Generalized Wintgen inequality for statistical submanifolds in statistical manifolds of constant curvature

M. Evren Aydin,Adela Mihai,Ion Mihai

doi:10.1007/s13373-016-0086-1

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. De Smet et al. (Arch. Math. (Brno) 35:115–128, 1999) conjectured a generalized Wintgen inequality for submanifolds of arbitrary dimension and codimension in Riemannian space forms. This conjecture was proved by Lu (J. Funct. Anal. 261:1284–1308, 2011) and by Ge and Tang (Pac. J. Math. 237:87–95, 2008), independently. In the present paper we establish a generalized Wintgen inequality for statistical submanifolds in statistical manifolds of constant curvature.

Highlights

In 1979, Wintgen [25] proved that the Gauss curvature G, the squared mean curvature H 2 and the normal curvature G⊥ of any surface M2 in E4 always satisfy the inequality
The Wintgen inequality was extended by Rouxel [20] and by Guadalupe and Rodriguez [10] independently, for surfaces M2 of arbitrary codimension m in real space forms M2+m(c); namely
In 1999, De Smet et al [7] formulated the conjecture on Wintgen inequality for submanifolds of real space forms, which is known as the DDVV conjecture

Summary

Introduction

For surfaces M2 of the Euclidean space E3, the Euler inequality G ≤ H 2 is fulfilled, where G is the (intrinsic) Gauss curvature of M2 and H 2 is the (extrinsic) squared mean curvature of M2. 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 [5]. In 1999, De Smet et al [7] formulated the conjecture on Wintgen inequality for submanifolds of real space forms, which is known as the DDVV conjecture. This conjecture was proven by the authors for submanifolds Mn of arbitrary dimension n ≥ 2 and codimension 2 in real space forms Mn+2(c) of constant sectional curvature c.

Statistical manifolds and their submanifolds

Statistical surfaces in statistical manifolds of constant curvature

Wintgen inequality for statistical submanifolds