In this paper,we investigate spacelike graphs defined over a domain Ω⊂M^(n) in the Lorentz manifold M^(n)×ℝ with the metric−ds^(2)+σ,where M^(n) is a complete Riemannian n-manifold with the metricσ,Ωhas piece...In this paper,we investigate spacelike graphs defined over a domain Ω⊂M^(n) in the Lorentz manifold M^(n)×ℝ with the metric−ds^(2)+σ,where M^(n) is a complete Riemannian n-manifold with the metricσ,Ωhas piecewise smooth boundary,and ℝ denotes the Euclidean 1-space.We prove an interesting stability result for translating spacelike graphs in M^(n)×ℝ under a conformal transformation.展开更多
Authors discover that a spacelike surface in Minkowski 3-space is related toa integrable system. They obtain a representation formula for spacelike surfaces withprescribed mean curvature. This representation formula i...Authors discover that a spacelike surface in Minkowski 3-space is related toa integrable system. They obtain a representation formula for spacelike surfaces withprescribed mean curvature. This representation formula is equivalent to that obtained byAkutagawa and Nishikawa.展开更多
We establish integral formulas of Minkowski’s type for compact spacelike hypersurfacesin de sitter space S<sub>1</sub><sup>n+1</sup>（1）and give their applications to the case of constant r-t...We establish integral formulas of Minkowski’s type for compact spacelike hypersurfacesin de sitter space S<sub>1</sub><sup>n+1</sup>（1）and give their applications to the case of constant r-th mean curvature（r=1,2,…,n-1）.When r=1 we recover Montiel’s result.展开更多
Spacelike Willmore surfaces in 4-dimensional Lorentzian space forms, a topic in Lorentzian conformal geometry which parallels the theory of Willmore surfaces in S4, are studied in this paper. We define two kinds of tr...Spacelike Willmore surfaces in 4-dimensional Lorentzian space forms, a topic in Lorentzian conformal geometry which parallels the theory of Willmore surfaces in S4, are studied in this paper. We define two kinds of transforms for such a surface, which produce the so-called left/right polar surfaces and the adjoint surfaces. These new surfaces are again conformal Willmore surfaces. For them the interesting duality theorem holds. As an application spacelike Willmore 2-spheres are classified. Finally we construct a family of homogeneous spacelike Willmore tori.展开更多
Let M be a connected n dimensional space form spacelike isometrically immersed in a (2n-1) dimensional indefinite space form. If M is maximal, we prove that either M is totally geodesic or M is a piece of the n dimens...Let M be a connected n dimensional space form spacelike isometrically immersed in a (2n-1) dimensional indefinite space form. If M is maximal, we prove that either M is totally geodesic or M is a piece of the n dimensional hyperbolic cylinder in the (2n-1) dimensional pseudo hyperbolic space.展开更多
The authors generalize the Fenchel theorem for strong spacelike closed curves of index 1 in the 3-dimensional Minkowski space, showing that the total curvature must be less than or equal to 2π. Here the strong spacel...The authors generalize the Fenchel theorem for strong spacelike closed curves of index 1 in the 3-dimensional Minkowski space, showing that the total curvature must be less than or equal to 2π. Here the strong spacelike condition means that the tangent vector and the curvature vector span a spacelike 2-plane at each point of the curve γ under consideration. The assumption of index 1 is equivalent to saying that γ winds around some timelike axis with winding number 1. This reversed Fenchel-type inequality is proved by constructing a ruled spacelike surface with the given curve as boundary and applying the Gauss-Bonnet formula. As a by-product, this shows the existence of a maximal surface with γ as the boundary.展开更多
In the light of great importance of curves and their frames in many differentbranches of science, especially differential geometry as well as geometric properties andthe uses in various fields, we are interested here ...In the light of great importance of curves and their frames in many differentbranches of science, especially differential geometry as well as geometric properties andthe uses in various fields, we are interested here to study a special kind of curves calledSmarandache curves in Lorentz 3-space. Then, we present some characterizations for thesecurves and calculate their Darboux invariants. Moreover, we classify TP, TU, PU andTPU-Smarandache curves of a spacelike curve according to the causal character of thevector, curve and surface used in the study. Besides, we give some of differential geometricproperties and important relations between that curves. Finally, to demonstrate ourtheoretical results a computational example is given with graph.展开更多
It is shown that a compact spacelike hypersurface which is contained in the chronological future (or past) of an equator of de Sitter space is a totally umbilical round sphere if the kth mean curvature function Hk is ...It is shown that a compact spacelike hypersurface which is contained in the chronological future (or past) of an equator of de Sitter space is a totally umbilical round sphere if the kth mean curvature function Hk is a linear combination of Hk+1,…, Hn. This is a new angle to characterize round spheres.展开更多
基金supported in part by the NSFC(11801496,11926352)the Fok Ying-Tung Education Foundation(China)the Hubei Key Laboratory of Applied Mathematics(Hubei University).
文摘In this paper,we investigate spacelike graphs defined over a domain Ω⊂M^(n) in the Lorentz manifold M^(n)×ℝ with the metric−ds^(2)+σ,where M^(n) is a complete Riemannian n-manifold with the metricσ,Ωhas piecewise smooth boundary,and ℝ denotes the Euclidean 1-space.We prove an interesting stability result for translating spacelike graphs in M^(n)×ℝ under a conformal transformation.
文摘Authors discover that a spacelike surface in Minkowski 3-space is related toa integrable system. They obtain a representation formula for spacelike surfaces withprescribed mean curvature. This representation formula is equivalent to that obtained byAkutagawa and Nishikawa.
基金Li Haizhong is supported by NNSFC No.19701017 Basic Science Research Foundation of Tsinghua University Chen Weihua is supported by NNSFC No.19571005
文摘We establish integral formulas of Minkowski’s type for compact spacelike hypersurfacesin de sitter space S<sub>1</sub><sup>n+1</sup>（1）and give their applications to the case of constant r-th mean curvature（r=1,2,…,n-1）.When r=1 we recover Montiel’s result.
基金supported by the National Natural Science Foundation of China (Grant No. 10771005)
文摘Spacelike Willmore surfaces in 4-dimensional Lorentzian space forms, a topic in Lorentzian conformal geometry which parallels the theory of Willmore surfaces in S4, are studied in this paper. We define two kinds of transforms for such a surface, which produce the so-called left/right polar surfaces and the adjoint surfaces. These new surfaces are again conformal Willmore surfaces. For them the interesting duality theorem holds. As an application spacelike Willmore 2-spheres are classified. Finally we construct a family of homogeneous spacelike Willmore tori.
基金the National Natural Science Foundationof China (No.1970 10 17
文摘Let M be a connected n dimensional space form spacelike isometrically immersed in a (2n-1) dimensional indefinite space form. If M is maximal, we prove that either M is totally geodesic or M is a piece of the n dimensional hyperbolic cylinder in the (2n-1) dimensional pseudo hyperbolic space.
基金supported by the National Natural Science Foundation of China(No.11471021)the Fundamental Research Funds for the Central Universities of China(No.531107050874)
文摘The authors generalize the Fenchel theorem for strong spacelike closed curves of index 1 in the 3-dimensional Minkowski space, showing that the total curvature must be less than or equal to 2π. Here the strong spacelike condition means that the tangent vector and the curvature vector span a spacelike 2-plane at each point of the curve γ under consideration. The assumption of index 1 is equivalent to saying that γ winds around some timelike axis with winding number 1. This reversed Fenchel-type inequality is proved by constructing a ruled spacelike surface with the given curve as boundary and applying the Gauss-Bonnet formula. As a by-product, this shows the existence of a maximal surface with γ as the boundary.
文摘In the light of great importance of curves and their frames in many differentbranches of science, especially differential geometry as well as geometric properties andthe uses in various fields, we are interested here to study a special kind of curves calledSmarandache curves in Lorentz 3-space. Then, we present some characterizations for thesecurves and calculate their Darboux invariants. Moreover, we classify TP, TU, PU andTPU-Smarandache curves of a spacelike curve according to the causal character of thevector, curve and surface used in the study. Besides, we give some of differential geometricproperties and important relations between that curves. Finally, to demonstrate ourtheoretical results a computational example is given with graph.
文摘It is shown that a compact spacelike hypersurface which is contained in the chronological future (or past) of an equator of de Sitter space is a totally umbilical round sphere if the kth mean curvature function Hk is a linear combination of Hk+1,…, Hn. This is a new angle to characterize round spheres.
