共查询到20条相似文献,搜索用时 31 毫秒
1.
Hirohiko Shima 《Transactions of the American Mathematical Society》1999,351(12):4713-4726
We characterize invariant projectively flat affine connections in terms of affine representations of Lie algebras, and show that a homogeneous space admits an invariant projectively flat affine connection if and only if it has an equivariant centro-affine immersion. We give a correspondence between semi-simple symmetric spaces with invariant projectively flat affine connections and central-simple Jordan algebras.
2.
We show that the well-known fact that the equivariant cohomology (with real coefficients) of a torus action is a torsion-free
module if and only if the map induced by the inclusion of the fixed point set is injective generalises to actions of arbitrary
compact connected Lie groups if one replaces the fixed point set by the set of points with isotropy rank equal to the rank
of the acting group. This is true essentially because the action on this set is always equivariantly formal. In case this
set is empty we show that the induced action on the set of points with highest occuring isotropy rank is Cohen-Macaulay. It
turns out that just as equivariant formality of an action is equivalent to equivariant formality of the action of a maximal
torus, the same holds true for equivariant injectivity and the Cohen-Macaulay property. In addition, we find a topological
criterion for equivariant injectivity in terms of orbit spaces. 相似文献
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5.
Fabio Podestà 《Monatshefte für Mathematik》1996,122(3):215-225
This work deals with positively curved compact Riemannian manifolds which are acted on by a closed Lie group of isometries whose principal orbits have codimension one and are isotropy irreducible homogeneous spaces. For such manifolds we can show that their universal covering manifold may be isometrically immersed as a hypersurface of revolution in an euclidean space. 相似文献
6.
Claudio Gorodski 《Bulletin of the Brazilian Mathematical Society》1996,27(1):1-22
We describe a method to construct embedded, minimal hyperspheres in rank two compact symmetric spaces which are equivariant under the isotropy action of the symmetric space, and we supply the details of the construction for the exceptional Lie groupG
2.Partially supported by CNPq (brazil) 相似文献
7.
We present a new class of high-order variational integrators on Lie groups. We show that these integrators are symplectic and momentum-preserving, can be constructed to be of arbitrarily high order, or can be made to converge geometrically. Furthermore, these methods are capable of taking very large time-steps. We demonstrate the construction of one such variational integrator for the rigid body and discuss how this construction could be generalized to other related Lie group problems. We close with several numerical examples which demonstrate our claims and discuss further extensions of our work. 相似文献
8.
Hamilton–Pontryagin Integrators on Lie Groups Part I: Introduction and Structure-Preserving Properties 总被引:1,自引:0,他引:1
In this paper, structure-preserving time-integrators for rigid body-type mechanical systems are derived from a discrete Hamilton–Pontryagin variational principle. From this principle, one can derive a novel class of variational partitioned Runge–Kutta methods on Lie groups. Included among these integrators are generalizations of symplectic Euler and Störmer–Verlet integrators from flat spaces to Lie groups. Because of their variational design, these integrators preserve a discrete momentum map (in the presence of symmetry) and a symplectic form. In a companion paper, we perform a numerical analysis of these methods and report on numerical experiments on the rigid body and chaotic dynamics of an underwater vehicle. The numerics reveal that these variational integrators possess structure-preserving properties that methods designed to preserve momentum (using the coadjoint action of the Lie group) and energy (for example, by projection) lack. 相似文献
9.
The Clebsch method provides a unifying approach for deriving variational principles for continuous and discrete dynamical
systems where elements of a vector space are used to control dynamics on the cotangent bundle of a Lie group via a velocity
map. This paper proves a reduction theorem which states that the canonical variables on the Lie group can be eliminated, if
and only if the velocity map is a Lie algebra action, thereby producing the Euler–Poincaré (EP) equation for the vector space
variables. In this case, the map from the canonical variables on the Lie group to the vector space is the standard momentum
map defined using the diamond operator. We apply the Clebsch method in examples of the rotating rigid body and the incompressible
Euler equations. Along the way, we explain how singular solutions of the EP equation for the diffeomorphism group (EPDiff)
arise as momentum maps in the Clebsch approach. In the case of finite-dimensional Lie groups, the Clebsch variational principle
is discretized to produce a variational integrator for the dynamical system. We obtain a discrete map from which the variables
on the cotangent bundle of a Lie group may be eliminated to produce a discrete EP equation for elements of the vector space.
We give an integrator for the rotating rigid body as an example. We also briefly discuss how to discretize infinite-dimensional
Clebsch systems, so as to produce conservative numerical methods for fluid dynamics.
相似文献
10.
We show that pseudo-Riemannian almost quaternionic homogeneous spaces with index 4 and an \(\mathbb {H}\)-irreducible isotropy group are locally isometric to a pseudo-Riemannian quaternionic Kähler symmetric space if the dimension is at least 16. In dimension 12 we give a non-symmetric example. 相似文献
11.
Motivated by developments in numerical Lie group integrators, we introduce a family of local coordinates on Lie groups denoted
generalized polar coordinates. Fast algorithms are derived for the computation of the coordinate maps, their tangent maps and the inverse tangent maps.
In particular we discuss algorithms for all the classical matrix Lie groups and optimal complexity integrators for n-spheres. 相似文献
12.
Silvio Reggiani 《Annals of Global Analysis and Geometry》2010,37(4):351-359
A very important class of homogeneous Riemannian manifolds are the so-called normal homogeneous spaces, which have associated
a canonical connection. In this study, we obtain geometrically the (connected component of the) group of affine transformations
with respect to the canonical connection for a normal homogeneous space. The naturally reductive case is also treated. This
completes the geometric calculation of the isometry group of naturally reductive spaces. In addition, we prove that for normal
homogeneous spaces the set of fixed points of the full isotropy is a torus. As an application of our results it follows that
the holonomy group of a homogeneous fibration is contained in the group of (canonically) affine transformations of the fibers;
in particular, this holonomy group is a Lie group (this is a result of Guijarro and Walschap). 相似文献
13.
On the Construction of Geometric Integrators in the RKMK Class 总被引:2,自引:0,他引:2
Kenth Engø 《BIT Numerical Mathematics》2000,40(1):41-61
We consider the construction of geometric integrators in the class of RKMK methods. Any differential equation in the form of an infinitesimal generator on a homogeneous space is shown to be locally equivalent to a differential equation on the Lie algebra corresponding to the Lie group acting on the homogeneous space. This way we obtain a distinction between the coordinate-free phrasing of the differential equation and the local coordinates used. In this paper we study methods based on arbitrary local coordinates on the Lie group manifold. By choosing the coordinates to be canonical coordinates of the first kind we obtain the original method of Munthe-Kaas [16]. Methods similar to the RKMK method are developed based on the different coordinatizations of the Lie group manifold, given by the Cayley transform, diagonal Padé approximants of the exponential map, canonical coordinates of the second kind, etc. Some numerical experiments are also given. 相似文献
14.
Volker Hauschild 《manuscripta mathematica》1980,32(3-4):365-379
Actions of compact Lie groups on the homogeneous spaces G/NT, G a compact connected semisimple Lie group, NTG the normalizor of a maximal torus T in G, are considered. If the acting group is a torus, the action is lifted to the universal covering G/T and the corresponding equivariant cohomology is computed for a coefficient field of characteristic O. The symmetry degree of all homogeneous spaces G/NT is computed confirming a conjecture of W. Y. Hsiang. The nonexistence of fixed points of semisimple compact Lie group actions on G/NT is proved in the case that the group acts differentiably and effectively. 相似文献
15.
Summary We use recent results on symplectic integration of Hamiltonian systems with constraints to construct symplectic integrators on cotangent bundles of manifolds by embedding the manifold in a linear space. We also prove that these methods are equivariant under cotangent lifts of a symmetry group acting linearly on the ambient space and consequently preserve the corresponding momentum. These results provide an elementary construction of symplectic integrators for Lie-Poisson systems and other Hamiltonian systems with symmetry. The methods are illustrated on the free rigid body, the heavy top, and the double spherical pendulum. 相似文献
16.
In this paper we consider transitive actions of Lie groups on analytic manifolds. We study three cases of analytic manifolds
and their corresponding transformation groups. Given a free action on the left, we define left orbit spaces and consider actions
on the right by maximal compact subgroups. We show that these actions are transitive and find the corresponding isotropy subgroups.
Further, we show that the left orbit spaces are reductive homogeneous spaces. This article thus forms the basis of a forthcoming
paper on invariant differential operators on homogeneous manifolds.
Partially supported by a Carver Research Initiative Grant. 相似文献
17.
We use techniques from homotopy theory, in particular the connection between configuration spaces and iterated loop spaces,
to give geometric explanations of stability results for the cohomology of the varieties of regular semisimple elements in
the simple complex Lie algebras of classical type A, B or C, as well as in the group . We show that the cohomology spaces of stable versions of these varieties have an algebraic stucture, which identifies them
as “free Poisson algebras” with suitable degree shifts. Using this, we are able to give explicit formulae for the corresponding
Poincaré series, which lead to power series identities by comparison with earlier work. The cases of type B and C involve ideas from equivariant homotopy theory. Our results may be interpreted in terms of the actions of a Weyl group on
its coinvariant algebra (i.e. the coordinate ring of the affine space on which it acts, modulo the invariants of positive
degree; this space coincides with the cohomology ring of the flag variety of the associated Lie group) and on the cohomology
of its associated complex discriminant variety.
Received August 31, 1998; in final form August 1, 1999 / Published online October 30, 2000 相似文献
18.
We prove a one-to-one correspondence between differential symmetry breaking operators for equivariant vector bundles over two homogeneous spaces and certain homomorphisms for representations of two Lie algebras, in connection with branching problems of the restriction of representations. We develop a new method (F-method) based on the algebraic Fourier transform for generalized Verma modules, which characterizes differential symmetry breaking operators by means of certain systems of partial differential equations. In contrast to the setting of real flag varieties, continuous symmetry breaking operators of Hermitian symmetric spaces are proved to be differential operators in the holomorphic setting. In this case, symmetry breaking operators are characterized by differential equations of second order via the F-method. 相似文献
19.
Arnaud Leyssens 《代数通讯》2013,41(4):2173-2183
We study the homogeneous spaces of a smooth group G over affine spaces or local k-algebra and gerbs locally banded by a semi-simple group H dèfine over k. In particular we show that every gerb locally banded by H and rationally trivial is trivial, and that an homogeneous space of G with semi-simple isotropy which is rationally trivial is trivial in the Zariski topology. This extends a result of Colliot-Thélène and Ojanguren concerning G-torsors. 相似文献
20.
We study curvatures of homogeneous Randers spaces. After deducing the coordinate-free formulas of the flag curvature and Ricci scalar of homogeneous Randers spaces, we give several applications. We first present a direct proof of the fact that a homogeneous Randers space is Ricci quadratic if and only if it is a Berwald space. We then prove that any left invariant Randers metric on a non-commutative nilpotent Lie group must have three flags whose flag curvature is positive, negative and zero, respectively. This generalizes a result of J.A. Wolf on Riemannian metrics. We prove a conjecture of J. Milnor on the characterization of central elements of a real Lie algebra, in a more generalized sense. Finally, we study homogeneous Finsler spaces of positive flag curvature and particularly prove that the only compact connected simply connected Lie group admitting a left invariant Finsler metric with positive flag curvature is SU(2). 相似文献