An Expression for the Riemann Tensor of a Submanifold Given by a System of Equations in a Riemannian Space

We derive an expression for the Riemann tensor of a submanifold given implicitly by a system of independent equations in a Riemannian space. In particular, we prove a formula for the internal curvature of a two-surface in a three-dimensional Riemannian space. Some applications of the formula are given.     

复射影空间中具有平坦法丛的一般极小子流形的注记

设Mn是复射影空间CPn+p/2中具有平坦法丛的一般极小子流形.该文研究了这种子流形的曲率性质与几何性质之间的关系.运用活动标架法,得到关于Ricci曲率和第二基本形式模长的刚性定理,完善了已有文献的相关结果.此外,该文还得到具有平坦法丛的一般子流形一个重要性质.     

Submanifolds of positive Ricci curvature in a Euclidean space

In this paper we study the role of constant vector fields on a Euclidean space R ^n+p in shaping the geometry of its compact submanifolds. For an n-dimensional compact submanifold M of the Euclidean space R ^n+p with mean curvature vector field H and a constant vector field on R ^n+p , the smooth function is used to obtain a characterization of sphere among compact submanifolds of positive Ricci curvature (cf. main Theorem).      

Classification of Casorati ideal Lagrangian submanifolds in complex space forms

In this paper, using optimization methods on Riemannian submanifolds, we establish two improved inequalities for generalized normalized δ-Casorati curvatures of Lagrangian submanifolds in complex space forms. We provide examples showing that these inequalities are the best possible and classify all Casorati ideal Lagrangian submanifolds (in the sense of B.-Y. Chen) in a complex space form. In particular, we generalize the recent results obtained in G.E. Vîlcu (2018) [34].     

The coefficient coalgebra of a symmetrized tensor space

The coefficient coalgebra of r-fold tensor space and its dual, the Schur algebra, are generalized in such a way that the role of the symmetric group Σ _r is played by an arbitrary subgroup of Σ _r . The dimension of the coefficient coalgebra of a symmetrized tensor space is computed and the dual of this coalgebra is shown to be isomorphic to the analogue of the Schur algebra.     

On the Riemannian curvature tensor of a real hypersurface in a complex projective space

We consider real hypersurfaces M in complex projective space equipped with both the Levi–Civita and generalized Tanaka–Webster connections and classify them when the covariant derivatives associated with both connections, either in the direction of the structure vector field or any direction of the maximal holomorphic distribution, coincide when applying to the Riemannian curvature tensor of the real hypersurface.     

Some special surfaces in the pseudo-Galilean space

We describe some special surfaces in pseudo-Galilean spaces such as helical surfaces, ruled screw surfaces, surfaces of revolution and in particular tori of revolution. We define special surfaces and find their main properties.      

复射影空间中迷向Kaehler子流形

讨论了复射影空间中迷向Kaehler流形,运用活动标架法获得关于截面曲率,Ricci曲率和第二基本形式模长的Pinching定理,将相关结果作了一定的推广.     

Invariant surfaces in the homogeneous space Sol with constant curvature

A surface in homogeneous space is said to be an invariant surface if it is invariant under some of the two 1‐parameter groups of isometries of the ambient space whose fix point sets are totally geodesic surfaces. In this work we study invariant surfaces that satisfy a certain condition on their curvatures. We classify invariant surfaces with constant mean curvature and constant Gaussian curvature. Also, we characterize invariant surfaces that satisfy a linear Weingarten relation.     

Some remarks on pseudo-umbilical totally real submanifolds in a complex projective space

This paper studies the relationship between the pseudo-umbilical totally real submanifolds and the minimal totally real submanifolds in a complex projective space. Two theo- rems which claim that some types of pseudo-umbilical totally real submanifolds must be minimal submanifolds are proved.     

Finsler空间上的Weyl曲率

The Weyl curvature of a Finsler metric is investigated. This curvature constructed from Riemannain curvature. It is an important projective invariant of Finsler metrics. The author gives the necessary conditions on Weyl curvature for a Finsler metric to be Randers metric and presents examples of Randers metrics with non-scalar curvature. A global rigidity theorem for compact Finsler manifolds satisfying such conditions is proved. It is showed that for such a Finsler manifold,if Ricci scalar is negative,then Finsler metric is of Randers type.     

Dynamical equations for the quantum product on a symplectic space in affine coordinates

We derive a system of dynamical equations for an associative noncommutative product of functions on a symplectic space. This system is explicitly solved in semiclassical approximation.Translated from Matematicheskie Zametki, vol. 77, no. 1, 2005, pp. 42–52.Original Russian Text Copyright © 2005 by O. N. Grigorev, M. V. Karasev.This revised version was published online in April 2005 with a corrected issue number.     

Tight surfaces in three-dimensional compact Euclidean space forms

In this paper we define and discuss tight surfaces -- smooth or polyhedral -- in three-dimensional compact Euclidean space forms and prove existence and non-existence results. It will be shown that all orientable and most of the non-orientable surfaces can be tightly immersed in any of these space forms.

     

Certain condition on the second fundamental form of CR submanifolds of maximal CR dimension of complex Euclidean space

We treat m-dimensional real submanifolds M of complex space forms ̿M when the maximal holomorphic tangent subspace is (m−1)-dimensional. On these manifolds there exists an almost contact structure F which is naturally induced from the ambient space and in this paper we study the condition h(FX,Y)−h(X,FY) = g(FX,Y)η, η∊ T⊥(M), on the structure F and on the second fundamental form h of these submanifolds. Especially when the ambient space ̿M is a complex Euclidean space, we obtain a complete classification of submanifolds M which satisfy these conditions.Mathematics Subject Classifications (2000): 53C15, 53C40, 53B20.     

A variational proof for the existence of a conformal metric with preassigned negative Gaussian curvature for compact Riemann surfaces of genus > 1

Given a smooth functionK < 0 we prove a result by Berger, Kazhdan and others that in every conformal class there exists a metric which attains this function as its Gaussian curvature for a compact Riemann surface of genusg > 1. We do so by minimizing an appropriate functional using elementary analysis. In particular forK a negative constant, this provides an elementary proof of the uniformization theorem for compact Riemann surfaces of genusg > 1. An erratum to this article is available at .     

Finding minimal spanning trees in a euclidean coordinate space

The minimal spanning tree problem of a point set in ak-dimensional Euclidean space is considered and a new version of the multifragmentMST-algorithm of Bentley and Friedman is given. The minimal spanning tree is found by repeatedly joining the minimal subtree with the closest subtree. Ak-d tree is used for choosing the connecting edges. Computation time of the algorithm depends on the configuration of the point set: for normally distributed random points the algorithm is very fast. Two extreme cases demandingO(n logn) andO(n ²) operations,n being the cardinality of the point set, are also given.     

Ricci Curvature of Certain Submanifolds in Kenmotsu Space Forms

In this paper, we obtain some sharp inequalities between the Ricci cur- vature and the squared mean curvature for bi-slant and semi-slant submanifolds in Kenmotsu space forms. Estimates of the scalar curvature and the k-Ricci curvature, in terms of the squared mean curvature, are also proved respectively.     

关于四元射影空间的正曲率全复子流形

本文讨论四元射影空间的全复子流形，证明了四元射影空间的正截面曲率紧致全复子流形一定是全测地的。     

On the tensor algebra of a non-abelian group and applications

We introduce the tensor algebra J() of a crossed complex and give its relations with the James construction for a filtered space. For a group G, we define via multi-derivations the derived algebra I _* G which is isomorphic to the algebra C(JG) of chains on JG We derive from a certain exact sequence for the homology H*JG applications to the homotopy of the suspended classifying space BG.