The equation \(X\nabla \text{det}X = \text{det}X\nabla \text{tr}X\), multiplication of cofactor pair systems, and the Levi-Civita equivalence problem
type: Item
instance of: scholarly article
title: The equation \(X\nabla \text{det}X = \text{det}X\nabla \text{tr}X\), multiplication of cofactor pair systems, and the Levi-Civita equivalence problem
zbMATH Open document ID: 1114.53036
author: Jens Jonasson
published in: Journal of Geometry and Physics
publication date: 2006-12-07T00:00:00Z
review text: The author is interested in studying a type of PDE related to some problems arising in dynamical systems, and to some previous works by Rauch-Wojciechowski, Marciniak, Lundmark, Waksjö and Benenti. More precisely he studies differential equations of the type \(A^{-1}\nabla V=\widetilde{A}^{-1}\nabla\widetilde{V}\) on an \(n\)-dimensional Riemannian manifold \((Q,g)\), where \(A\) and \(\widetilde{A}\) are nonsingular tensors of rank two, \(\nabla\) is the gradient operator, and \(V\), \(\widetilde{V}\) are two \({\mathbb R}\)-valued functions on \(Q\). Furthermore, he considers suitable algebraic structures on the space of solutions of such equations and obtains methods for constructing new systems with multiplicative structure. He relates the role of the multiplication with the theory of equivalent dynamical systems on Riemannian manifolds. The paper, after a concise introduction, splits into four additional sections: 2. Quasi Cauchy-Riemann equations; 3. Multiplication; 4. The equation \(X\nabla\det X=\det X\nabla \text{TR} X\); 5. Multiplication and Levi-Civita equivalence. The paper ends with some conclusions and interesting questions to examine in the future.
reviewed by: Agostino Prástaro
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 70E40
Mathematics Subject Classification ID: 70H03
Mathematics Subject Classification ID: 70S05
zbMATH DE Number: 5078795
zbMATH Keywords: Riemannian geometry PDE's
zbMATH Keywords: multiplication cofactor pair
zbMATH Keywords: Levi-Civita equivalence
MaRDI profile type: Publication
full work available at URL: j.geomphys.2006.03.001
OpenAlex ID: W1485397452
cites work: Special symmetric two-tensors, equivalent dynamical systems, cofactor and bi-cofactor systems
cites work: Conformal Killing tensors with vanishing torsion and the separation of variables in the Hamilton--Jacobi equation
cites work: A class of nonconservative Lagrangian systems on Riemannian manifolds.
cites work: Q3244523
cites work: On the equation grad f = M grad g
cites work: Higher-Dimensional Integrable Newton Systems with Quadratic Integrals of Motion
cites work: Driven Newton equations and separable time-dependent potentials
cites work: Quasi-Lagrangian systems of Newton equations
cites work: A Treatise on the Analytical Dynamics of Particles and Rigid Bodies
cites work: The Poincaré Lemma Associated with Certain Derivations
cites work: Q3783649
DOI: 10.1016/J.GEOMPHYS.2006.03.001
description: scientific article; zbMATH DE number 5078795