Geodesics Closed On The Projective Plane

Given a C ^∞ Riemannian metric g on P ² we prove that (, g) has constant curvature iff all geodesics are closed. Therefore is the first non-trivial example of a manifold such that the smooth Riemannian metrics which involve that all geodesics are closed are unique up to isometries and scaling. This remarkable phenomenon is not true on the 2-sphere, since there is a large set of C ^∞ metrics whose geodesics are all closed and have the same period 2π (called Zoll metrics), but no metric of this set can be obtained from another metric of this set via an isometry and scaling. As a corollary we conclude that all two-dimensional P-manifolds are SC-manifolds. Received: April 2007; Revision: September 2007; Accepted: September 2007     

Isometries, Geodesics and Jacobi Fields of Lorentzian Heisenberg Group

The paper is mainly devoted to determine the groups of isometries of the Heisenberg group endowed with each of the three left invariant Lorentzian metrics which are possible on it; also, an explicit computation of all the isometries for the (two) non flat Lorentzian metrics is done. Moreover, explicit formulas for the geodesic curves and the Jacobi vector fields for each of these three Lorentzian metrics are computed.     

Geodesics on the Group of Semi-Affine Transformations of the Euclidean Plane

We obtain parameterized representations of geodesics of a sub-Riemannian metric on the three-dimensional Lie group of semi-affine transformations of the Euclidean plane, i.e., those that act as orientation preserving affine mappings on one axis, and as translations on the other one.     

On Manifolds Admitting Continuous Foliations by Geodesics

We show that complete, simply connected Riemannian manifolds admitting continuous foliations by geodesics with integrable orthogonal distributions are homeomorphic to products F×R. Moreover, the geodesics in the foliation are global minimizers.     

Homogeneous Sub-Riemannian Geodesics on a Group of Motions of the Plane

Differential Equations - Homogeneous sub-Riemannian geodesics are described for the standard sub-Riemannian structure on the group $${mathrm {SE}}(2)$$ of proper motions of the plane. It is shown...     

The Geodesics of a Sub-Riemannian Metric on the Group of Semiaffine Transformations of the Euclidean Plane

Siberian Mathematical Journal - We obtain the parametrized representations of the geodesics of a left-invariant sub-Riemannian metric on the group of semiaffine transformations of the Euclidean...     

On the Existence of Homogeneous Geodesics in Homogeneous Riemannian Manifolds

It is proved that every homogeneous Riemannian manifold admits a geodesic which is an orbit of a one-parameter group of isometries.     

几种特殊曲面上的测地线

具体刻画了柱面、锥面、旋转曲面上测地线的几何特征,所得结果一方面匡正了某些文献关于锥面上测地线的错误断言,一方面推广了现有文献关于旋转曲面上测地线几何性质的描述.     

Generalized Symplectic Action and Symplectomorphism Groups of Coadjoint Orbits

Let M be a quantizable symplectic manifold. If ψ_t is a loop in the group {Ham}(M) of Hamiltonian symplectomorphisms of M and A is a 2k-cycle in M, we define a symplectic action κ_A(ψ)∊ U(1) around ψ_t(A), which is invariant under deformations of ψ, and such that κ_A(ψ) depends only on the homology class of A. Using properties of κ_A( ) we determine a lower bound for ♯π₁(Ham(O)), where O is a quantizable coadjoint orbit of a compact Lie group. In particular we prove that ♯π₁(Ham(CPⁿ)) ≥ n+1. Mathematics Subject Classifications (2000): 53D05, 57S05, 57R17, 57T20.     

On the Existence of Focal Points near Closed Geodesics on Surfaces

We establish some criteria for the existence or nonexistence of focal points near closed geodesics on surfaces. These criteria are in terms of the curvature of the manifold along the closed geodesic and the average values of the partial derivatives of the curvature in the direction perpendicular to the geodesic. Our criteria lead to a new family of examples of surfaces with no focal points. We also show that if S is a compact surface with no focal points and an inequality relating the curvature of the surface to the curvature of the horocycles holds, then the horocycles (considered as curves in S) are uniformly C ^2+Lipschitz.     

Geodesics in minimal surfaces

Abstract—In this paper, we consider connected minimal surfaces in R³ with isothermal coordinates and with a family of geodesic coordinates curves, these surfaces will be called GICM-surfaces. We give a classification of the GICM-surfaces. This class of minimal surfaces includes the catenoid, the helicoid and Enneper’s surface. Also, we show that one family of this class of minimal surfaces has at least one closed geodesic and one 1-periodic family of this class has finite total curvature. As application we show other characterization of catenoid and helicoid. Finally, we show that the class of GICM-surfaces coincides with the class of minimal surfaces whose the geodesic curvature k _g ¹ and k _g ² of the coordinates curves satisfy αk _g ¹ + βk _g ² = 0, α, β ∈ R.     

双曲测地线与拟圆

设D是R2中的有界Jordan域,证明了D是拟圆当且仅当存在常数M≥1,对D中任意 两条不相交的闭双曲测地线段α1,α2,恒有mod(△(α1,α2;R2))≤Mmod(△(α1,α2;D)).     

Geodesics in Heisenberg Groups

We obtain equations of geodesic lines in Heisenberg groups H²ⁿ⁺¹and prove that the ideal boundary of the Heisenberg group H²ⁿ⁺¹is a sphere S^2n-1with a natural CR-structure and corresponding Carnot-Carathéodory metric, i.e. it is a one-point compactification of the Heisenberg group H^2n-1of the next dimension in a row.     

The Hunter-Saxton System and the Geodesics on a Pseudosphere

We show that the two-component Hunter-Saxton system with negative coupling constant describes the geodesic flow on an infinite-dimensional pseudosphere. This approach yields explicit solution formulae for the Hunter-Saxton system. Using this geometric intuition, we conclude by constructing global weak solutions. The main novelty compared with similar previous studies is that the metric is indefinite.     

Closed Geodesics on Incomplete Surfaces

We consider the problem of finding embedded closed geodesics on the two-sphere with an incomplete metric defined outside a point. Various techniques including curve shortening methods are used The authors acknowledge the support of the Australian Research Council