Some properties related to the NUT-Taub-like spacetime, such as the surface of infinite red-shift, horizon, singularity and the area of the NUT-Taub-like black hole are discussed. Furthermore, the geodesics in the NUT...Some properties related to the NUT-Taub-like spacetime, such as the surface of infinite red-shift, horizon, singularity and the area of the NUT-Taub-like black hole are discussed. Furthermore, the geodesics in the NUT-Taub-like spacetime are obtained in some special cases. Specifically, the circular orbits for a massive particle are derived, which can reduce to the cases of the Schwarzschild spacetime and the NUT-Taub spacetime when m^＊ = 0 and m^＊ 〈〈 M, respectively.展开更多
We study circular time-like geodesics in the spacetime of a black hole including global monopole. We show that when the range of parameter changed the properties of the circular geodesics and the radiation of accretio...We study circular time-like geodesics in the spacetime of a black hole including global monopole. We show that when the range of parameter changed the properties of the circular geodesics and the radiation of accretion disks are different. It follows that the properties of the accretion disk around black hole including global monopole can be different from that of a disk around Schwarzschild black hole.展开更多
Using Weierstrassian elliptic functions the exact geodesics in the Schwarzschild metric are expressed in a simple and most transparent form. The results are useful for analytical and numerical applications. For exampl...Using Weierstrassian elliptic functions the exact geodesics in the Schwarzschild metric are expressed in a simple and most transparent form. The results are useful for analytical and numerical applications. For example we calculate the perihelion precession and the light deflection in the post-Einsteinian approximation. The bounded orbits are computed in the post-Newtonian order. As a topical application we calculate the gravitational red shift for a star moving in the Schwarzschild field.展开更多
We review the concept of congruence of null geodesics, the Raychaudhuri equation for the expansion, its harmonic oscillator version and associated “quantum” propagator, the role of the equation in the derivation of ...We review the concept of congruence of null geodesics, the Raychaudhuri equation for the expansion, its harmonic oscillator version and associated “quantum” propagator, the role of the equation in the derivation of the Penrose singularity theorem, the definition of trapped surfaces, and the derivation of the theorem itself.展开更多
In this paper, we get necessary and sufficient conditions for a Finsler space en- dowed with an （α,β）-metric where its geodesic coefficients G^i （x, y） and the reverse of geodesic coefficients G^i（x,-y） have t...In this paper, we get necessary and sufficient conditions for a Finsler space en- dowed with an （α,β）-metric where its geodesic coefficients G^i （x, y） and the reverse of geodesic coefficients G^i（x,-y） have the same Douglas curvature. They are the conditions such that （α,β）-metrics have reversible geodesics.展开更多
We study the null bulk geodesic motion in the brane world in which the bulk metric has an un-stabilized extra spatial dimension. We find that the null bulk geodesic motion as observed on the 3-brane with Z<SUB>2...We study the null bulk geodesic motion in the brane world in which the bulk metric has an un-stabilized extra spatial dimension. We find that the null bulk geodesic motion as observed on the 3-brane with Z<SUB>2</SUB> symmetry would be a timelike geodesic motion even though there exists an extra non-gravitational force in contrast with the case of the stabilized extra spatial dimension. In other words the presence of the extra non-gravitational force would not violate the Z<SUB>2</SUB> symmetry.展开更多
In this paper we address the implementation issue of the geodesics method with constraints on Heisenberg manifolds. First we present more details on the method in order to facilitate its implementation and second we c...In this paper we address the implementation issue of the geodesics method with constraints on Heisenberg manifolds. First we present more details on the method in order to facilitate its implementation and second we consider Mathema-tica as a software tool for the simulation. This implementation is of great importance since it allows easy and direct determination of Ricci tensor, which plays a fundamental role in the Heisenberg manifold metric.展开更多
Let ind(c) be the Morse index of a closed geodesic c in an(n+1)-dimensional Riemannian manifold M.We prove that an oriented closed geodesic c is unstable if n + ind(c) is odd and a non-oriented closed geodesic c is un...Let ind(c) be the Morse index of a closed geodesic c in an(n+1)-dimensional Riemannian manifold M.We prove that an oriented closed geodesic c is unstable if n + ind(c) is odd and a non-oriented closed geodesic c is unstable if n + ind(c) is even.Our result is a generalization of the famous theorem due to Poincar'e which states that the closed minimizing geodesic on a Riemann surface is unstable.展开更多
Abstract The non-uniqueness on geodesics and geodesic disks in the universal asymptotic Teichmfiller space AT（D） are studied in this paper. It is proved that if # is asymptotically extremal in [[#]] with h （μ） ...Abstract The non-uniqueness on geodesics and geodesic disks in the universal asymptotic Teichmfiller space AT（D） are studied in this paper. It is proved that if # is asymptotically extremal in [[#]] with h （μ） 〈 h＊ （μ） for some point ζ∈D, then there exist infinitely many geodesic segments joining [[0]] and [[μ]], and infinitely many holomorphic geodesic disks containing [[0]] and [μ]] in AT（D）.展开更多
Let B denote the set of those points p in an infinite dimensional Teichmiiller space such that a geodesic between the zero point and p is not unique. Some properties of the set B are given.
Let E be the Engel group and D be a bracket generating left invariant distribution with a Lorentzian metric, which is a nondegenerate metric of index 1. In this paper, the author constructs a parametrization of a quas...Let E be the Engel group and D be a bracket generating left invariant distribution with a Lorentzian metric, which is a nondegenerate metric of index 1. In this paper, the author constructs a parametrization of a quasi-pendulum equation by Jacobi functions, and then gets the space-like Hamiltonian geodesics in the Engel group with a sub-Lorentzian metric.展开更多
In this paper, the authors construct a class of unitary invariant strongly pseudoconvex complex Finsler metrics which are of the form F =√[ rf(s- t)[, where r = ||v||~ 2, s =| z,v |~2/r, t =|| z||~ 2, f(w) is a real-...In this paper, the authors construct a class of unitary invariant strongly pseudoconvex complex Finsler metrics which are of the form F =√[ rf(s- t)[, where r = ||v||~ 2, s =| z,v |~2/r, t =|| z||~ 2, f(w) is a real-valued smooth positive function of w ∈ R,and z is in a unitary invariant domain M C^n. Complex Finsler metrics of this form are unitary invariant. We prove that F is a class of weakly complex Berwald metrics whose holomorphic curvature and Ricci scalar curvature vanish identically and are independent of the choice of the function f. Under initial value conditions on f and its derivative f, we prove that all the real geodesics of F =√[rf(s- t)] on every Euclidean sphere S^(2n-1) M are great circles.展开更多
If all prime closed geodesics on(S^n, F) with an irreversible Finsler metric F are irrationally elliptic,there exist either exactly 2 [(n+1)/2] or infinitely many distinct closed geodesics. As an application, we show ...If all prime closed geodesics on(S^n, F) with an irreversible Finsler metric F are irrationally elliptic,there exist either exactly 2 [(n+1)/2] or infinitely many distinct closed geodesics. As an application, we show the existence of three distinct closed geodesics on bumpy Finsler(S^3, F) if any prime closed geodesic has non-zero Morse index.展开更多
Let(RP^(n),F)be a bumpy and irreversible Finsler n-dimensional real projective space with reversibilityλand flag curvature K satisfying(λ/1+λ)^(2)<K≤1 when n is odd,and K≥0 when n is even.We show that if there...Let(RP^(n),F)be a bumpy and irreversible Finsler n-dimensional real projective space with reversibilityλand flag curvature K satisfying(λ/1+λ)^(2)<K≤1 when n is odd,and K≥0 when n is even.We show that if there exist exactly 2[n+1/2]prime closed geodesics on such(RP^(n),F),then all of them are non-contractible,which coincides with the Katok’s examples.展开更多
Let M=S^(n)/Γand h be a nontrivial element of finite order p inπ_(1)(Μ),where the integers n,p≥2,Γis a finite abelian group which acts freely and isometrically on the n-sphere and therefore M is diffeomorphic to ...Let M=S^(n)/Γand h be a nontrivial element of finite order p inπ_(1)(Μ),where the integers n,p≥2,Γis a finite abelian group which acts freely and isometrically on the n-sphere and therefore M is diffeomorphic to a compact space form.In this paper,we prove that there are infinitely many non-contractible closed geodesics of class[h]on the compact space form with C^(r)-generic Finsler metrics,where 4≤r≤∞.The conclusion also holds for Cr-generic Riemannian metrics for 2≤r≤∞.The proof is based on the resonance identity of non-contractible closed geodesics on compact space forms.展开更多
The quasinormal mode frequencies can be understood from the massless particles trapped at the unstable circular null geodesics and slowly leaking out to infinity. Based on this viewpoint, in this paper, we semiclassic...The quasinormal mode frequencies can be understood from the massless particles trapped at the unstable circular null geodesics and slowly leaking out to infinity. Based on this viewpoint, in this paper, we semiclassically construct the entropy spectrum of the static and stationary black holes from the null geodesics. The result shows that the spacing of the entropy spectrum only depends on the property of the black hole in the eikonal limit. Moreover, for a black hole far from the extremal case, the spacing is found to be smaller than 2π for any dimension, which is very different from the result of the previous work by using the usual quasinormal mode frequencies.展开更多
3D face similarity is a critical issue in computer vision, computer graphics and face recognition and so on. Since Fr@chet distance is an effective metric for measuring curve similarity, a novel 3D face similarity mea...3D face similarity is a critical issue in computer vision, computer graphics and face recognition and so on. Since Fr@chet distance is an effective metric for measuring curve similarity, a novel 3D face similarity measure method based on Fr^chet distances of geodesics is proposed in this paper. In our method, the surface similarity between two 3D faces is measured by the similarity between two sets of 3D curves on them. Due to the intrinsic property of geodesics, we select geodesics as the comparison curves. Firstly, the geodesics on each 3D facial model emanating from the nose tip point are extracted in the same initial direction with equal angular increment. Secondly, the Fr@chet distances between the two sets of geodesics on the two compared facial models are computed. At last, the similarity between the two facial models is computed based on the Fr6chet distances of the geodesics obtained in the second step. We verify our method both theoretically and practically. In theory, we prove that the similarity of our method satisfies three properties： reflexivity, symmetry, and triangle inequality. And in practice, experiments are conducted on the open 3D face database GavaDB, Texas 3D Face Recognition database, and our 3D face database. After the comparison with iso-geodesic and Hausdorff distance method, the results illustrate that our method has good discrimination ability and can not only identify the facial models of the same person, but also distinguish the facial models of any two different persons.展开更多
In this paper, the relationship between the existence of closed geodesics and the volume growth of complete noncompact Riemannian manifolds is studied. First the authors prove a diffeomorphic result of such an n-m2nif...In this paper, the relationship between the existence of closed geodesics and the volume growth of complete noncompact Riemannian manifolds is studied. First the authors prove a diffeomorphic result of such an n-m2nifold with nonnegative sectional curvature, which improves Marenich-Toponogov＇s theorem. As an application, a rigidity theorem is obtained for nonnegatively curved open manifold which contains a clesed geodesic. Next the authors prove a theorem about the nonexistence of closed geodesics for Riemannian manifolds with sectional curvature bounded from below by a negative constant.展开更多
基金Project supported by the National Natural Science Foundation of China （Grant No 10475036）, the Natural Science Foundation of Liaoning Province, China （Grant No 20032012） and the Scientific Research Foundation of the Higher Education Institute of Liaoning Province, China （Grant No 05L215）.
文摘Some properties related to the NUT-Taub-like spacetime, such as the surface of infinite red-shift, horizon, singularity and the area of the NUT-Taub-like black hole are discussed. Furthermore, the geodesics in the NUT-Taub-like spacetime are obtained in some special cases. Specifically, the circular orbits for a massive particle are derived, which can reduce to the cases of the Schwarzschild spacetime and the NUT-Taub spacetime when m^＊ = 0 and m^＊ 〈〈 M, respectively.
基金supported by the National Natural Science Foundation of China(Grant No.10873004)the State Key Development Program for Basic Research Program of China(Grant No.2010CB832803)the Program for Changjiang Scholars and Innovative Research Team in University(Grant No.IRT0964)
文摘We study circular time-like geodesics in the spacetime of a black hole including global monopole. We show that when the range of parameter changed the properties of the circular geodesics and the radiation of accretion disks are different. It follows that the properties of the accretion disk around black hole including global monopole can be different from that of a disk around Schwarzschild black hole.
文摘Using Weierstrassian elliptic functions the exact geodesics in the Schwarzschild metric are expressed in a simple and most transparent form. The results are useful for analytical and numerical applications. For example we calculate the perihelion precession and the light deflection in the post-Einsteinian approximation. The bounded orbits are computed in the post-Newtonian order. As a topical application we calculate the gravitational red shift for a star moving in the Schwarzschild field.
文摘We review the concept of congruence of null geodesics, the Raychaudhuri equation for the expansion, its harmonic oscillator version and associated “quantum” propagator, the role of the equation in the derivation of the Penrose singularity theorem, the definition of trapped surfaces, and the derivation of the theorem itself.
基金Partially supported by NSF (10801079)Partially supported by RFDP (20080551002)+1 种基金Partially supported by LPMC of MOE of ChinaPartially supported by the 973 Program of MOST, NNSF, MCME, RFDP, LPMC of MOE of China, S. S. Chern Foundation, and Nankai University
文摘In this paper, the concavity of closed geodesics proposed by M. Morse in 1930s is studied.
基金Supported by the National Natural Science Foundation of China(Grant No.11471246)the Jiangxi Provincial Science and Technology Project(Grant No.20161BAB211021)
文摘In this paper, we get necessary and sufficient conditions for a Finsler space en- dowed with an （α,β）-metric where its geodesic coefficients G^i （x, y） and the reverse of geodesic coefficients G^i（x,-y） have the same Douglas curvature. They are the conditions such that （α,β）-metrics have reversible geodesics.
基金The project supported by National Natural Science Foundation of China under Grant No.10175070National Basic Research Project of China under Grant No.2003CB716300
文摘We study the null bulk geodesic motion in the brane world in which the bulk metric has an un-stabilized extra spatial dimension. We find that the null bulk geodesic motion as observed on the 3-brane with Z<SUB>2</SUB> symmetry would be a timelike geodesic motion even though there exists an extra non-gravitational force in contrast with the case of the stabilized extra spatial dimension. In other words the presence of the extra non-gravitational force would not violate the Z<SUB>2</SUB> symmetry.
文摘In this paper we address the implementation issue of the geodesics method with constraints on Heisenberg manifolds. First we present more details on the method in order to facilitate its implementation and second we consider Mathema-tica as a software tool for the simulation. This implementation is of great importance since it allows easy and direct determination of Ricci tensor, which plays a fundamental role in the Heisenberg manifold metric.
基金supported by National Natural Science Foundation of China (Grant Nos.10801127,10731080)
文摘Let ind(c) be the Morse index of a closed geodesic c in an(n+1)-dimensional Riemannian manifold M.We prove that an oriented closed geodesic c is unstable if n + ind(c) is odd and a non-oriented closed geodesic c is unstable if n + ind(c) is even.Our result is a generalization of the famous theorem due to Poincar'e which states that the closed minimizing geodesic on a Riemann surface is unstable.
基金Supported by National Natural Science Foundation of China(Grant No.11371045)
文摘Abstract The non-uniqueness on geodesics and geodesic disks in the universal asymptotic Teichmfiller space AT（D） are studied in this paper. It is proved that if # is asymptotically extremal in [[#]] with h （μ） 〈 h＊ （μ） for some point ζ∈D, then there exist infinitely many geodesic segments joining [[0]] and [[μ]], and infinitely many holomorphic geodesic disks containing [[0]] and [μ]] in AT（D）.
文摘Let B denote the set of those points p in an infinite dimensional Teichmiiller space such that a geodesic between the zero point and p is not unique. Some properties of the set B are given.
基金supported by the Science and Technology Development Fund of Nanjing Medical University(No.2017NJMU005).
文摘Let E be the Engel group and D be a bracket generating left invariant distribution with a Lorentzian metric, which is a nondegenerate metric of index 1. In this paper, the author constructs a parametrization of a quasi-pendulum equation by Jacobi functions, and then gets the space-like Hamiltonian geodesics in the Engel group with a sub-Lorentzian metric.
基金supported by the National Natural Science Foundation of China(Nos.11271304,11171277)the Program for New Century Excellent Talents in University(No.NCET-13-0510)+1 种基金the Fujian Province Natural Science Funds for Distinguished Young Scholars(No.2013J06001)the Scientific Research Foundation for the Returned Overseas Chinese Scholars,State Education Ministry
文摘In this paper, the authors construct a class of unitary invariant strongly pseudoconvex complex Finsler metrics which are of the form F =√[ rf(s- t)[, where r = ||v||~ 2, s =| z,v |~2/r, t =|| z||~ 2, f(w) is a real-valued smooth positive function of w ∈ R,and z is in a unitary invariant domain M C^n. Complex Finsler metrics of this form are unitary invariant. We prove that F is a class of weakly complex Berwald metrics whose holomorphic curvature and Ricci scalar curvature vanish identically and are independent of the choice of the function f. Under initial value conditions on f and its derivative f, we prove that all the real geodesics of F =√[rf(s- t)] on every Euclidean sphere S^(2n-1) M are great circles.
基金National Natural Science Foundation of China (Grant Nos. 11131004, 11471169 and 11401555)the Key Laboratory of Pure Mathematics and Combinatorics of Ministry of Education of China and Nankai University, China Postdoctoral Science Foundation (Grant No. 2014T70589)Chinese Universities Scientific Fund (Grant No. WK0010000037)
文摘If all prime closed geodesics on(S^n, F) with an irreversible Finsler metric F are irrationally elliptic,there exist either exactly 2 [(n+1)/2] or infinitely many distinct closed geodesics. As an application, we show the existence of three distinct closed geodesics on bumpy Finsler(S^3, F) if any prime closed geodesic has non-zero Morse index.
基金The first author was partially supported by National Key R&D Program of China(Grant No.2020YFA0713300)NSFC(Grant Nos.11671215 and 11790271)+1 种基金LPMC of MOE of China and Nankai Universitythe second author was partially supported by NSFC(Grant Nos.11771341 and 12022111)。
文摘Let(RP^(n),F)be a bumpy and irreversible Finsler n-dimensional real projective space with reversibilityλand flag curvature K satisfying(λ/1+λ)^(2)<K≤1 when n is odd,and K≥0 when n is even.We show that if there exist exactly 2[n+1/2]prime closed geodesics on such(RP^(n),F),then all of them are non-contractible,which coincides with the Katok’s examples.
基金supported by NSFC(Grant Nos.12371195,12022111)the Fundamental Research Funds for the Central Universities(Grant No.2042023kf0207)+1 种基金the second author was partially supported by NSFC(Grant No.11831009)Fundings of Innovating Activities in Science and Technology of Hubei Province。
文摘Let M=S^(n)/Γand h be a nontrivial element of finite order p inπ_(1)(Μ),where the integers n,p≥2,Γis a finite abelian group which acts freely and isometrically on the n-sphere and therefore M is diffeomorphic to a compact space form.In this paper,we prove that there are infinitely many non-contractible closed geodesics of class[h]on the compact space form with C^(r)-generic Finsler metrics,where 4≤r≤∞.The conclusion also holds for Cr-generic Riemannian metrics for 2≤r≤∞.The proof is based on the resonance identity of non-contractible closed geodesics on compact space forms.
基金supported by the National Natural Science Foundation of China(Grant Nos.1120507411375075 and 11522541)the Fundamental Research Funds for the Central Universities(Grant No.lzujbky-2015-jl01)
文摘The quasinormal mode frequencies can be understood from the massless particles trapped at the unstable circular null geodesics and slowly leaking out to infinity. Based on this viewpoint, in this paper, we semiclassically construct the entropy spectrum of the static and stationary black holes from the null geodesics. The result shows that the spacing of the entropy spectrum only depends on the property of the black hole in the eikonal limit. Moreover, for a black hole far from the extremal case, the spacing is found to be smaller than 2π for any dimension, which is very different from the result of the previous work by using the usual quasinormal mode frequencies.
基金This work was supported by the National Natural Science Foundation of China under Grant Nos. 61702293, 61772294, and 61572078, the Open Research Fund of the Ministry of Education Engineering Research Center of Virtual Reality Application of China under Grant No. MEOBNUEVRA201601. It was also partially supported by the National High Technology Research and Development 863 Program of China under Grant No. 2015AA020506, and the National Science and Technology Pillar Program during the 12th Five-Year Plan Period of China under Grant No. 2013BAI01B03.
文摘3D face similarity is a critical issue in computer vision, computer graphics and face recognition and so on. Since Fr@chet distance is an effective metric for measuring curve similarity, a novel 3D face similarity measure method based on Fr^chet distances of geodesics is proposed in this paper. In our method, the surface similarity between two 3D faces is measured by the similarity between two sets of 3D curves on them. Due to the intrinsic property of geodesics, we select geodesics as the comparison curves. Firstly, the geodesics on each 3D facial model emanating from the nose tip point are extracted in the same initial direction with equal angular increment. Secondly, the Fr@chet distances between the two sets of geodesics on the two compared facial models are computed. At last, the similarity between the two facial models is computed based on the Fr6chet distances of the geodesics obtained in the second step. We verify our method both theoretically and practically. In theory, we prove that the similarity of our method satisfies three properties： reflexivity, symmetry, and triangle inequality. And in practice, experiments are conducted on the open 3D face database GavaDB, Texas 3D Face Recognition database, and our 3D face database. After the comparison with iso-geodesic and Hausdorff distance method, the results illustrate that our method has good discrimination ability and can not only identify the facial models of the same person, but also distinguish the facial models of any two different persons.
基金Project supported by the National Natural Science Foundation of China(Nos.10971055,11171096)the Research Fund for the Doctoral Program of Higher Education of China(No.20104208110002)the Funds for Disciplines Leaders of Wuhan(No.Z201051730002)
文摘In this paper, the relationship between the existence of closed geodesics and the volume growth of complete noncompact Riemannian manifolds is studied. First the authors prove a diffeomorphic result of such an n-m2nifold with nonnegative sectional curvature, which improves Marenich-Toponogov＇s theorem. As an application, a rigidity theorem is obtained for nonnegatively curved open manifold which contains a clesed geodesic. Next the authors prove a theorem about the nonexistence of closed geodesics for Riemannian manifolds with sectional curvature bounded from below by a negative constant.
