Symmetric subvarieties in compactifications and the Radon transform on Riemannian symmetric spaces of noncompact type

We investigate the totally geodesic Radon transform which assigns a function to its integration over totally geodesic symmetric submanifolds in Riemannian symmetric spaces of noncompact type. Our main concern is focused on the injectivity and support theorem. Our approach is based on the projection slice theorem relating the totally geodesic Radon transform and the Fourier transforms on symmetric spaces. Our approach also uses the study of geometry concerned with the totally geodesic symmetric subvarieties in Riemannian symmetric spaces in terms of the cell structure of the Satake compactifications.     

纤维丛的全测地子流形

本文证明了底空间Ｍ是纤维丛Ｐ的全测地子流形；并且在ｄｉｍＰ－ｄｉｍＭ＝２时证明了若Ｐ是平坦的，则Ｐ的每一纤维也是全测地子流形．     

On some kinds of fuzzy connected spaces

In this paper we introduce new results in fuzzy connected spaces. Among the results obtained we can mention the good extension of local connectedness. Also we prove that in a T ₁-fuzzy compact space the notions c-zero dimensional, strong c-zero dimensional and totally c_i-disconnected are equivalent.     

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.     

关于CPn中共圆平坦全实极小子流形的一点注记

Let M be a concircularly flat totally real minimal submanifold in CP~4. The infimum V_m of the volume V(M) of M is obtained, also the necessary and sufficient conditions of "V(M)=V_m" is given.     

单层介质膜保偏全反射直角三棱镜镀膜材料的选择及其微分相位特性研究

入射角、基底玻璃折射率、膜介质折射率、膜厚等因素会影响单层介质膜保偏全反射直角三棱镜的保偏特性，且各因素的影响程度并不相同。本文分析了上述各因素的微小变化对保偏特性的影响，并提出了选择膜介质的方法     

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.     

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 .