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 isomorphic to either Z_3 or SO(2). We classify such hypersurfaces and show how they can be constructed starting from 2-dimensional positive definite affine spheres.
In this article we obtain a classification of strictly locally convex affine hypersurfaces in A^{n+1} for which the geometrical structure is pointwise invariant under the group SO(n-1) represented by rotations around a fixed axis in the tangent space.
An affine hypersurface $M$ is said to admit a pointwise symmetry, if there exists a subgroup $G$ of ${\rm Aut}(T_p M)$ for all $p\in M$, which preserves (pointwise) the affine metric $h$, the difference tensor $K$ and the affine shape operator $S$. Here, we consider 3-dimensional indefinite affine hyperspheres, i.e. $S= HId$ (and thus $S$ is trivially preserved). In Part 1 we found the possible symmetry groups $G$ and gave for each $G$ a canonical form of $K$. We started a classification by showing that hyperspheres admitting a pointwise ${\mathbb Z}_2\times {\mathbb Z}_2$ resp. ${\mathbb R}$-…
An affine hypersurface is said to admit a pointwise symmetry, if there exists a subgroup of the automorphism group of the tangent space, which preserves (pointwise) the affine metric h, the difference tensor K and the affine shape operator S. In this paper, we deal with positive definite affine hypersurfaces of dimension three. First we solve an algebraic problem. We determine the non-trivial stabilizers G of the pair (K,S) under the action of SO(3) on an Euclidean vectorspace (V,h) and find a representative (canonical form of K and S) of each (SO(3)/G)-orbit. Then, we classify hypersurfaces a…
We study constant mean curvature spacelike hypersurfaces and in particular maximal hypersurfaces immersed in pp-wave spacetimes satisfying the timelike convergence condition. We prove the non-existence of compact spacelike hypersurfaces whose constant mean curvature is non-zero and also that every compact maximal hypersurface is totally geodesic. Moreover, we give an extension of the classical Calabi-Bernstein theorem to this class of pp-wave spacetimes.
We prove that a Riemannian product of type M x R (where R denotes the Euclidean line) admits totally umbilical hypersurfaces if and only if M has locally the structure of a warped product and we give a complete description of the totally umbilical hypersurfaces in this case. Moreover, we give a necessary and sufficient condition under which a Riemannian three-manifold carrying a unit Killing field admits totally geodesic surfaces and we study local and global properties of three-manifolds satisfying this condition.
An affine hypersurface M is said to admit a pointwise symmetry, if there exists a subgroup G of Aut(T_p M) for all p in M, which preserves (pointwise) the affine metric h, the difference tensor K and the affine shape operator S. Here, we consider 3-dimensional indefinite affine hyperspheres, i.e. S= H Id (and thus S is trivially preserved). First we solve an algebraic problem. We determine the non-trivial stabilizers G of a traceless cubic form on a Lorentz-Minkowski space R^3_1 under the action of the isometry group SO(1,2) and find a representative of each SO(1,2)/G-orbit. Since the affine c…
Formula for the force field of Newtonian dynamical systems admitting the normal shift of hypersurfaces in Riemannian manifolds is considered. Problem of globalization for geometric structures associated with this formula is studied.
Problem of global integration of geometric structures arising in the theory of dynamical systems admitting the normal shift is considered. In the case when such integration is possible the problem of globalization for shift maps is studied.
Let $M$ be a real hypersurface of a complex space form with constant curvature $c$. In this paper, we study the hypersurface $M$ admitting Miao-Tam critical metric, i.e. the induced metric $g$ on $M$ satisfies the equation:$-(Δ_gλ)g+\nabla^2_gλ-λRic=g$, where $λ$ is a smooth function on $M$. At first, for the case where $M$ is Hopf, $c=0$ and $c\neq0$ are considered respectively. For the non-Hopf case, we prove that the ruled real hypersurfaces of non-flat complex space forms do not admit Miao-Tam critical metrics. Finally, it is proved that a compact hypersurface of a complex Euclidean space …
The object of the present paper is to characterize two classes of almost Kenmotsu manifolds admitting Ricci-Yamabe soliton. It is shown that a $(k,μ)'$-almost Kenmotsu manifold admitting a Ricci-Yamabe soliton or gradient Ricci-Yamabe soliton is locally isometric to the Riemannian product $\mathbb{H}^{n+1}(-4) \times \mathbb{R}^n$. For the later case, the potential vector field is pointwise collinear with the Reeb vector field. Also, a $(k,μ)$-almost Kenmotsu manifold admitting certain Ricci-Yamabe soliton with the curvature property $Q \cdot P = 0$ is locally isometric to the hyperbolic space…
In this paper, we study locally strongly convex Tchebychev hypersurfaces, namely the {\it centroaffine totally umbilical hypersurfaces}, in the $(n+1)$-dimensional affine space $\mathbb{R}^{n+1}$. We first make an ordinary-looking observation that such hypersurfaces are characterized by having a Riemannian structure admitting a canonically defined closed conformal vector field. Then, by taking the advantage of properties about Riemannian manifolds with closed conformal vector fields, we show that the ellipsoids are the only centroaffine Tchebychev hyperovaloids. This solves the longstanding pr…
In this work, we study various properties of embedded hypersurfaces in $1+1+2$ decomposed spacetimes with a preferred spatial direction, denoted $e^μ$, which are orthogonal to the fluid flow velocity of the spacetime and admit a proper conformal transformation. To ensure a non-vanishing positivity scalar curvature of the induced metric, we impose that the scalar curvature of the conformal metric is non-negative and that the associated conformal factor satisfies $\hatφ^2+2\hat{\hatφ}>0$, where $\hat{\ast}$ denotes derivative along $e^μ$. Firstly, it is demonstrated that such hypersurface is eit…
Generalized Berwald manifolds are Finsler manifolds admitting linear connections such that the parallel transports preserve the Finslerian length of tangent vectors. By the fundamental result of the theory \cite{V5} such a linear connection must be metrical with respect to the averaged Riemannian metric given by integration of the Riemann-Finsler metric on the indicatrix hypersurfaces. Therefore the linear connection is uniquely determined by its torsion tensor. If the torsion is zero then we have a classical Berwald manifolds. Otherwise the torsion is a strange data we need to express in term…
Using minimal hypersurfaces, we obtain topological obstructions to admitting complete metrics with positive scalar curvature on a given class of non-compact n-manifolds with n less than 8. We show that the Liouville theorem for a locally conformally flat n-manifold of non-negative scalar curvature follows from the impossibility of there being a positive scalar curvature metric on its connect sum with the n-torus. With the recent work of Chodosh-Li, the Liouville theorem is now proved in all remaining cases. Finally, using MOTS instead of minimal hypersurfaces, we show an Initial Data Set versi…
Geometric scattering for nonlinear wave equations on the Schwarzschild metric
Thang Long Institute of Mathematics and Applied Sciences (TIMAS), Thang Long University（长龙数学与应用科学研究所（TIMAS），长龙大学）；Faculty of Mathematics and Informatics, Hanoi University of Science and Technology（河内科技大学数学与信息学院）
It is shown that in a class of maximal globally hyperbolic spacetimes admitting two local Killing vectors, the past (defined with respect to an appropriate time orientation) of any compact constant mean curvature hypersurface can be covered by a foliation of compact constant mean curvature hypersurfaces. Moreover, the mean curvature of the leaves of this foliation takes on arbitrarily negative values and so the initial singularity in these spacetimes is a crushing singularity. The simplest examples occur when the spatial topology is that of a torus, with the standard global Killing vectors, bu…
