In this paper, we introduce the notion of the second kind twisted surfaces in Minkowski 3-space. We classify all non-degenerate second kind twisted surfaces in terms of flat, minimal, constant Gaussian and constant mean curvature surfaces, with respect to a chosen lightlike transversal bundle. We also prove that a lightlike second kind twisted surfaces, with respect to a chosen lightlike transversal vector bundle, are the lightcones, the lightlike binormal surfaces over pseudo null base curve and the lightlike ruled surfaces with null rulings whose base curve lies on lightcone.

Mihai obtained the Wintgen inequality, also known as the generalized Wintgen inequality, for Lagrangian submanifolds in complex space forms and also characterized the corresponding equality case. Submanifolds M which satisfy the equality in these optimal general inequalities are called generalized Wintgen ideal submanifolds in the ambient spaceM. For generalized Wintgen ideal Lagrangian submanifolds M n in complex space formsM n (4c), we will show some properties concerning different kinds of their pseudosymmetry in the sense of Deszcz.

In this paper we discuss δ(2, 2) Chen ideal submanifolds M 4 in the Euclidean space E 6 , and we find the necessary and sufficient conditions under which such a submanifold M 4 is semi-symmetric, i.e. it satisfies the condition R(X, Y) • R = 0. 1. Chen ideal submanifolds of Euclidean spaces Let M n be an n-dimensional Riemannian submanifold of an (n + m)-dimensional Euclidean space E n+m , (n ≥ 2, m ≥ 1) and let , ∇ and , ∇ be the Riemannian metric and the corresponding Levi-Civita connection on M n and on E n+m , respectively. Tangent vector fields on M n will be written as X, Y,. .. and normal vector fields on M n in E n+m will be written as ξ, η,. .. The formulae of Gauss and Weingarten, concerning the decomposition of the vector fields ∇ X Y and ∇ X ξ, respectively, into their tangential and normal components along M n in E n+m , are given by ∇ X Y = ∇ X Y + h(X, Y) and ∇ X ξ = −A ξ (X) + ∇ ⊥ X ξ, respectively, whereby h is the second fundamental form and A ξ is the shape operator or Weingarten map of M n with respect to the normal vector field ξ, such that (h(X, Y), ξ) = (A ξ (X), Y), and ∇ ⊥ is the connection in the normal bundle. The mean curvature vector field H is defined by H = 1 n tr h and its length H = H is the (extrinsic) mean curvature of M n in E n+m. A submanifold M n in E n+m is totally geodesic when h = 0, totally umbilical when h = H, minimal when H = 0 and pseudo-umbilical when H is an umbilical normal direction [2]. Let {E 1 ,. .. , E n , ξ 1 ,. .. , ξ m } be any adapted orthonormal local frame field on the submanifold M n in E n+m , denoted for short also as {E i , ξ α }, whereby i ∈ {1, 2,. .. , n} and α ∈ {1, 2,. .. , m}. By the equation of Gauss, the (0, 4) Riemann-Christoffel curvature tensor of a submanifold M n in E n+m is given by R(X, Y, Z, W) = (h(Y, Z), h(X, W)) − (h(X, Z), h(Y, W)). The (0, 2) Ricci curvature tensor of M n is defined by S(X, Y) = i R(X, E i , E i , Y) and the metrically corresponding (1, 1) tensor or Ricci operator will also be denoted by S : (S(X), Y) = S(X, Y). The scalar curvature of a Riemannian manifold M n is defined by τ = i< j K(E i ∧ E j) whereby K(E i ∧ E j) = R(E i , E j , E j , E i) is the sectional curvature for the plane section π = E i ∧ E j , (i j). By the equation of Ricci, the normal curvature tensor of a submanifold M n in E n+m is given by R ⊥ (X, Y, ξ, η) = ([A ξ , A η ](X), Y), whereby [A ξ , A η ] = A ξ A η − A η A ξ , which, as already observed by Cartan [1], implies that the normal connection is flat or trivial if and only if all shape operators A ξ are simultaneously diagonalisable. The function inf K : M n → R is defined by (inf K)(p) = inf{K(p, π) |π is a plane section of T p (M n)}. In [3], B.-Y. Chen introduced the δ(2)-curvature as δ(2) = τ − inf K, which clearly is a Riemannian scalar invariant