Periodic solutions of Hilbert’s fourth problem

2021 ◽  
pp. 1-20
Author(s):  
J. C. Álvarez Paiva ◽  
J. Barbosa Gomes
Keyword(s):  
Euclidean Space ◽  
Symmetric Space ◽  
Periodic Solutions ◽  
Space Form ◽  
Finsler Metric ◽  
Riemannian Symmetric Space ◽  
Closed Formula ◽  
Flat Metric ◽  
Straight Lines

It is shown that a possibly irreversible [Formula: see text] Finsler metric on the torus, or on any other compact Euclidean space form, whose geodesics are straight lines is the sum of a flat metric and a closed [Formula: see text]-form. This is used to prove that if [Formula: see text] is a compact Riemannian symmetric space of rank greater than one and [Formula: see text] is a reversible [Formula: see text] Finsler metric on [Formula: see text] whose unparametrized geodesics coincide with those of [Formula: see text], then [Formula: see text] is a Finsler symmetric space.

scholarly journals Distribution of length spectrum of circles on a complex hyperbolic space

1999 ◽  
Vol 153 ◽  
pp. 119-140 ◽  
Author(s):  
Toshiaki Adachi
Keyword(s):  
Symmetric Space ◽  
Hyperbolic Space ◽  
Real Line ◽  
Isometry Group ◽  
Space Form ◽  
Geodesic Curvature ◽  
Real Space ◽  
Complex Hyperbolic Space ◽  
Riemannian Symmetric Space ◽  
Rank One

AbstractIt is well-known that all geodesics on a Riemannian symmetric space of rank one are congruent each other under the action of isometry group. Being concerned with circles, we also know that two closed circles in a real space form are congruent if and only if they have the same length. In this paper we study how prime periods of circles on a complex hyperbolic space are distributed on a real line and show that even if two circles have the same length and the same geodesic curvature they are not necessarily congruent each other.

scholarly journals On the least positive eigenvalue of the Laplacian for the compact quotient of a certain Riemannian symmetric space

1980 ◽  
Vol 78 ◽  
pp. 137-152 ◽  
Author(s):  
Hajime Urakawa
Keyword(s):  
Fixed Point ◽  
Euclidean Space ◽  
Symmetric Space ◽  
Lie Group ◽  
Discrete Subgroup ◽  
Riemannian Symmetric Space ◽  
Compact Type ◽  
Identity Component ◽  
Rank One ◽  
Quotient Manifold

Let (, g) be the standard Euclidean space or a Riemannian symmetric space of non-compact type of rank one. Let G be the identity component of the Lie group of all isometries of (, g). Let Γ be a discrete subgroup of G acting fixed point freely on whose quotient manifold MΓ is compact.

scholarly journals Minimal Maslov number of R-spaces canonically embedded in Einstein-Kähler C-spaces

2019 ◽  
Vol 6 (1) ◽  
pp. 303-319
Author(s):  
Yoshihiro Ohnita
Keyword(s):  
Symmetric Space ◽  
Floer Homology ◽  
Lagrangian Submanifolds ◽  
Symplectic Manifolds ◽  
Lagrangian Submanifold ◽  
Riemannian Symmetric Space ◽  
Canonical Embedding ◽  
Real Form ◽  
Isotropy Representation ◽  
Totally Geodesic

AbstractAn R-space is a compact homogeneous space obtained as an orbit of the isotropy representation of a Riemannian symmetric space. It is known that each R-space has the canonical embedding into a Kähler C-space as a real form, and thus a compact embedded totally geodesic Lagrangian submanifold. The minimal Maslov number of Lagrangian submanifolds in symplectic manifolds is one of invariants under Hamiltonian isotopies and very fundamental to study the Floer homology for intersections of Lagrangian submanifolds. In this paper we show a Lie theoretic formula for the minimal Maslov number of R-spaces canonically embedded in Einstein-Kähler C-spaces, and provide some examples of the calculation by the formula.

The topological-Euclidean space form problem

1978 ◽  
Vol 45 (2) ◽  
pp. 181-192 ◽  
Author(s):  
F. T. Farrell ◽  
W. C. Hsiang
Keyword(s):  
Euclidean Space ◽  
Space Form ◽  
Form Problem

CLASSES MINIMALES DE RÉSEAUX ET RÉTRACTIONS GÉOMÉTRIQUES ÉQUIVARIANTES DANS LES ESPACES SYMÉTRIQUES

2001 ◽  
Vol 64 (2) ◽  
pp. 275-286 ◽  
Author(s):  
CHRISTOPHE BAVARD
Keyword(s):  
Finite Group ◽  
Euclidean Space ◽  
Symmetric Space ◽  
Group Action ◽  
Symmetric Spaces ◽  
Classification Problem ◽  
Finite Group Action ◽  
Natural Geometry ◽  
A Finite Group

Equivariant and cocompact retractions of certain symmetric spaces are constructed. These retractions are defined using the natural geometry of symmetric spaces and in relation to the theory of lattices of euclidean space. The following cases are considered: the symmetric space corresponding to lattices endowed with a finite group action, from which is obtained some information relating to the classification problem of these lattices, and the Siegel space Sp2g(R)/Ug, for which a natural Sp2g(Z)-equivariant cocompact retract of codimension 1 is obtained.

scholarly journals Vassiliev invariants from symmetric spaces

2016 ◽  
Vol 25 (10) ◽  
pp. 1650055 ◽  
Author(s):  
Indranil Biswas ◽  
Niels Leth Gammelgaard
Keyword(s):  
Symmetric Space ◽  
Lie Algebra ◽  
Curvature Tensor ◽  
Symmetric Spaces ◽  
Weight System ◽  
Riemannian Symmetric Space ◽  
Riemann Curvature ◽  
Riemann Curvature Tensor ◽  
Vassiliev Invariants ◽  
Chord Diagrams

We construct a natural framed weight system on chord diagrams from the curvature tensor of any pseudo-Riemannian symmetric space. These weight systems are of Lie algebra type and realized by the action of the holonomy Lie algebra on a tangent space. Among the Lie algebra weight systems, they are exactly characterized by having the symmetries of the Riemann curvature tensor.

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