Pseudo-umbilical Submanifolds Research Articles

Characterizations of pseudo-umbilical submanifolds of Kaehler manifolds by means of specific types

Characterizations of pseudo-umbilical submanifolds of Kaehler manifolds by means of specific types

Classification of f-biharmonic submanifolds in Lorentz space forms

Abstract In this paper, f-biharmonic submanifolds with parallel normalized mean curvature vector field in Lorentz space forms are discussed. When f f is a constant, we prove that such submanifolds have parallel mean curvature vector field with the minimal polynomial of the shape operator of degree ≤ 2 \le 2 . When f f is a function, we completely classify such pseudo-umbilical submanifolds.

On pseudo-umbilical spacelike submanifolds in indefinite space form $$\mathcal {M}_p^{n+p}(c)$$

An intrinsic condition for Pseudo-umbilical spacelike submanifold immersed in indefinite space form is derived, which is used to show that such submanifold is totally geodesic. Next, using a result of Aiyama (Tokyo J Math 18:81–90, 1995), it is proved that Pseudo-umbilical spacelike submanifold is totally umbilical.

Pseudo-umbilical space-like submanifolds in locally symmetric pseudo-Riemannian manifolds

Pseudo-umbilical space-like submanifolds in locally symmetric pseudo-Riemannian manifolds

Biharmonic Submanifolds with Parallel Normalized Mean Curvature Vector Field in Pseudo-Riemannian Space Forms

In this paper, we investigate biharmonic submanifolds with parallel normalized mean curvature vector field in pseudo-Riemannian space forms and classify completely such pseudo-umbilical submanifolds. Also, we prove that such submanifolds have parallel mean curvature vector field under the assumption that they have diagonalizable shape operator with at most two distinct principal curvatures in the direction of the mean curvature vector field, and apply it to obtain a partial classification result.

Umbilical Properties of Spacelike Co-dimension Two Submanifolds

For Riemannian submanifolds of a semi-Riemannian manifold, we introduce the concepts of total shear tensor and shear operators as the trace-free part of the corresponding second fundamental form and shape operators. The relationship between these quantities and the umbilical properties of the submanifold is shown. Several novel notions of umbilical submanifolds are then considered along with the classical concepts of totally umbilical and pseudo-umbilical submanifolds. Then we focus on the case of co-dimension 2, and we present necessary and sufficient conditions for the submanifold to be umbilical with respect to a normal direction. Moreover, we prove that the umbilical direction, if it exists, is unique —unless the submanifold is totally umbilical— and we give a formula to compute it explicitly. When the ambient manifold is Lorentzian we also provide a way of determining its causal character. We end the paper by illustrating our results on the Lorentzian geometry of the Kerr black hole.

A Note on Pseudo-Umbilical Submanifolds of Hessian Manifolds with Constant Hessian Sectional Curvature

The geometry of Hessian manifold, as a branch of statistics, physics, Kaehlerian, and affine differential geometry, is deeply fruitful and a new field for scientists. However, inspite of its importance submanifolds and curvature conditions of it have not been so well known yet. In this paper, we focus on the pseudo-umbilical submanifolds on Hessian manifold with constant Hessian sectional curvature and using sectional curvature conditions we obtain new results on it.

Biharmonic Riemannian maps

We give necessary and sufficient conditions for Riemannian maps to be biharmonic. We also define pseudo-umbilical Riemannian maps as a generalization of pseudo-umbilical submanifolds and show that such Riemannian maps put some restrictions on the target m

On Lagrangian surfaces in CP2(c̃)

Chen and Ogiue completely classified totally umbilical submanifolds in a non-flat complex-space-form. However, the classification problem of pseudo-umbilical submanifolds in a non-flat complex-space-form is still open. Very recently, Chen introduced the notion of Lagrangian H-umbilical submanifolds which is the simplest totally real submanifolds next to the totally geodesic ones in a complex-space-form and and classified Lagrangian H-umbilical submanifolds in a complex-space-form. The author proved that a Lagrangian H-umbilical surface M in a complex 2-dimensional complex projective space CP 2 (c) is an isotropic surface in CP 2 (c) if and only if M is a minimal surface in CP 2 (c). In this paper, firstly, we prove that a Lagrangian surface M in CP 2 (c) is an isotropic surface in CP 2 (c) if and only if M is a minimal surface in CP 2 (c). Secondly, we classify Lagrangian non-totally geodesic pseudo-umbilical surfaces in CP 2 (c).

Pseudo-umbilical submanifolds of constant curvature Riemannian manifolds

Bai [1] gave an intrinsic integral inequality for compact minimal submanifolds of constant curvature Riemannian manifolds. In this paper, we extend Bai's result to the case of pseudo-umbilical submanifolds.

Totally real pseudo-umbilical submanifolds of a quaternion space form

Let M(c) denote a 4n-dimensional quaternion space form of quaternion sectional curvature c, and let P(H) denote the 4n-dimensional quaternion projective space of constant quaternion sectional curvature 4. Let N be an n-dimensional Riemannian manifold isometrically immersed in M(c). We call N a totally real submanifold of M(c) if each tangent 2-plane of N is mapped into a totally real plane in M (c). B. Y. Chen and C. S. Houh proved in [1].

Totally Real Pseudo-Umbilical Submanifolds of a Complex Projective Space

In this paper, we study the totally real pseudo-umbilical submanifolds in the complex projective space, and obtain two integral inequalities and some conditions under which the submanifold has parallel second fundamental form.

Pseudo-umbilical submanifolds of a space form $N^{n+p}(C)$

Pseudo-umbilical submanifolds of a space form $N^{n+p}(C)$

Finite type pseudo-umbilical submanifolds in a hypersphere

The notion of finite type submanifolds was introduced by B.Y. Chen. In this article we study 2- and 3-type pseudo-umbilical submanifolds in a hypersphere. Two theorems in this respect are obtained.

Geometric applications of euclidean Jordan triple systems

The article presents a result about the trace form of Jordan triple systems and as application the classification of the pseudo-umbilical symmetric submanifolds of hyperbolic space.

Pseudo-umbilical submanifolds of codimension $3$ with constant mean curvature

Pseudo-umbilical submanifolds of codimension $3$ with constant mean curvature

Pseudo-umbilical submanifolds of a Riemannian manifold of constant curvature, II

Pseudo-umbilical submanifolds of a Riemannian manifold of constant curvature, II

A note on pseudo-umbilical submanifolds with $M$-index $1$ and codimension $2$ in Euclidean spaces

A note on pseudo-umbilical submanifolds with $M$-index $1$ and codimension $2$ in Euclidean spaces

Pseudo-umbilical submanifolds of codimension $2$

Pseudo-umbilical submanifolds of codimension $2$

Pseudo-umbilical submanifolds with $M$-index $łeq 1$ in Euclidean spaces

Pseudo-umbilical submanifolds with $M$-index $łeq 1$ in Euclidean spaces