On the complete integrability of the geodesic flow of manifolds all of whose geodesics are closed

1997 ◽  
Vol 17 (6) ◽  
pp. 1359-1370
Author(s):  
CARLOS E. DURÁN
Keyword(s):  
Finite Group ◽  
Symmetric Space ◽  
Geodesic Flow ◽  
Complete Integrability ◽  
Integrals Of Motion ◽  
Completely Integrable ◽  
Rank One ◽  
A Finite Group

We show that the geodesic flow of a metric all of whose geodesics are closed is completely integrable, with tame integrals of motion. Applications to classical examples are given; in particular, it is shown that the geodesic flow of any quotient $M/\Gamma$ of a compact, rank one symmetric space $M$ by a finite group acting freely by isometries is completely integrable by tame integrals.

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.

On a conjecture of Green

1997 ◽  
Vol 17 (1) ◽  
pp. 247-252
Author(s):  
CHENGBO YUE
Keyword(s):  
Riemannian Manifold ◽  
Symmetric Space ◽  
Sectional Curvature ◽  
Geodesic Flow ◽  
Rigidity Theorem ◽  
Locally Symmetric Space ◽  
Closed Riemannian Manifold ◽  
Rank One ◽  
The Mean ◽  
Negative Sectional Curvature

Green [5] conjectured that if $M$ is a closed Riemannian manifold of negative sectional curvature such that the mean curvatures of the horospheres through each point depend only on the point, then $V$ is a locally symmetric space of rank one. He proved this in dimension two. In this paper we prove that under Green's assumption, $M$ must be asymptotically harmonic and that the geodesic flow on $M$ is $C^{\infty}$ conjugate to that of a locally symmetric space of rank one. Combining this with the recent rigidity theorem of Besson–Courtois–Gallot [1], it follows that Green's conjecture is true for all dimensions.

scholarly journals TOPOLOGICAL STABLE RANK OF INCLUSIONS OF UNITAL C*-ALGEBRAS

2006 ◽  
Vol 17 (01) ◽  
pp. 19-34 ◽  
Author(s):  
HIROYUKI OSAKA ◽  
TAMOTSU TERUYA
Keyword(s):  
Finite Group ◽  
Tensor Product ◽  
Finite Type ◽  
Stable Rank ◽  
C Algebras ◽  
Rank One ◽  
Topological Stable Rank ◽  
Crossed Product Algebra ◽  
A Finite Group ◽  
Product Algebra

Let 1 ∈ A ⊂ B be an inclusion of C*-algebras of C*-index-finite type with depth 2. We try to compute the topological stable rank of B (= tsr (B)) when A has topological stable rank one. We show that tsr (B) ≤ 2 when A is a tsr boundedly divisible algebra, in particular, A is a C*-minimal tensor product UHF ⊗ D with tsr (D) = 1. When G is a finite group and α is an action of G on UHF, we know that a crossed product algebra UHF ⋊α G has topological stable rank less than or equal to two. These results are affirmative data to a generalization of a question by Blackadar in 1988.

scholarly journals Adapted complex structures and geometric quantization

1999 ◽  
Vol 154 ◽  
pp. 171-183 ◽  
Author(s):  
Róbert Szőke
Keyword(s):  
Symmetric Space ◽  
Tangent Bundle ◽  
Geodesic Flow ◽  
Symmetric Spaces ◽  
Complex Structure ◽  
Geometric Quantization ◽  
Riemannian Symmetric Space ◽  
Complex Structures ◽  
Rank One ◽  
Push Forward

AbstractA compact Riemannian symmetric space admits a canonical complexification. This so called adapted complex manifold structure JA is defined on the tangent bundle. For compact rank-one symmetric spaces another complex structure JS is defined on the punctured tangent bundle. This latter is used to quantize the geodesic flow for such manifolds. We show that the limit of the push forward of JA under an appropriate family of diffeomorphisms exists and agrees with JS.

scholarly journals Complete integrability of geodesic motion in Sasaki–Einstein toric Yp,q spaces

2015 ◽  
Vol 30 (33) ◽  
pp. 1550180 ◽  
Author(s):  
Elena Mirela Babalic ◽  
Mihai Visinescu
Keyword(s):  
Geodesic Flow ◽  
Einstein Manifold ◽  
Complete Integrability ◽  
Geodesic Motion ◽  
Integrals Of Motion ◽  
Killing Vectors ◽  
Einstein Spaces ◽  
Constants Of Motion ◽  
Complete Set

We construct explicitly the constants of motion for geodesics in the five-dimensional Sasaki–Einstein spaces [Formula: see text]. To carry out this task, we use the knowledge of the complete set of Killing vectors and Killing–Yano tensors on these spaces. In spite of the fact that we generate a multitude of constants of motion, only five of them are functionally independent implying the complete integrability of geodesic flow on [Formula: see text] spaces. In the particular case of the homogeneous Sasaki–Einstein manifold [Formula: see text] the integrals of motion have simpler forms and the relations between them are described in detail.

scholarly journals An Anosov action on the bundle of Weyl chambers

1985 ◽  
Vol 5 (4) ◽  
pp. 587-593 ◽  
Author(s):  
Hans-Christoph Im Hof
Keyword(s):  
Symmetric Space ◽  
Geodesic Flow ◽  
Riemannian Symmetric Space ◽  
Compact Type ◽  
Rank One

AbstractWe introduce an Anosov action on the bundle of Weyl chambers of a riemannian symmetric space of non-compact type, which for rank one spaces coincides with the geodesic flow.

scholarly journals Integrability of the geodesic flow on the resolved conifolds over Sasaki–Einstein space T1,1

2018 ◽  
Vol 33 (19) ◽  
pp. 1850107
Author(s):  
Mihai Visinescu
Keyword(s):  
Geodesic Flow ◽  
Hamiltonian Dynamics ◽  
Complete Integrability ◽  
Einstein Space ◽  
Geodesic Motion ◽  
Completely Integrable ◽  
Constants Of Motion ◽  
Small Resolution

Methods of Hamiltonian dynamics are applied to study the geodesic flow on the resolved conifolds (rcs) over Sasaki–Einstein space [Formula: see text]. We construct explicitly the constants of motion and prove complete integrability of geodesics in the five-dimensional Sasaki–Einstein space [Formula: see text] and its Calabi–Yau metric cone. The singularity at the apex of the metric cone can be smoothed out in two different ways. Using the small resolution, the geodesic motion on the rc remains completely integrable. Instead, in the case of the deformation of the conifold, the complete integrability is lost.

Hypoelliptic Laplacian and Orbital Integrals (AM-177)

2011 ◽  
Author(s):  
Jean-Michel Bismut
Keyword(s):  
Fixed Point ◽  
Symmetric Space ◽  
Heat Kernel ◽  
Geodesic Flow ◽  
Casimir Operator ◽  
Index Theory ◽  
Weighted Sum ◽  
Orbital Integrals ◽  
Hypoelliptic Heat Kernel ◽  
Wave Kernel

This book uses the hypoelliptic Laplacian to evaluate semisimple orbital integrals in a formalism that unifies index theory and the trace formula. The hypoelliptic Laplacian is a family of operators that is supposed to interpolate between the ordinary Laplacian and the geodesic flow. It is essentially the weighted sum of a harmonic oscillator along the fiber of the tangent bundle, and of the generator of the geodesic flow. In this book, semisimple orbital integrals associated with the heat kernel of the Casimir operator are shown to be invariant under a suitable hypoelliptic deformation, which is constructed using the Dirac operator of Kostant. Their explicit evaluation is obtained by localization on geodesics in the symmetric space, in a formula closely related to the Atiyah-Bott fixed point formulas. Orbital integrals associated with the wave kernel are also computed. Estimates on the hypoelliptic heat kernel play a key role in the proofs, and are obtained by combining analytic, geometric, and probabilistic techniques. Analytic techniques emphasize the wavelike aspects of the hypoelliptic heat kernel, while geometrical considerations are needed to obtain proper control of the hypoelliptic heat kernel, especially in the localization process near the geodesics. Probabilistic techniques are especially relevant, because underlying the hypoelliptic deformation is a deformation of dynamical systems on the symmetric space, which interpolates between Brownian motion and the geodesic flow. The Malliavin calculus is used at critical stages of the proof.

ON HIGHER FROBENIUS–SCHUR INDICATORS

2021 ◽  
pp. 1-11
Author(s):  
YANJUN LIU ◽  
WOLFGANG WILLEMS
Keyword(s):  
Finite Group ◽  
Representation Theory ◽  
Irreducible Characters ◽  
A Finite Group

Abstract Similarly to the Frobenius–Schur indicator of irreducible characters, we consider higher Frobenius–Schur indicators $\nu _{p^n}(\chi ) = |G|^{-1} \sum _{g \in G} \chi (g^{p^n})$ for primes p and $n \in \mathbb {N}$ , where G is a finite group and $\chi $ is a generalised character of G. These invariants give answers to interesting questions in representation theory. In particular, we give several characterisations of groups via higher Frobenius–Schur indicators.

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