Totally Geodesic Subsets in the Manifold of Directions of Physical Space

Let M ₀ be the Minkowski 4-space, let ₂(M ₀) denote the second exterior power of M ₀ equipped with a structure of a pseudo-Euclidean space with signature (3,3), let K ₀(M ₀) ₂ M ₀ be the light cone, and let G _{1 2}(M ₀) be the set of the oriented 2-planes meeting the interior of K ₀(M ₀). Four types of totally geodesic two-manifolds in G ₁ are described such that manifolds of one type are pairwise congruent as subsets in ₂(M ₀), while manifolds of different types are not. Models of such manifolds in the disk D ³ are constructed. An explicit formula for the curvature tensor of G ₁ is given. Bibliography: 6 titles.     

Absence of Three-Dimensional Totally Geodesic Submanifolds in the Manifold of Directions of Physical Space

It is proved that the manifold of directions of physical space contains no three-dimensional totally geodesic submanifolds. Bibliography: 3 titles.     

Totally Geodesic Invariant Subm anifolds of Contact Metric Manifold

§１．　ＰｒｅｌｉｍｉｎａｒｉｅｓＬｅｔＭｂｅａ（２ｎ＋１）－ｄｉｍｅｎｓｉｏｎａｌｃｏｎｔａｃｔｍｅｔｒｉｃｍａｎｉｆｏｌｄｗｉｔｈｓｔｒｕｃｔｕｒｅｔｅｎｓｏｒｓ（Φ－，ξ－，η－，ｇ）．ＴｈｅｎｔｈｅｙｓａｔｉｓｆｙΦ－ξ－＝０，η－（ξ－）＝１，Φ－２＝－Ｉ＋η－ξ－，η－（Ｘ）＝ｇ（Ｘ，ξ－），　　　ｇ（Φ－Ｘ，Φ－Ｙ）＝ｇ（Ｘ，Ｙ）－η－（Ｘ）η－（Ｙ），ｇ（Ｘ，Φ－Ｙ）＝ｄη－（Ｘ，Ｙ）（１．１）ＦｏｒａｎｙｖｅｃｔｏｒｆｉｅｌｄｓＸａｎｄＹ…     

Totally Geodesic Seifert Surfaces in Hyperbolic Knot and Link Complements I

The first examples of totally geodesic Seifert surfaces are constructed for hyperbolic knots and links, including both free and totally knotted surfaces. Then it is proved that two bridge knot complements cannot contain totally geodesic orientable surfaces.     

Totally Geodesic Orbits of Isometries

We study cohomogeneity one Riemannian manifolds and we establish some simple criterium to test when a singular orbit is totally geodesic. As an application, we classify compact, positively curved Riemannian manifolds which are acted on isometrically by a non-semisimple Lie group with an hypersurface orbit.     

二维复空间型的全实曲面

本文有兴趣于二维复空间型的全实曲面的研究．得到了一些有趣结果，并将［１，２］中的有关定理推广到具平行平均曲率向量的全实曲面的情形．     

Characterization of Totally Geodesic Totally Real 3-dimensional Submanifolds in the 6-sphere

In this paper, we study Lagrangian submanifolds of the nearly Kaehler 6-sphere. We derive a pinching result for the Ricci curvature of such submanifolds thus providing a characterisation of the totally geodesic submanifold. Our pinching result improves a previous result obtained by H. Li.     

Totally Geodesic and Parallel Hypersurfaces of Four-Dimensional Oscillator Groups

We obtain the complete classification and explicitly describe totally geodesic and parallel hypersurfaces of four-dimensional oscillator groups, equipped with a one-parameter family of left-invariant Lorentzian metrics.     

Totally Umbilical Hypersurfaces in a Locally Symmetric Manifold

In this paper we obtain some formulas for totally umbilical hypersurfaces in a locally symmetric manifold, and derive some local results on the hypersurfaces from these formulas.     

局部对称流形的全脐子流形

In this paper we obtain some formulas for totally umbilical submanifolds in a locally symmetric manifold, and derive some local results on the submanifolds from these formulas.     

Totally Umbilical Subrnanifolds in a Locally Symmetric Manifold

In this paper we obtain some formulas for totally umbilical submanifolds in a localiy symmetric manifold, and dcrivc some local rcsults on the submanifolds from these formulas.     

A Gap Theorem for Free Boundary Minimal Surfaces in Geodesic Balls of Hyperbolic Space and Hemisphere

In this paper we provide a pinching condition for the characterization of the totally geodesic disk and the rotational annulus among minimal surfaces with free boundary in geodesic balls of three-dimensional hyperbolic space and hemisphere. The pinching condition involves the length of the second fundamental form, the support function of the surface, and a natural potential function in hyperbolic space and hemisphere.     

Stationary Curvatures of Two-Dimensional Totally Geodesic Submanifolds in the Grassmannian of Bivectors

An infinitesimal criterion indicating when a two-dimensional submanifold of a Riemannian symmetric space is totally geodesic is given. As an application, the classification of two-dimensional totally geodesic submanifolds of the Grassmannian of bivectors is given in a new way, and it is proved that the sectional curvature takes stationary values on tangent spaces of such submanifolds. Bibliography: 9 titles.     

The Totally Geodesic Coisotropic Submanifolds in Kähler Manifolds

In this paper, we consider the coisotropic submanifolds in a Kähler manifold of nonnegative holomorphic curvature. We prove an intersection theorem for compact totally geodesic coisotropic submanifolds and discuss some topological obstructions for the existence of such submanifolds. Our results apply to Lagrangian submanifolds and real hypersurfaces since the class of coisotropic submanifolds includes them. As an application, we give a fixed-point theorem for compact Kähler manifolds with positive holomorphic curvature. Also, our results can be further extended to nearly Kähler manifolds.     

Classification of Timelike Constant Slope Surfaces in 3-Dimensional Minkowski Space

A surface in the Minkowski 3-space is called a constant slope surface if its position vector makes a constant angle with the normal at each point on the surface. In this paper, we give a complete classification of timelike constant slope surfaces in the three dimensional Minkowski space.     

Classification of $$\eta $$-Biharmonic Surfaces in Non-flat Lorentz Space Forms

In this paper, we prove that \(\eta \)-biharmonic surfaces in non-flat three-dimensional Lorentz space forms are isoparametric and give full classification results.     

On Geodesic Curves on Quotient Manifold of Nondegenerate Affinor Fields

We consider the quotient manifold of the manifold of nondegenerate affinor fields on a compact manifold with respect to the action of the group of nowhere vanishing functions. This manifold is endowed with a structure of infinite-dimensional Lie group. On this Lie group, we construct an object of linear connection with respect to which all left-invariant vector fields are covariantly constant (the Cartan connection). We also find the geodesics of the Cartan connection.