First problem of globalization in the theory of dynamical systems admitting the normal shift of hypersurfaces

3-dimensional affine hypersurfaces admitting a pointwise SO(2)- or Z_3-symmetry

In (equi-)affine differential geometry, the most important algebraic invariants are the affine (Blaschke) metric h, the affine shape operator S and the difference tensor K. A hypersurface is said to admit a pointwise symmetry if at every point there exists a linear transformation preserving the affine metric, the affine shape operator and the difference tensor K. The study of submanifolds which admit pointwise isometries was initiated by Bryant (math.DG/0007128). In this paper, we consider the 3-dimensional positive definite hypersurfaces for which at each point the group of symmetries is isom…

在允许非平凡共形向量场的黎曼流形中的等周问题

The isoperimetric problem in the Riemannian manifold admitting a non-trivial conformal vector field

本文在具有非平凡共形向量场的黎曼流形中引入保体积、降面积的平均曲率型流，证明其长时间存在与收敛，并建立星形区域等周不等式。

Dynamical systems admitting normal shift and wave equations

High frequency limit for most of wave phenomena is known as quasiclassical limit or ray optics limit. Propagation of waves in this limit is described in terms of wave fronts and rays. Wave front is a surface of constant phase whose points are moving along rays. As it appears, their motion can be described by Hamilton equations being special case for Newton's equations. In simplest cases (e. g. light propagation in isotropic non-homogeneous refracting media) the dynamics of points preserves orthogonality of wave front and rays. This property was generalized in the theory of dynamical systems ad…

Real hypersurfaces with $^{*}$-Ricci solitons of non-flat complex space forms

Kaimakamis and Panagiotidou in \cite{KP} introduced the notion of $^*$-Ricci soliton and studied the real hypersurfaces of a non-flat complex space form admitting a $^*$-Ricci soliton whose potential vector field is the structure vector field. In this article, we consider that a real hypersurface of a non-flat complex space form admits a $^*$-Ricci soliton whose potential vector field belongs to the principal curvature space and the holomorphic distribution.

Newtonian normal shift in multidimensional Riemannian geometry

Explicit description for arbitrary Newtonian dynamical system admitting the normal shift in Riemannian manifold of the dimension $n\geq 3$ is found. On the base of this result the kinematics of normal shift of hypersurfaces along trajectories of such system is studied.

零超曲面数据与环境向量场：零阶/一阶 Killing 视界

Null hypersurface data and ambient vector fields: Killing horizons of order zero/one

本文研究带切向零向量场 η 的零超曲面，建立其形变张量与几何张量的恒等式，推广表面引力并提出三类抽象视界，涵盖一般拓扑与不动点情形。

乘积空间A类超曲面上的近Ricci孤立子

Almost Ricci Solitons on Class $\mathcal A$ Hypersurfaces of Product Spaces

本文对乘积空间中切向分量为主方向的超曲面进行分类，给出其容许近Ricci孤立子的充要条件，并证明此类孤立子仅存在于旋转超曲面上。

On the concept of normal shift in non-metric geometry

Theory of Newtonian dynamical systems admitting normal shift of hypersurfaces was first developed for the case of Riemannian manifolds. Recently it was generalized for manifolds geometric equipment of which is given by some regular Lagrangian or, equivalently, by some regular Hamiltonian dynamical system. In present paper we consider further generalization of this theory for the case, when geometry of manifold is given by generalized Legendre transformation.

On the subset of normality equations describing generalized Legendre transformation

Normality equations describe Newtonian dynamical systems admitting normal shift of hypersurfaces. They were first derived in Euclidean geometry, then in Riemannian geometry. Recently they were rederived in more general case, when geometry of manifold is given by generalized Legendre transformation. As appears, in this case some part of normality equations describe generalized Legendre transformation itself irrespective to that Newtonian dynamical system, for which others are written. In present paper this smaller part of normality equations is studied.

V-representation for normality equations in geometry of generalized Legendre transformation

Normality equations describe Newtonian dynamical systems admitting normal shift of hypersurfaces. These equations were first derived in Euclidean geometry. Then very soon they were rederived in Riemannian and in Finslerian geometry. Recently I have found that normality equations can be derived in geometry given by classical and/or generalized Legendre transformation. However, in this case they appear to be written in p-representation, i. e. in terms of momentum covector and its components. The goal of present paper is to transform normality equations back to v-representation, which is more nat…

The Cauchy problem for parallel spinors as first-order symmetric hyperbolic system

We prove that a smooth Riemannian manifold admitting an imaginary generalized Killing spinor whose Dirac current satisfies an additional algebraic constraint condition can be embedded as spacelike Cauchy hypersurface in a smooth Lorentzian manifold on which the given spinor extends to a null parallel spinor. This is in contrast to a corresponding Cauchy problem for real generalized Killing spinors into Riemannian manifolds. The construction is based on first order symmetric hyperbolic PDE-methods. In fact, the coupled evolution equations for metric and spinor as considered here extend and gene…