摘要 :
It is well-known that Vrancken-Li-Simon classified locally strongly convex affine hyperspheres in Rn+1 whose affine metric are of constant sectional curvatures, but on the other side it is still a difficult problem to classify n-d...
展开
It is well-known that Vrancken-Li-Simon classified locally strongly convex affine hyperspheres in Rn+1 whose affine metric are of constant sectional curvatures, but on the other side it is still a difficult problem to classify n-dimensional locally strongly convex affine hyperspheres whose affine metrics are Einstein. In this paper, we have solved the problem in case n = 4. (C) 2016 Elsevier B.V. All rights reserved.
收起
摘要 :
We introduce a new family of affine metrics on a locally strictly convex surface M in affine 4-space. Then, we define the symmetric and antisymmetric equiaffine planes associated with each metric. We show that if M is immersed in ...
展开
We introduce a new family of affine metrics on a locally strictly convex surface M in affine 4-space. Then, we define the symmetric and antisymmetric equiaffine planes associated with each metric. We show that if M is immersed in a locally strictly convex hyperquadric, then the symmetric and the antisymmetric planes coincide and contain the affine normal of the hyperquadric. In particular, any surface immersed in a locally strictly convex hyperquadric is affine semiumbilical with respect to the symmetric or antisymmetric equiaffine planes.
收起
摘要 :
Conditions for the existence of a gyroscope in spaces with affine connections and metrics are found. They appear as special types of Fermi-Walker transports for vector fields, lying in a subspace, orthogonal to the velocity vector...
展开
Conditions for the existence of a gyroscope in spaces with affine connections and metrics are found. They appear as special types of Fermi-Walker transports for vector fields, lying in a subspace, orthogonal to the velocity vector field (a non-null contravariant vector field) of an observer.
收起
摘要 :
Two metrics on a manifold are geodesically equivalent if the sets of their unparameterized geodesics coincide. We show that if two G-invariant metrics of arbitrary signature on homogenous space G/H are geodesically equivalent, the...
展开
Two metrics on a manifold are geodesically equivalent if the sets of their unparameterized geodesics coincide. We show that if two G-invariant metrics of arbitrary signature on homogenous space G/H are geodesically equivalent, they are affinely equivalent, i.e. they have the same Levi-Civita connection. We also prove that the existence of nonproportional, geodesically equivalent, G-invariant metrics on homogenous space G/H implies that their holonomy algebra cannot be full. We give an algorithm for finding all left invariant metrics geodesically equivalent to a given left invariant metric on a Lie group. Using that algorithm we prove that no two left invariant metrics of any signature on sphere S-3 are geodesically equivalent. However, we present examples of Lie groups that admit geodesically equivalent, nonproportional, left-invariant metrics.
收起
摘要 :
We consider locally strictly convex surfaces M in affine 4-space. By using the metric of the transversal vector field on M we introduce a new affine normal plane and the familly of affine distance functions on M. We show that the ...
展开
We consider locally strictly convex surfaces M in affine 4-space. By using the metric of the transversal vector field on M we introduce a new affine normal plane and the familly of affine distance functions on M. We show that the singularities of the family of affine distance functions appear at points on the affine normal plane and the affine focal points correspond to degenerate singularities of this family. Moreover we show that if M is immersed in a locally strictly convex hypersurface, then the affine normal plane contains the affine normal vector to the hypersurface and conclude that any surface immersed in a locally strictly convex hypersphere is affine semiumbilical.
收起
摘要 :
We use a new approach to study locally strongly convex hypersurfaces with constant sectional curvature in the affine space Rn+1. We prove a nice relation involving the eigenvalues of the shape operator S and the difference tensor ...
展开
We use a new approach to study locally strongly convex hypersurfaces with constant sectional curvature in the affine space Rn+1. We prove a nice relation involving the eigenvalues of the shape operator S and the difference tensor K of the affine hypersurface. This is achieved by making full use of the Codazzi equations for both the shape operator and the difference tensor and the Ricci identity in an indirect way. Starting from this relation, we give a classification of locally strongly convex hypersurface with constant sectional curvature whose shape operator S has at most one eigenvalue of multiplicity one.
收起
摘要 :
We review some particular solutions of the vacuum Cartan equation for the non-Riemannian part of the connection in Metric Affine Gravity, by exploiting a variational approach. As application we show how a quite general non Riemann...
展开
We review some particular solutions of the vacuum Cartan equation for the non-Riemannian part of the connection in Metric Affine Gravity, by exploiting a variational approach. As application we show how a quite general non Riemannian model gives a Proca type equation for the trace of the non metricity 1-form Q.
收起
摘要 :
We review some particular solutions of the vacuum Cartan equation for the non-Riemannian part of the connection in Metric Affine Gravity, by exploiting a variational approach. As application we show how a quite general non Riemann...
展开
We review some particular solutions of the vacuum Cartan equation for the non-Riemannian part of the connection in Metric Affine Gravity, by exploiting a variational approach. As application we show how a quite general non Riemannian model gives a Proca type equation for the trace of the non metricity 1-form Q.
收起
摘要 :
We prove that the affine deformation of a Minkowski norm is a Minkowski norm. Some important classical norms are derived by using the affine deformation recipe. Applications are done for some special Finsler manifolds.
摘要 :
The goal of this short paper is to give conditions for the completeness of the Binet–Legendre metric in Finsler geometry. The case of the Funk and Hilbert metrics in a convex domain are discussed.
原文获取