The set of spatial rigid body motions forms a Lie group known as the special Euclidean group in three dimensions, (3). Chasles’s theorem states that there exists a screw motion between two arbitrary elements of (3). In this paper we investigate whether there exist a Riemannian metric whose geodesics are screw motions. We prove that no Riemannian metric with such geodesics exists and we show tha...

We use a construction which we call generalized cylinders to give a new proof of the fundamental theorem of hypersurface theory. It has the advantage of being very simple and the result directly extends to semi-Riemannian manifolds and to embeddings into spaces of constant curvature. We also give a new way to identify spinors for different metrics and to derive the variation formula for the Dir...

It is a well-known fact that on a bounded spectral interval the Dirac spectrum can described locally by a non-decreasing sequence of continuous functions of the Riemannian metric. In the present article we extend this result to a global version. We think of the spectrum of a Dirac operator as a function Z → R and endow the space of all spectra with an arsinh-uniform metric. We prove that the sp...

The comparison theory for the Riccati equation satisfied by the shape operator of parallel hypersurfaces is generalized to semi–Riemannian manifolds of arbitrary index, using one–sided bounds on the Riemann tensor which in the Riemannian case correspond to one–sided bounds on the sectional curvatures. Starting from 2–dimensional rigidity results and using an inductive technique, a new class of ...

We study umbilic (space-like) submanifolds of pseudo-Riemannian space forms, then define totally semi-umbilic space-like submanifold of pseudo Euclidean space and relate this notion to umbilicity. Finally we give characterization of total semi-umbilicity for space-like submanifolds contained in pseudo sphere or pseudo hyperbolic space or the light cone.A pseudo-Riemannian submanifold M in (a...

The comparison theory for the Riccati equation satisfied by the shape operator of parallel hypersurfaces is generalized to semi–Riemannian manifolds of arbitrary index, using one–sided bounds on the Riemann tensor which in the Riemannian case correspond to one–sided bounds on the sectional curvatures. Starting from 2–dimensional rigidity results and using an inductive technique, a new class of ...

We initiate the study of X-ray tomography on sub-Riemannian manifolds, for which Heisenberg group exhibits simplest nontrivial example. With language Fourier transform, we prove an operator-valued incarnation Slice Theorem, and apply this new tool to show that a sufficiently regular function is determined by its line integrals over geodesics. also consider family taming metrics g? approximating...

‎The notion of quasi-Einstein metric in physics is equivalent to the notion of Ricci soliton in Riemannian spaces‎. ‎Quasi-Einstein metrics serve also as solution to the Ricci flow equation‎. ‎Here‎, ‎the Riemannian metric is replaced by a Hessian matrix derived from a Finsler structure and a quasi-Einstein Finsler metric is defined‎. ‎In compact case‎, ‎it is proved that the quasi-Einstein met...

In this paper, we study a special class of generalized Douglas-Weyl metrics whose Douglas curvature is constant along any Finslerian geodesic. We prove that for every Landsberg metric in this class of Finsler metrics, ? = 0 if and only if H = 0. Then we show that every Finsler metric of non-zero isotropic flag curvature in this class of metrics is a Riemannian if and only if ? = 0.

the general relatively isotropic mean landsberg metrics contain the general relativelyisotropic landsberg metrics. a class of finsler metrics is given, in which the mentioned two conceptsare equivalent. in this paper, an interpretation of general relatively isotropic mean landsberg metrics isfound by using c-conformal transformations. some necessary conditions for a general relativelyisotropic ...