The aim of the present paper is to investigate intrinsically the notion of a concircular π-vector field in Finsler geometry. This generalizes the concept of a concircular vector field in Riemannian geometry and the c...The aim of the present paper is to investigate intrinsically the notion of a concircular π-vector field in Finsler geometry. This generalizes the concept of a concircular vector field in Riemannian geometry and the concept of concurrent vector field in Finsler geometry. Some properties of concircular π-vector fields are obtained. Different types of recurrence are discussed. The effect of the existence of a concircular π-vector field on some important special Finsler spaces is investigated. Almost all results obtained in this work are formulated in a coordinate-free form.展开更多
When a closed Finsler manifold admits continuous isometric actions,estimating the number of orbits of prime closed geodesics seems a more reasonable substitution for estimating the number of prime closed geodesics.To ...When a closed Finsler manifold admits continuous isometric actions,estimating the number of orbits of prime closed geodesics seems a more reasonable substitution for estimating the number of prime closed geodesics.To extend the results of Duan,Long,Rademacher,Wang and others on the existence of two prime closed geodesics to the equivariant situation,we propose the question if a closed Finsler manifold has only one orbit of prime closed geodesics if and only if it is a compact rank-one Riemannian symmetric space.In this paper,we study this problem in homogeneous Finsler geometry,and get a positive answer when the dimension is even or the metric is reversible.We guess the rank inequality and the algebraic techniques in this paper may continue to play an important role for discussing our question in the non-homogeneous situation.展开更多
In this paper, we study isoparametric hypersurfaces in Finsler space forms by investigating focal points, tubes and parallel hypersurfaces of submanifolds. We prove that the focal submanifolds of isoparametric hypersu...In this paper, we study isoparametric hypersurfaces in Finsler space forms by investigating focal points, tubes and parallel hypersurfaces of submanifolds. We prove that the focal submanifolds of isoparametric hypersurfaces are anisotropic-minimal and obtain a general Cartan-type formula in a Finsler space form with vanishing reversible torsion, from which we give some classifications on the number of distinct principal curvatures or their multiplicities.展开更多
Kinematics in Finsler space is used to study the propagation of ultra high energy cosmic rays particles through the cosmic microwave background radiation. We find that the GZK threshold is lifted dramatically in Rande...Kinematics in Finsler space is used to study the propagation of ultra high energy cosmic rays particles through the cosmic microwave background radiation. We find that the GZK threshold is lifted dramatically in Randers-Finsler space. A tiny deformation of spacetime from Minkowskian to Finslerian allows more ultra-high energy cosmic rays particles to arrive at the earth. It is suggested that the lower bound of particle mass is related with the negative second invariant speed in Randers-Finsler space.展开更多
The geometric characterization and structure of Finsler manifolds with constant flag curvature (CFC) are studied. It is proved that a Finsler space has constant flag curvature 1 (resp. 0) if and only if the Ricci curv...The geometric characterization and structure of Finsler manifolds with constant flag curvature (CFC) are studied. It is proved that a Finsler space has constant flag curvature 1 (resp. 0) if and only if the Ricci curvature along the Hilbert form on the projective sphere bundle attains identically its maximum (resp. Ricci scalar). The horizontal distribution H of this bundle is integrable if and only if M has zero flag curvature. When a Finsler space has CFC, Hilbert form’s orthogonal complement in the horizontal distribution is also integrable. Moreover, the minimality of its foliations is equivalent to given Finsler space being Riemannian, and its first normal space is vertical.展开更多
Finsler geometry is a natural and fundamental generalization of Riemann geometry. The Finsler structure depends on both coordinates and velocities. It is defined as a function on tangent bundle of a manifold. We use t...Finsler geometry is a natural and fundamental generalization of Riemann geometry. The Finsler structure depends on both coordinates and velocities. It is defined as a function on tangent bundle of a manifold. We use the Bianchi identities satisfied by the Chern curvature to set up a gravitation theory in Berwald-Finsler space. The geometric part of the gravitational field equation is nonsymmetric in general. This indicates that the local Lorentz invariance is violated.展开更多
A theory invoking concepts from differential geometry of generalized Finsler space in conjunction with diffuse interface modeling is described and implemented in finite element(FE)simulations of dual-phase polycrystal...A theory invoking concepts from differential geometry of generalized Finsler space in conjunction with diffuse interface modeling is described and implemented in finite element(FE)simulations of dual-phase polycrystalline ceramic microstructures.Order parameters accounting for fracture and other structural transformations,notably partial dislocation slip,twinning,or phase changes,are dimensionless entries of an internal state vector of generalized pseudo-Finsler space.Ceramics investigated in computations are a boron carbide-titanium diboride(B4C-TiB2)composite and a diamond-silicon carbide(C-SiC)composite.Deformation mechanisms-in addition to elasticity and cleavage fracture in grains of any phase-include restricted dislocation glide(TiB2 phase),deformation twinning(B4C and-SiC phases),and stress-induced amorphization(B4C phase).The metric tensor of generalized Finsler space is scaled conformally according to dilatation induced by cavitation or other fracture modes and densification induced by phase changes.Simulations of pure shear consider various morphologies and lattice orientations.Effects of microstructure on overall strength of each composite are reported.In B4C-TiB2,minor improvements in shear strength and ductility are observed with an increase in the second phase from 10 to 18%by volume,suggesting that residual stresses or larger-scale crack inhibition may be responsible for toughness gains reported experimentally.In diamond-SiC,a composite consisting of diamond crystals encapsulated in a nano-crystalline SiC matrix shows improved strength and ductility relative to a two-phase composite with isolated bulk SiC grains.展开更多
The geometry of Teichmuller metric in an asymptotic Teichmuller space is studled in this article. First, a binary infinitesimal form of Teichmuller metric on AT(X) is proved. Then, the notion of angles between two g...The geometry of Teichmuller metric in an asymptotic Teichmuller space is studled in this article. First, a binary infinitesimal form of Teichmuller metric on AT(X) is proved. Then, the notion of angles between two geodesic curves in the asymptotic Teichmuller space AT(X) is introduced. The existence of such angles is proved and the explicit formula is obtained. As an application, a sufficient condition for non-uniqueness geodesics in AT(X) is obtained.展开更多
文摘The aim of the present paper is to investigate intrinsically the notion of a concircular π-vector field in Finsler geometry. This generalizes the concept of a concircular vector field in Riemannian geometry and the concept of concurrent vector field in Finsler geometry. Some properties of concircular π-vector fields are obtained. Different types of recurrence are discussed. The effect of the existence of a concircular π-vector field on some important special Finsler spaces is investigated. Almost all results obtained in this work are formulated in a coordinate-free form.
基金supported by National Natural Science Foundation of China(Grant Nos.11821101 and 11771331)Beijing Natural Science Foundation(Grant No.1182006)。
文摘When a closed Finsler manifold admits continuous isometric actions,estimating the number of orbits of prime closed geodesics seems a more reasonable substitution for estimating the number of prime closed geodesics.To extend the results of Duan,Long,Rademacher,Wang and others on the existence of two prime closed geodesics to the equivariant situation,we propose the question if a closed Finsler manifold has only one orbit of prime closed geodesics if and only if it is a compact rank-one Riemannian symmetric space.In this paper,we study this problem in homogeneous Finsler geometry,and get a positive answer when the dimension is even or the metric is reversible.We guess the rank inequality and the algebraic techniques in this paper may continue to play an important role for discussing our question in the non-homogeneous situation.
基金supported by National Natural Science Foundation of China (Grant Nos. 11971253 and 11471246)AnHui Natural Science Foundation (Grant No. 1608085MA03)。
文摘In this paper, we study isoparametric hypersurfaces in Finsler space forms by investigating focal points, tubes and parallel hypersurfaces of submanifolds. We prove that the focal submanifolds of isoparametric hypersurfaces are anisotropic-minimal and obtain a general Cartan-type formula in a Finsler space form with vanishing reversible torsion, from which we give some classifications on the number of distinct principal curvatures or their multiplicities.
文摘Kinematics in Finsler space is used to study the propagation of ultra high energy cosmic rays particles through the cosmic microwave background radiation. We find that the GZK threshold is lifted dramatically in Randers-Finsler space. A tiny deformation of spacetime from Minkowskian to Finslerian allows more ultra-high energy cosmic rays particles to arrive at the earth. It is suggested that the lower bound of particle mass is related with the negative second invariant speed in Randers-Finsler space.
文摘The geometric characterization and structure of Finsler manifolds with constant flag curvature (CFC) are studied. It is proved that a Finsler space has constant flag curvature 1 (resp. 0) if and only if the Ricci curvature along the Hilbert form on the projective sphere bundle attains identically its maximum (resp. Ricci scalar). The horizontal distribution H of this bundle is integrable if and only if M has zero flag curvature. When a Finsler space has CFC, Hilbert form’s orthogonal complement in the horizontal distribution is also integrable. Moreover, the minimality of its foliations is equivalent to given Finsler space being Riemannian, and its first normal space is vertical.
文摘Finsler geometry is a natural and fundamental generalization of Riemann geometry. The Finsler structure depends on both coordinates and velocities. It is defined as a function on tangent bundle of a manifold. We use the Bianchi identities satisfied by the Chern curvature to set up a gravitation theory in Berwald-Finsler space. The geometric part of the gravitational field equation is nonsymmetric in general. This indicates that the local Lorentz invariance is violated.
文摘A theory invoking concepts from differential geometry of generalized Finsler space in conjunction with diffuse interface modeling is described and implemented in finite element(FE)simulations of dual-phase polycrystalline ceramic microstructures.Order parameters accounting for fracture and other structural transformations,notably partial dislocation slip,twinning,or phase changes,are dimensionless entries of an internal state vector of generalized pseudo-Finsler space.Ceramics investigated in computations are a boron carbide-titanium diboride(B4C-TiB2)composite and a diamond-silicon carbide(C-SiC)composite.Deformation mechanisms-in addition to elasticity and cleavage fracture in grains of any phase-include restricted dislocation glide(TiB2 phase),deformation twinning(B4C and-SiC phases),and stress-induced amorphization(B4C phase).The metric tensor of generalized Finsler space is scaled conformally according to dilatation induced by cavitation or other fracture modes and densification induced by phase changes.Simulations of pure shear consider various morphologies and lattice orientations.Effects of microstructure on overall strength of each composite are reported.In B4C-TiB2,minor improvements in shear strength and ductility are observed with an increase in the second phase from 10 to 18%by volume,suggesting that residual stresses or larger-scale crack inhibition may be responsible for toughness gains reported experimentally.In diamond-SiC,a composite consisting of diamond crystals encapsulated in a nano-crystalline SiC matrix shows improved strength and ductility relative to a two-phase composite with isolated bulk SiC grains.
基金supported by National Natural Science Foundation of China(11371045,11301248)
文摘The geometry of Teichmuller metric in an asymptotic Teichmuller space is studled in this article. First, a binary infinitesimal form of Teichmuller metric on AT(X) is proved. Then, the notion of angles between two geodesic curves in the asymptotic Teichmuller space AT(X) is introduced. The existence of such angles is proved and the explicit formula is obtained. As an application, a sufficient condition for non-uniqueness geodesics in AT(X) is obtained.