . In 1979, P. Wintgen [28] proved that the Gauss curvature K, the squared mean curvature H 2 and the normal curvature K ⊥ of any surface M 2 in E 4 always satisfy the inequality K ≤ H 2 − |K ⊥ |; the equality holds if and only if the ellipse of curvature of M 2 in E 4 is a circle. The Whitney 2-sphere satisfies the equality case of the Wintgen inequality identically. A survey containing recent results on surfaces satisfying identically the equality case of Wintgen inequality can be read in [4]. Later, the Wintgen inequality was extended by B. Rouxel [24] and by I.V. Guadalupe and L. Rodriguez [11] independently, for surfaces M 2 of arbitrary codimension m in real space forms M 2+m (c); namely K ≤ H 2 − |K ⊥ | + c.

On totally real statistical submanifolds

“…for any X, Y ∈ Γ(TM). Then K 2 ∈ Γ(TM (1,2) ) satisfies three conditions of Lemma 2.6 as in Example 2.8, and hence (M, ∇ := ∇ g + K 2 , g, J) becomes a holomorphic statistical manifold.…”

Hyperelastic curves along immersions

“…Ikawa expressed a characterization of a helix by a differential equation in a Riemannian manifold and obtained the necessary and sufficient condition as depending upon the mean curvature vector field that a helix in a Riemannian submanifold corresponds to a helix in ambient manifold [12,17,18]. One of the most active area of differential geometry has been the theory of isometric immersions [9] and this theory is still active area, see for instance [2] and [4]. The aim of this paper is to consider hyperelastic curves along an immersion.…”

Statistical Solitons and Inequalities for Statistical Warped Product Submanifolds

“…Several sharp inequalities between extrinsic and intrinsic curvatures for different submanifolds in real, complex, and quaternionic space forms endowed with various connections have been obtained (e.g., [14][15][16][17][18][19][20][21]). Such inequalities with a pair of conjugate affine connections involving the normalized scalar curvature of statistical submanifolds in different ambient spaces were obtained in [22][23][24][25][26].…”