scholarly journals Reduction by group symmetry of second order variational problems on a semidirect product of Lie groups with positive definite Riemannian metric

2004 ◽  
Vol 10 (4) ◽  
pp. 526-548 ◽  
Author(s):  
Claudio Altafini
Keyword(s):  
Lie Groups ◽  
Semidirect Product ◽  
Variational Problems ◽  
Riemannian Metric ◽  
Positive Definite ◽  
Second Order ◽  
Group Symmetry

Second order duality for variational problems involving generalized convexity

OPSEARCH ◽  
2014 ◽  
Vol 52 (3) ◽  
pp. 582-596 ◽  
Author(s):  
Anurag Jayswal ◽  
I. M. Stancu-Minasian ◽  
Sarita Choudhury
Keyword(s):  
Generalized Convexity ◽  
Variational Problems ◽  
Second Order ◽  
Second Order Duality

Second-order-optimal filters on Lie groups

2013 ◽  
Author(s):  
Alessandro Saccon ◽  
Jochen Trumpf ◽  
Robert Mahony ◽  
A. Pedro Aguiar
Keyword(s):  
Lie Groups ◽  
Second Order ◽  
Optimal Filters

Characteristic functions of semigroups in semi-simple Lie groups

2019 ◽  
Vol 31 (4) ◽  
pp. 815-842
Author(s):  
Luiz A. B. San Martin ◽  
Laercio J. Santos
Keyword(s):  
Laplace Transform ◽  
Lie Groups ◽  
Lie Algebra ◽  
Lie Group ◽  
Convex Cone ◽  
Convex Set ◽  
Flag Manifold ◽  
Riemannian Metric ◽  
Iwasawa Decomposition ◽  
Characteristic Functions

Abstract Let G be a noncompact semi-simple Lie group with Iwasawa decomposition {G=KAN} . For a semigroup {S\subset G} with nonempty interior we find a domain of convergence of the Helgason–Laplace transform {I_{S}(\lambda,u)=\int_{S}e^{\lambda(\mathsf{a}(g,u))}\,dg} , where dg is the Haar measure of G, {u\in K} , {\lambda\in\mathfrak{a}^{\ast}} , {\mathfrak{a}} is the Lie algebra of A and {gu=ke^{\mathsf{a}(g,u)}n\in KAN} . The domain is given in terms of a flag manifold of G written {\mathbb{F}_{\Theta(S)}} called the flag type of S, where {\Theta(S)} is a subset of the simple system of roots. It is proved that {I_{S}(\lambda,u)<\infty} if λ belongs to a convex cone defined from {\Theta(S)} and {u\in\pi^{-1}(\mathcal{D}_{\Theta(S)}(S))} , where {\mathcal{D}_{\Theta(S)}(S)\subset\mathbb{F}_{\Theta(S)}} is a B-convex set and {\pi:K\rightarrow\mathbb{F}_{\Theta(S)}} is the natural projection. We prove differentiability of {I_{S}(\lambda,u)} and apply the results to construct of a Riemannian metric in {\mathcal{D}_{\Theta(S)}(S)} invariant by the group {S\cap S^{-1}} of units of S.

scholarly journals Lagrangians, Gauge Functions, and Lie Groups for Semigroup of Second-Order Differential Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-11 ◽  
Author(s):  
Z. E. Musielak ◽  
N. Davachi ◽  
M. Rosario-Franco
Keyword(s):  
Differential Equations ◽  
Lie Groups ◽  
Lie Group ◽  
Algebraic Structure ◽  
Second Order ◽  
Lagrangian Formalism ◽  
Group Approach ◽  
Second Order Differential Equations ◽  
Gauge Functions ◽  
The Relationship

A set of linear second-order differential equations is converted into a semigroup, whose algebraic structure is used to generate novel equations. The Lagrangian formalism based on standard, null, and nonstandard Lagrangians is established for all members of the semigroup. For the null Lagrangians, their corresponding gauge functions are derived. The obtained Lagrangians are either new or generalization of those previously known. The previously developed Lie group approach to derive some equations of the semigroup is also described. It is shown that certain equations of the semigroup cannot be factorized, and therefore, their Lie groups cannot be determined. A possible solution of this problem is proposed, and the relationship between the Lagrangian formalism and the Lie group approach is discussed.

scholarly journals Prolongations of G-Structures to Tangent Bundles

1968 ◽  
Vol 32 ◽  
pp. 67-108 ◽  
Author(s):  
Akihiko Morimoto
Keyword(s):  
Riemannian Metrics ◽  
Riemannian Metric ◽  
Positive Definite ◽  
The Other ◽  
Tensor Fields ◽  
Almost Complex Structures ◽  
Almost Complex ◽  
The One ◽  
Lie Subgroup ◽  
Tangent Bundles

The purpose of the present paper is to study the prolongations of G-structures on a manifold M to its tangent bundle T(M), G being a Lie subgroup of GL(n,R) with n = dim M. Recently, K. Yano and S. Kobayashi [9] studied the prolongations of tensor fields on M to T(M) and they proposed the following question: Is it possible to associate with each G-structure on M a naturally induced G-structure on T(M), where G′ is a certain subgroup of GL(2n,R)? In this paper we give an answer to this question and we shall show that the prolongations of some special tensor fields by Yano-Kobayashi — for instance, the prolongations of almost complex structures — are derived naturally by our prolongations of the classical G-structures. On the other hand, S. Sasaki [5] studied a prolongation of Riemannian metrics on M to a Riemannian metric on T(M), while the prolongation of a (positive definite) Riemannian metric due to Yano-Kobayashi is always pseudo-Riemannian on T(M) but never Riemannian. We shall clarify the circumstances for this difference and give the reason why the one is positive definite Riemannian and the other is not.

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