Let （M, g） be an n-dimensional Riemannian manifold and T＊M be its cotan-gent bundle equipped with the rescaled Sasaki type metric. In this paper, we firstly study the paraholomorphy property of the rescaled Sasaki ...Let （M, g） be an n-dimensional Riemannian manifold and T＊M be its cotan-gent bundle equipped with the rescaled Sasaki type metric. In this paper, we firstly study the paraholomorphy property of the rescaled Sasaki type metric by using some compati-ble paracomplex structures on T＊M. Second, we construct locally decomposable Golden Riemannian structures on T＊M . Finally we investigate curvature properties of T＊M .展开更多
In this note we give a geometrical presentation to the 4D Riemannian curvature as it relates to the Newtonian gravity in the 4D Lorentz manifold. The compacting of the proper time as is necessary for the unification w...In this note we give a geometrical presentation to the 4D Riemannian curvature as it relates to the Newtonian gravity in the 4D Lorentz manifold. The compacting of the proper time as is necessary for the unification with the Maxwell electrodynamics, as given by Einstein and Kaluza-Klein, should the universe be only of 4D space-time, led to the concept of gravitational field singularity sinks known as black holes, that would not be acceptable under a 5D homogeneous manifold through which the 4D Lorentz manifold evolved by application of the Perelman-Ricci Flow entropy mapping, which is consistent with both Maxwell suggested magnetic monopole, the quantum Higgs vacuum theory and the Gell-Mann standard model for hadrons.展开更多
In this paper, we provide simple and explicit formulas for computing Riemannian curvatures, mean curvature vectors, principal curvatures and principal directions for a 2-dimensional Riemannian manifold embedded in IRk...In this paper, we provide simple and explicit formulas for computing Riemannian curvatures, mean curvature vectors, principal curvatures and principal directions for a 2-dimensional Riemannian manifold embedded in IRk with k > 3.展开更多
This paper aims to study the Berger type deformed Sasaki metric g_(BS)on the second order tangent bundle T^(2)M over a bi-Kählerian manifold M.The authors firstly find the Levi-Civita connection of the Berger typ...This paper aims to study the Berger type deformed Sasaki metric g_(BS)on the second order tangent bundle T^(2)M over a bi-Kählerian manifold M.The authors firstly find the Levi-Civita connection of the Berger type deformed Sasaki metric g_(BS)and calculate all forms of Riemannian curvature tensors of this metric.Also,they study geodesics on the second order tangent bundle T^(2)M and bi-unit second order tangent bundle T^(2)_(1,1)M,and characterize a geodesic of the bi-unit second order tangent bundle in terms of geodesic curvatures of its projection to the base.Finally,they present some conditions for a sectionσ:M→T^(2)M to be harmonic and study the harmonicity of the different canonical projections and inclusions of(T^(2)M,g_(BS)).Moreover,they search the harmonicity of the Berger type deformed Sasaki metric g_(BS)and the Sasaki metric g_(S) with respect to each other.展开更多
文摘Let （M, g） be an n-dimensional Riemannian manifold and T＊M be its cotan-gent bundle equipped with the rescaled Sasaki type metric. In this paper, we firstly study the paraholomorphy property of the rescaled Sasaki type metric by using some compati-ble paracomplex structures on T＊M. Second, we construct locally decomposable Golden Riemannian structures on T＊M . Finally we investigate curvature properties of T＊M .
文摘In this note we give a geometrical presentation to the 4D Riemannian curvature as it relates to the Newtonian gravity in the 4D Lorentz manifold. The compacting of the proper time as is necessary for the unification with the Maxwell electrodynamics, as given by Einstein and Kaluza-Klein, should the universe be only of 4D space-time, led to the concept of gravitational field singularity sinks known as black holes, that would not be acceptable under a 5D homogeneous manifold through which the 4D Lorentz manifold evolved by application of the Perelman-Ricci Flow entropy mapping, which is consistent with both Maxwell suggested magnetic monopole, the quantum Higgs vacuum theory and the Gell-Mann standard model for hadrons.
基金The first author was supported in part by NSF (10241004) of ChinaNational Innovation Fund 1770900+2 种基金 Chinese Academy of Sciencesthe second author was supported in part by NSF grants CCR 9732306KDI-DMS-9873326.
文摘In this paper, we provide simple and explicit formulas for computing Riemannian curvatures, mean curvature vectors, principal curvatures and principal directions for a 2-dimensional Riemannian manifold embedded in IRk with k > 3.
文摘This paper aims to study the Berger type deformed Sasaki metric g_(BS)on the second order tangent bundle T^(2)M over a bi-Kählerian manifold M.The authors firstly find the Levi-Civita connection of the Berger type deformed Sasaki metric g_(BS)and calculate all forms of Riemannian curvature tensors of this metric.Also,they study geodesics on the second order tangent bundle T^(2)M and bi-unit second order tangent bundle T^(2)_(1,1)M,and characterize a geodesic of the bi-unit second order tangent bundle in terms of geodesic curvatures of its projection to the base.Finally,they present some conditions for a sectionσ:M→T^(2)M to be harmonic and study the harmonicity of the different canonical projections and inclusions of(T^(2)M,g_(BS)).Moreover,they search the harmonicity of the Berger type deformed Sasaki metric g_(BS)and the Sasaki metric g_(S) with respect to each other.
