Abstract
PurposeIn 1979, P. Wintgen obtained a basic relationship between the extrinsic normal curvature the intrinsic Gauss curvature, and squared mean curvature of any surface in a Euclidean 4-space with the equality holding if and only if the curvature ellipse is a circle. In 1999, P. J. De Smet, F. Dillen, L. Verstraelen and L. Vrancken gave a conjecture of Wintgen inequality, named as the DDVV-conjecture, for general Riemannian submanifolds in real space forms. Later on, this conjecture was proven to be true by Z. Lu and by Ge and Z. Tang independently. Since then, the study of Wintgen’s inequalities and Wintgen ideal submanifolds has attracted many researchers, and a lot of interesting results have been found during the last 15 years. The main purpose of this paper is to extend this conjecture of Wintgen inequality for bi-slant submanifold in conformal Sasakian space form endowed with a quarter symmetric metric connection.Design/methodology/approachThe authors used standard technique for obtaining generalized Wintgen inequality for bi-slant submanifold in conformal Sasakian space form endowed with a quarter symmetric metric connection.FindingsThe authors establish the generalized Wintgen inequality for bi-slant submanifold in conformal Sasakian space form endowed with a quarter symmetric metric connection, and also find conditions under which the equality holds. Some particular cases are also stated.Originality/valueThe research may be a challenge for new developments focused on new relationships in terms of various invariants, for different types of submanifolds in that ambient space with several connections.
Highlights
A conformal change of the metric g leads to a metric which is no more compatible with the almost contact structure (w, ξ, η)
Wintgen [2] proved that the Gauss curvature K, the normal curvature K⊥ and the squared mean curvature kHk2 for any surface M~ 2 in E4 satisfy the inequality [3] as follows: kHk2 ≥ K þ jK⊥j and the equality holds if and only if the ellipse of curvature of M~ 2 in E4 is a circle
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
Summary
P. Wintgen [2] proved that the Gauss curvature K, the normal curvature K⊥ and the squared mean curvature kHk2 for any surface M~ 2 in E4 satisfy the inequality [3] as follows: kHk2 ≥ K þ jK⊥j and the equality holds if and only if the ellipse of curvature of M~ 2 in E4 is a circle. Wintgen [2] proved that the Gauss curvature K, the normal curvature K⊥ and the squared mean curvature kHk2 for any surface M~ 2 in E4 satisfy the inequality [3] as follows: kHk2 ≥ K þ jK⊥j and the equality holds if and only if the ellipse of curvature of M~ 2 in E4 is a circle We discuss such inequality for various slant cases as an application of the obtained inequality
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