共查询到20条相似文献,搜索用时 171 毫秒
1.
Gautier Berck 《Mathematische Annalen》2009,343(4):955-973
Using the symplectic definition of the Holmes-Thompson volume we prove that totally geodesic submanifolds of a Finsler manifold
are minimal for this volume. Thanks to well suited technics the minimality of totally geodesic hypersurfaces (see álvarez
Paiva and Berck in Adv Math 204(2):647–663, 2006) and 2-dimensional totally geodesic surfaces (see álvarez Paiva and Berck
in Adv Math 204(2):647–663, 2006, Ivanov in Algebra i Analiz 13(1)26–38, 2001) had already been proved. However the corresponding
statement for the Hausdorff measure is known to be wrong even in the simplest case of totally geodesic 2-dimensional surfaces
in a 3-dimensional Finsler manifold (see álvarez Paiva and Berck in Adv Math 204(2):647–663, 2006). 相似文献
2.
In [T. Kimura, M.S. Tanaka, Totally geodesic submanifold in compact symmetric spaces of rank two, Tokyo J. Math., in press], the authors obtained the global classification of the maximal totally geodesic submanifolds in compact connected irreducible symmetric spaces of rank two. In this paper, we determine their stability as minimal submanifolds in compact symmetric spaces of rank two. 相似文献
3.
We study Finsler PL spaces, that is simplicial complexes glued out of simplices cut off from some normed spaces. We are interested in the class of Finsler PL spaces featuring local uniqueness of geodesics (for complexes made of Euclidean simplices, this property is equivalent to local CAT(0)). Though non-Euclidean normed spaces never satisfy CAT(0), it turns out that they share many common features. In particular, a globalization theorem holds: in a simply-connected Finsler PL space local uniqueness of geodesics implies the global one. However the situation is more delicate here: some basic convexity properties do not extend to the PL Finsler case. 相似文献
4.
We prove that a four-dimensional generalized symmetric space does not admit any non-degenerate hypersurfaces with parallel second fundamental form, in particular non-degenerate totally geodesic hypersurfaces, unless it is locally symmetric. However, spaces which are known as generalized symmetric spaces of type C do admit non-degenerate parallel hypersurfaces and we verify that they are indeed symmetric. We also give a complete and explicit classification of all non-degenerate totally geodesic hypersurfaces of spaces of this type. 相似文献
5.
《Mathematische Nachrichten》2017,290(2-3):474-481
A geodesic in a homogeneous Finsler space is called a homogeneous geodesic if it is an orbit of a one‐parameter subgroup of G . A homogeneous Finsler space is called Finsler g.o. space if its all geodesics are homogeneous. Recently, the author studied Finsler g.o. spaces and generalized some geometric results on Riemannian g.o. spaces to the Finslerian setting. In the present paper, we investigate homogeneous geodesics in homogeneous spaces, and obtain the sufficient and necessary condition for an space to be a g.o. space. As an application, we get a series of new examples of Finsler g.o. spaces. 相似文献
6.
Here, it is shown that every vector field on a Finsler space which keeps geodesic circles invariant is conformal. A necessary and sufficient condition for a vector field to keep geodesic circles invariant, known as concircular vector fields, is obtained. This leads to a significant definition of concircular vector fields on a Finsler space. Finally, complete Finsler spaces admitting a special conformal vector field are classified. 相似文献
7.
《Comptes Rendus de l'Academie des Sciences Series IIA Earth and Planetary Science》1997,324(10):1127-1132
É. Cartan introduced in 1926 the Riemannian locally symmetric spaces, as the spaces whose curvature tensor is parallel. They also owe their name to the fact that, for each point, the geodesic reflexion is a local isometry. The aim of this Note is to announce a strong rigidity result for Finsler spaces. Namely, we show that a negatively curved locally symmetric (in the first sense above) Finsler space is isometric to a Riemann locally symmetric space. 相似文献
8.
This paper is devoted to a study on closed geodesics on Finsler and Riemannian spheres. We call a prime closed geodesic on a Finsler manifold rational, if the basic normal form decomposition (cf. [Y. Long, Bott formula of the Maslov-type index theory, Pacific J. Math. 187 (1999) 113-149]) of its linearized Poincaré map contains no 2×2 rotation matrix with rotation angle which is an irrational multiple of π, or irrational otherwise. We prove that if there exists only one prime closed geodesic on a d-dimensional irreversible Finsler sphere with d?2, it cannot be rational. Then we further prove that there exist always at least two distinct prime closed geodesics on every irreversible Finsler 3-dimensional sphere. Our method yields also at least two geometrically distinct closed geodesics on every reversible Finsler as well as Riemannian 3-dimensional sphere. We prove also such results hold for all compact simply connected 3-dimensional manifolds with irreversible or reversible Finsler as well as Riemannian metrics. 相似文献
9.
We show that certain mechanical systems, including a geodesic flow in any dimension plus a quasi-periodic perturbation by a potential, have orbits of unbounded energy.The assumptions we make in the case of geodesic flows are:
- (a)
- The metric and the external perturbation are smooth enough.
- (b)
- The geodesic flow has a hyperbolic periodic orbit such that its stable and unstable manifolds have a tranverse homoclinic intersection.
- (c)
- The frequency of the external perturbation is Diophantine.
- (d)
- The external potential satisfies a generic condition depending on the periodic orbit considered in (b).
10.
Barbora Batíková 《Applied Categorical Structures》2007,15(5-6):483-491
The category of all Hausdorff complete t-semi-uniform spaces is shown to be epireflective in the category of all Hausdorff
t-semi-uniform spaces but the reflection arrows need not be embeddings since there is no nontrivial epireflective subcategory
of the category of all Hausdorff t-semi-uniform spaces in which all reflection arrows are embeddings (t-semi-uniform spaces
are those semi-uniform spaces inducing a topology). On the other hand for every t-semi-uniform space X there exist a minimal and a maximal completion containing X as a dense subspace. The second one is an almost reflection in complete spaces, i.e., every uniformly continuous mapping
on X to a complete semi-uniform space can be extended (as a uniformly continuous map) onto the completion.
相似文献
11.
Sebastian Klein 《Differential Geometry and its Applications》2008,26(1):79-96
In this article, relations between the root space decomposition of a Riemannian symmetric space of compact type and the root space decompositions of its totally geodesic submanifolds (symmetric subspaces) are described. These relations provide an approach to the classification of totally geodesic submanifolds in Riemannian symmetric spaces; this is exemplified by the classification of the totally geodesic submanifolds in the complex quadric Qm:=SO(m+2)/(SO(2)×SO(m)) obtained in the second part of the article. The classification shows that the earlier classification of totally geodesic submanifolds of Qm by Chen and Nagano (see [B.-Y. Chen, T. Nagano, Totally geodesic submanifolds of symmetric spaces, I, Duke Math. J. 44 (1977) 745-755]) is incomplete. More specifically, two types of totally geodesic submanifolds of Qm are missing from [B.-Y. Chen, T. Nagano, Totally geodesic submanifolds of symmetric spaces, I, Duke Math. J. 44 (1977) 745-755]: The first type is constituted by manifolds isometric to CP1×RP1; their existence follows from the fact that Q2 is (via the Segre embedding) holomorphically isometric to CP1×CP1. The second type consists of 2-spheres of radius which are neither complex nor totally real in Qm. 相似文献
12.
Jun Ichi Fujii 《Linear algebra and its applications》2011,434(2):542-558
As a generalization of the Hiai-Petz geometries, we discuss two types of them where the geodesics are the quasi-arithmetic means and the quasi-geometric means respectively. Each derivative of such a geodesic might determine a new relative operator entropy. Also in these cases, the Finsler metric can be induced by each unitarily invariant norm. If the norm is strictly convex, then the geodesic is the shortest. We also give examples of the shortest paths which are not the geodesics when the Finsler metrics are induced by the Ky Fan k-norms. 相似文献
13.
Bucataru Ioan Milkovszki Tamás Muzsnay Zoltán 《Mediterranean Journal of Mathematics》2016,13(6):4567-4580
In this paper, we study the invariant metrizability and projective metrizability problems for the special case of the geodesic spray associated to the canonical connection of a Lie group. We prove that such canonical spray is projectively Finsler metrizable if and only if it is Riemann metrizable. This result means that this structure is rigid in the sense that considering left invariant metrics, the potentially much larger class of projective Finsler metrizable canonical sprays, corresponding to Lie groups, coincides with the class of Riemann metrizable canonical sprays. Generalisation of these results for geodesic orbit spaces are given.
相似文献14.
The distance function \({\varrho(p, q) ({\rm or} d(p, q))}\) of a distance space (general metric space) is not differentiable in general. We investigate such distance spaces over \({\mathbb{R}^n}\), whose distance functions are differentiable like in case of Finsler spaces. These spaces have several good properties, yet they are not Finsler spaces (which are special distance spaces). They are situated between general metric spaces (distance spaces) and Finsler spaces. We will investigate such curves of differentiable distance spaces, which possess the same properties as geodesics do in Finsler spaces. So these curves can be considered as forerunners of Finsler geodesics. They are in greater plenitude than Finsler geodesics, but they become geodesics in a Finsler space. We show some properties of these curves, as well as some relations between differentiable distance spaces and Finsler spaces. We arrive to these curves and to our results by using distance spheres, and using no variational calculus. We often apply direct geometric considerations. 相似文献
15.
In this paper two metric properties on geodesic length spaces are introduced by means of the metric projection, studying their validity on Alexandrov and Busemann NPC spaces. In particular, we prove that both properties characterize the non-positivity of the sectional curvature on Riemannian manifolds. Further results are also established on reversible/non-reversible Finsler–Minkowski spaces. 相似文献
16.
Hiroshi Tamaru 《Mathematische Annalen》2011,351(1):51-66
In this paper, we study the solvmanifolds constructed from any parabolic subalgebras of any semisimple Lie algebras. These
solvmanifolds are naturally homogeneous submanifolds of symmetric spaces of noncompact type. We show that the Ricci curvatures
of our solvmanifolds coincide with the restrictions of the Ricci curvatures of the ambient symmetric spaces. Consequently,
all of our solvmanifolds are Einstein, which provide a large number of new examples of noncompact homogeneous Einstein manifolds.
We also show that our solvmanifolds are minimal, but not totally geodesic submanifolds of symmetric spaces. 相似文献
17.
We construct spectral triples and, in particular, Dirac operators, for the algebra of continuous functions on certain compact metric spaces. The triples are countable sums of triples where each summand is based on a curve in the space. Several fractals, like a finitely summable infinite tree and the Sierpinski gasket, fit naturally within our framework. In these cases, we show that our spectral triples do describe the geodesic distance and the Minkowski dimension as well as, more generally, the complex fractal dimensions of the space. Furthermore, in the case of the Sierpinski gasket, the associated Dixmier-type trace coincides with the normalized Hausdorff measure of dimension log3/log2. 相似文献
18.
Chikako Mese 《manuscripta mathematica》1999,100(3):375-389
In this paper, we study the structure of locally compact metric spaces of Hausdorff dimension 2. If such a space has non-positive
curvautre and a local cone structure, then every simple closed curve bounds a conformal disk. On a surface (a topological
manifold of dimension 2), a distance function with non-positive curvature and whose metric topology is equivalent to the surface
topology gives a structure of a Riemann surface. The construction of conformal disks in these spaces uses minimal surface
theory; in particular, the solution of the Plateau Problem in metric spaces of non-positive curvature.
Received: 18 November 1997/ Revised versions: 15 January and 7 June 1999 相似文献
19.
S. V. Ivanov 《Proceedings of the Steklov Institute of Mathematics》2011,273(1):176-190
We prove that every Riemannian metric on the 2-disc such that all its geodesics are minimal is a minimal filling of its boundary
(within the class of fillings homeomorphic to the disc). This improves an earlier result of the author by removing the assumption
that the boundary is convex. More generally, we prove this result for Finsler metrics with area defined as the two-dimensional
Holmes-Thompson volume. This implies a generalization of Pu’s isosystolic inequality to Finsler metrics, both for the Holmes-Thompson
and Busemann definitions of the Finsler area. 相似文献
20.
《Nonlinear Analysis: Real World Applications》2007,8(4):1132-1143
In [G. Munteanu, Complex Spaces in Finsler, Lagrange and Hamilton Geometries, vol. 141, Kluwer Academic Publishers, Dordrecht, FTPH, 2004.] we underlined the motifs of a remarkable class of complex Finsler subspaces, namely the holomorphic subspaces. With respect to the Chern–Finsler complex connection (see [M. Abate, G. Patrizio, Finsler Metrics—A Global Approach, Lecture Notes in Mathematics, vol. 1591, Springer, Berlin, 1994.]) we studied in [G. Munteanu, The equations of a holomorphic subspace in a complex Finsler space, Publicationes Math. Debrecen, submitted for publication.] the Gauss, Codazzi and Ricci equations of a holomorphic subspace, the aim being to determine the interrelation between the holomorphic sectional curvature of the Chern–Finsler connection and that of its induced tangent connection.In the present paper, by means of the complex Berwald connection, we study totally geodesic holomorphic subspaces. With respect to complex Berwald connection the equations of the holomorphic subspace have simplified expressions. The totally geodesic subspace request is characterized by using the second fundamental form of complex Berwald connection. 相似文献