Adela Mihai

Geometry of the Solutions of Localized Induction Equation in the Pseudo-Galilean Space

We study the surfaces corresponding to solutions of the localized induction equation in the pseudo-Galilean space . We classify such surfaces with null curvature and characterize some special curves on these surfaces in .

Riemannian manifolds carrying a pair of skew symmetric conformal vector fields

We deal with a Riemannian manifoldM carrying a pair of skew symmetric conformal vector fields (X, Y). The existence of such a pairing is determined by an exterior differential system in involution (in the sense of Cartan). In this case,M is foliated by 3-dimensional totally geodesic submanifolds. Additional geometric properties are proved.

Ruled surfaces generated by elliptic cylindrical curves in the isotropic space

Abstract In this paper we study the ruled surfaces generated by elliptic cylindrical curves in the isotropic 3-space 𝕀 3 {\mathbb{I}^{3}} . We classify such surfaces in 𝕀 3 {\mathbb{I}^{3}} with constant curvature and satisfying an equation in terms of the components of the position vector field and the Laplacian operator. Several examples are given and illustrated by figures.

CR-Submanifolds in Complex and Sasakian Space Forms

We survey recent results on CR-submanifolds in complex space forms and contact CR-submanifolds in Sasakian space forms, including a few contributions of the present authors. The Ricci curvature and k-Ricci curvature of such submanifolds are estimated in terms of the squared mean curvature. A Wintgen-type inequality for totally real surfaces in complex space forms is proved. The equality case is shown to hold if and only if the ellipse of curvature is a circle at every point of the surface; an example of a totally real surface in \(\mathbf{C}^2\) satisfying the equality case identically is provided. A generalized Wintgen inequality for Lagrangian submanifolds in complex space forms is established. A Wintgen-type inequality for CR-submanifolds in complex space forms is stated. Geometric inequalities for warped product submanifolds in complex space forms are proved, the equality cases are characterized and examples for the equality cases are given. Also some obstructions to the minimality of warped product CR-submanifolds in complex space forms are derived. The scalar curvature of such submanifolds is estimated and classifications of submanifolds in complex space forms satisfying the equality case are given. The survey ends with results on contact CR-submanifolds in Sasakian space forms. Geometric inequalities for the Ricci curvature and k-Ricci curvature of contact CR-submanifolds in Sasakian space forms are stated. A recent result by the second author (i.e., a generalized Wintgen inequality for C-totally real submanifolds in Sasakian space forms) is extended to contact CR-submanifolds in Sasakian space forms.

Inequalities on the Ricci curvature

We improve Chen-Ricci inequalities for a Lagrangian submanifold Mn of dimension n (n 2) in a 2n -dimensional complex space form ̃ M2n(4c) of constant holomorphic sectional curvature 4c with a semi-symmetric metric connection and a Legendrian submanifold Mn in a Sasakian space form ̃ M2n+1(c) of constant φ -sectional curvature c with a semi-symmetric metric connection, respectively. Mathematics subject classification (2010): 53C40, 53B05, 53B15.

Lorentzian Manifolds having the Killing Property

We deal with a Lorentzian n -dimensional manifold M carrying two skew-symmetric Killing null vector fields ξ 1 and ξ n . They define a commutative left invariant pairing. The ( n – 2)-dimensional spatial distribution orthogonal to ξ 1 and ξ n is involutive and its leaves are totally geodesic and pseudo-isotropic. If n = 4, then it is shown that the general space-time M is of type D in Petrovs classification and the spatial surfaces which foliate M are totally geodesic and pseudo-isotropic. Properties of the congruence of Debever are pointed-out.