Spaces of Constant Curvature

Wolf, Joseph A.

doi:10.1090/chel/372

Cited by 463 publications

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“…Hence there is no a nearly integrable irreducible SO(3)-structure on a five-dimensional hyperbolic manifold (H 5 , g 0 ) [55]. This completes the proof of the theorem.…”

Section: Theorem 47 Let the Triplet (M G T ) Be A Statistical Manisupporting

confidence: 56%

A five-dimensional Riemannian manifold with an irreducible SO(3)-structure as a model of abstract statistical manifold

Mikeš

Stepanova

2013

Ann Glob Anal Geom

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In the present paper, we consider a five-dimensional Riemannian manifold with an irreducible SO(3)-structure as an example of an abstract statistical manifold. We prove that if a five-dimensional Riemannian manifold with an irreducible SO(3)-structure is a statistical manifold of constant curvature, then the metric of the Riemannian manifold is an Einstein metric. In addition, we show that a five-dimensional Euclidean sphere with an irreducible SO(3)-structure cannot be a conjugate symmetric statistical manifold. Finally, we show some results for a five-dimensional Riemannian manifold with a nearly integrable SO(3)-structure. For example, we prove that the structure tensor of a nearly integrable SO(3)-structure on a five-dimensional Riemannian manifold is a harmonic symmetric tensor and it defines the first integral of third order of the equations of geodesics. Moreover, we consider some topological properties of five-dimensional compact and conformally flat Riemannian manifolds with irreducible SO(3)-structure.

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“…Hence there is no a nearly integrable irreducible SO(3)-structure on a five-dimensional hyperbolic manifold (H 5 , g 0 ) [55]. This completes the proof of the theorem.…”

Section: Theorem 47 Let the Triplet (M G T ) Be A Statistical Manisupporting

confidence: 56%

A five-dimensional Riemannian manifold with an irreducible SO(3)-structure as a model of abstract statistical manifold

Mikeš

Stepanova

2013

Ann Glob Anal Geom

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“…For example, cyclic groups are fixed-point free, but also admit non fixed-point free representations: Fixed-point free groups have been completely classified by the works of Burnside and later Vincent [40] and Zassenhaus [44]. The classification can be found, for example, in [42]. For our purposes we will require only the following result, easily achieved from the classification: Our proof uses the construction of semidirect product of groups.…”

Section: Definition 213 Let G Be a Finite Group And F A Field A Rementioning

confidence: 99%

Multi-linear Secret-Sharing Schemes

Beimel

Ben-Efraim

Padrö

et al. 2014

Theory of Cryptography

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Abstract. Multi-linear secret-sharing schemes are the most common secret-sharing schemes. In these schemes the secret is composed of some field elements and the sharing is done by applying some fixed linear mapping on the field elements of the secret and some randomly chosen field elements. If the secret contains one field element, then the scheme is called linear. The importance of multi-linear schemes is that they provide a simple non-interactive mechanism for computing shares of linear combinations of previously shared secrets. Thus, they can be easily used in cryptographic protocols.In this work we study the power of multi-linear secret-sharing schemes. On one hand, we prove that ideal multi-linear secret-sharing schemes in which the secret is composed of p field elements are more powerful than schemes in which the secret is composed of less than p field elements (for every prime p). On the other hand, we prove super-polynomial lower bounds on the share size in multi-linear secret-sharing schemes. Previously, such lower bounds were known only for linear schemes.

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“…As is well-known (e.g. [61] p. 295), a connected Riemannian manifold is isotropic if and only if it is two point homogeneous (its isometry group is transitive on equidistant pairs of points), or else if and only if it is a Euclidean space or a symmetric space of rank one. Then D(s) H , resp.…”

Section: The Case Of An Isotropic Riemannian Symmetric Spacementioning

confidence: 98%

Symmetric Spaces and the Kashiwara-Vergne Method

Rouvière

2014

Lecture Notes in Mathematics

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CONTENTS AbstractThe Kashiwara-Vergne method reduces the proof of a deep result in analysis on a Lie group (transferring convolution of invariant distributions from the group to its Lie algebra, by means of the exponential mapping) to checking two formal Lie brackets identities linked to the Campbell-Hausdor¤ formula. First, we expound this method and a proof, for quadratic or solvable Lie algebras, of the Kashiwara-Vergne conjecture allowing to apply the method to those cases. We then extend it to a general symmetric space S = G=H. This leads to introduce a function e(X; Y ) of two tangent vectors X; Y at the origin of S, allowing to make explicit, in the exponential chart, G-invariant di¤erential operators of S, the structure of the algebra of all such operators, and an expansion of mean value operators and spherical functions. For Riemannian symmetric spaces of the noncompact type, otherwise well-known from the work of Harish-Chandra and Helgason, we compare this approach with the classical one. For rank one spaces (the hyperbolic spaces), we give an explicit formula for e(X; Y ).Finally, we explain a construction of e for a general symmetric space by means of Lie series linked to the Campbell-Hausdor¤ formula, in the spirit of the original Kashiwara-Vergne method. Proved from this construction, the main properties of e thus link the fundamental tools of H-invariant analysis on a symmetric space to its in…nitesimal structure.The results extend to line bundles over symmetric spaces. PrefaceLet G=K denote a Riemannian symmetric space of the noncompact type.« The action of the G-invariant di¤ erential operators on G=K on the radial functions on G=K is isomorphic with the action of certain di¤ erential operators with constant coe¢ cients. The isomorphism in question is used in Harish-Chandra's work on the Fourier analysis on G and is related to the Radon transform on G=K.For the case when G is complex a more direct isomorphism of this type is given, again as a consequence of results of HarishChandra.» This is a quote from Sigurður Helgason, Fundamental solutions of invariant di¤ erential operators on symmetric spaces, Amer. J. Maths. 86 (1964), p. 566, one of the …rst mathematical papers I have studied. As a consequence he showed that every (non-zero) G-invariant di¤erential operator D on G=K has a fundamental solution, whence the solvability of the di¤erential equation Du = f . Those results were fascinating to me, they still are and they fostered my interest in invariant di¤erential operators as well as Radon transforms. Yet I was dreaming simpler proofs could be given, without relying on HarishChandra's deep study of semisimple Lie groups... In the beginning of 1977 Michel Du ‡o [18] gave an analytic interpretation of the celebrated Du ‡o isomorphism he had exhibited six years before by means of algebraic constructions in an enveloping algebra. As a consequence he proved that every (non-zero) bi-invariant di¤erential operator on a Lie group has a local fundamental solution and is locally solvable. His work us...

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Spaces of Constant Curvature

Cited by 463 publications

References 0 publications

A five-dimensional Riemannian manifold with an irreducible SO(3)-structure as a model of abstract statistical manifold

A five-dimensional Riemannian manifold with an irreducible SO(3)-structure as a model of abstract statistical manifold

Multi-linear Secret-Sharing Schemes

Symmetric Spaces and the Kashiwara-Vergne Method

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