On a family of functional equations

V. A. Kyrov

Vladikavkazskij matematičeskij žurnal, Tome 20 (2018) no. 3, pp. 69-77

Résumé

The $(n+1)$-dimensional geometry of local maximum mobility is given by some non-degenerate and differentiable function $f$ of the pair of points on the manifold $M$, which is а motion group invariant of dimension $(n+1)(n+2)/2$. There is no complete classification of such geometries of dimension $n + 1$, but some examples are well known: Euclidean geometry, symplectic geometry, constant curvature geometry. Recently, some previously unknown geometries of local maximum mobility has been found using the embedding method. The embedding method enables one to find functions $f$ that define $(n+1)$-dimensional geometries of local maximum mobility by functions $\theta$ of known $n$-dimensional geometry of local maximum mobility. This problem is reduced to the solution of functional equations of a special type, which are a consequence of the invariance of the function $f$ of the pair of points with respect to the motion group. Such equations are solved in this paper. By differentiation, they are first reduced to functional differential equations, from which we pass to differential equations by separating the variables. Then the solutions of the latter are substituted into the original functional equations, after which we get the final result.

Bibliographie

[1] Kyrov V. A., “Functional equations in pseudo-Euclidean geometry”, Journal of Applied and Industrial Mathematics, 13:4 (2010), 38–51 (in Russian) | MR | Zbl

[2] Kyrov V. A., “Functional equations in simplicial geometry”, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 16, no. 2, 2010, 149–153 (in Russian)

[3] Kyrov V. A., “On some class of functional-differential equation”, J. Samara State Tech. Univ., Ser. Phys. Math. Sci., 26:1 (2012), 31–38 (in Russian) | DOI | MR

[4] Ovsyannikov L. V., Group Analysis of Differential Aquation, Nauka, M., 1978, 400 pp. (in Russian)

[5] Kyrov V. A., “Solving of functional equations associated with the scalar product”, Chelyabinskiy Fiziko-Matematicheskiy Zhurnal, 2:1 (2017), 30–45 (in Russian) | MR

[6] Kyrov V. A., Mikhailichenko G. G., “The analytic method of embedding symplectic geometry”, Siberian Electronic Mathematical Reports, 14 (2017), 657–672 (in Russian) | DOI | MR | Zbl

[7] Kyrov V. A., Mikhailichenko G. G., “An analytic method for the embedding of the Euclidean and pseudo-Euclidean geometries”, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 23, no. 2, 2017, 167–181 (in Russian) | MR

[8] Mikhailichenko G. G., The Mathematical Basics and Results of the Theory of Physical Structures, arXiv: 1602.02795 [math-ph] | MR