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
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.
scientific article; zbMATH DE number 5078795
scientific article