Generalized Wintgen inequality for slant submanifolds in metallic Riemannian space forms

Metallic Structures for Tangent Bundles over Almost Quadratic ϕ-Manifolds

“…where J denotes a tensor field of type (1,1), I is the identity vector field, and p, q are natural numbers, named as a metallic structure. The structure (M, J) is called a metallic manifold [26][27][28][29][30][31].…”

On Golden Lorentzian Manifolds Equipped with Generalized Symmetric Metric Connection

“…We identify a Riemannian manifold with the help of (M n , g) and of a (1, 1)-tensor field with G satisfying [4,6,10,11].…”

“…The Riemannian curvature tensor R of the locally golden space form M = M 1 (c 1 ) × M 2 (c 2 ) is written according to [11] as…”

“…The study of golden structures was also carried out on semi-Riemannian manifolds by M. Ozkan [7]. One can refer to [8][9][10][11], etc., and the references therein for recent developments in golden differential geometry.…”

An Optimal Inequality for the Normal Scalar Curvature in Metallic Riemannian Space Forms

“…They revealed that, for a submanifold M n of real space form Mn+m (c), the following hold ρ + ρ ⊥ ≤ ||H|| 2 + c, ρ denotes the normalised scalar curvature and ρ ⊥ stands for the normalised normal scalar curvature of M. This inequality is also known as the generalized Wintgen inequality or the normal scalar curvature conjecture and was proved independently by Ge and Tang [7], and Lu [8]. Recently generalized Wintgen inequalities have been established for submanifolds in Golden Riemannian manifolds [9], complex space forms [10], Sasakian space form [11], (κ, µ)-space forms [12], etc. For further literature about the DDVV inequality, one can refer to [13] and the references therein.…”