Special Symmetric Two-tensors, Equivalent Dynamical Systems, Cofactor and Bi-cofactor Systems

A general analysis of special classes of symmetric two-tensor on Riemannian manifolds is provided. These tensors arise in connection with special topics in differential geometry and analytical mechanics: geodesic equivalence and separation of variables. It is shown that they play an important role in the theory of correspondent (or equivalent) dynamical systems of Levi-Civita. By applying some new developments of this theory, it is shown that the recent notions of cofactor and cofactor-pair systems arise in a natural way, as non-Lagrangian systems having a Lagrangian equivalent. This circumstance extends the Hamiltonian methods, including the separation of variables of the Hamilton–Jacobi equation, to a special class of nonconservative systems. In this extension the case of indefinite metrics, may occur. Hence, it is shown that also pseudo-Riemannian geometry plays an important role also in classical mechanics.Research sponsored by the Dept. of Mathematics, University of Turin, and by INDAM-GNFM.     

Letter: Solar Gravitational Deflection of a Graviton

The solar gravitational deflection angle of a graviton is calculated through the scattering cross section of the graviton by the sun and shown to be equal to the light deflection angle as calculated from the null geodesic equation of general relativity.     

可变焦列阵柱面透镜均匀线聚焦系统

用两个列阵柱面透镜与非球面透镜组成变焦线聚焦系统，能将入射激光束会聚成辐照均匀、长度连续可调的焦线，它可用于Ｘ光激光实验研究．文中详细分析了焦线上光场分布的情况，给出了物理光学分析的结果，并与巳有技术作了比较．     

The Patterson–Sullivan Embedding and Minimal Volume Entropy for Outer Space

Motivated by Bonahon’s result for hyperbolic surfaces, we construct an analogue of the Patterson–Sullivan–Bowen–Margulis map from the Culler–Vogtmann outer space CV (F _k ) into the space of projectivized geodesic currents on a free group. We prove that this map is a continuous embedding and thus obtain a new compactification of the outer space. We also prove that for every k ≥ 2 the minimum of the volume entropy of the universal covers of finite connected volume-one metric graphs with fundamental group of rank k and without degree-one vertices is equal to (3k − 3) log 2 and that this minimum is realized by trivalent graphs with all edges of equal lengths, and only by such graphs. Received: December 2005, Accepted: March 2006     

The Geodesics of Metric Connections with Vectorial Torsion

The present note deals with the dynamics of metric connections with vectorial torsion, as already described by E. Cartan in 1925. We show that the geodesics of metric connections with vectorial torsion defined by gradient vector fields coincide with the Levi-Civita geodesics of a conformally equivalent metric. By pullback, this yields a systematic way of constructing invariants of motion for such connections from isometries of the conformally equivalent metric, and we explain in as much this result generalizes the Mercator projection which maps sphere loxodromes to straight lines in the plane. An example shows that Beltrami's theorem fails for this class of connections. We then study the system of differential equations describing geodesics in the plane for vector fields which are not gradients, and show among others that the Hopf–Rinow theorem does also not hold in general.     

Affine surfaces in R 4 with planar geodesics with respect to the affine metric

It is proved that non-degenerate surfaces in R ⁴ with planar geodesics are only the complex paraboloid and the product of two parabolas. An erratum to this article is available at .     

带自由边界调和映射的Liouville型定理

本文给出一类带由边界的调和映射的Liouville型定理,这种类型的定理在微分几何的一些问题中有十分重要的应用.我们通过对调和映射的能量选取特殊的变分族,得到任意从半空间的简单流形到一黎曼流形的带自由边界的调和映射在如果满足适当的条件(见定理)必为常值映射的结果.     

A global rigidity theorem for weakly Landsberg manifolds

In this paper we study a global rigidity property for weakly Landsberg manifolds and prove that a closed weakly Landsberg manifold with the negative flag curvature must be Riemannian.     

Chernoff's Theorem and Discrete Time Approximations of Brownian Motion on Manifolds

Let be a one-parameter family of positive integral operators on a locally compact space . For a possibly non-uniform partition of define a finite measure on the path space by using a) for the transition between any two consecutive partition times of distance and b) a suitable continuous interpolation scheme (e.g. Brownian bridges or geodesics). If necessary normalize the result to get a probability measure. We prove a version of Chernoff's theorem of semigroup theory and tightness results which yield convergence in law of such measures as the partition gets finer. In particular let be a closed smooth submanifold of a manifold . We prove convergence of Brownian motion on , conditioned to visit at all partition times, to a process on whose law has a density with respect to Brownian motion on which contains scalar, mean and sectional curvatures terms. Various approximation schemes for Brownian motion on are also given.      

Geodesic precession in the (Ω,A ab)-field theory

This paper aims to determine the geodesic precession in Yu’s (Ω, A_ab)-field theory[1], and to compare the result with that of the Schwarzschild orbit.