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We construct the classical mechanics associated with a conformally flat Riemannian metric on a compact, n-dimensional manifold without boundary. The corresponding gradient Ricci flow equation turns out to equal the time-dependent…
A quasi-product on the normed space is defined. In addition, the notions of the eigenvectors of a linear operator can be extended for the nonlinear operator. Based on the quasi-product and the generalized eigenvectors, the spectral theorems…
On any space-like W-surface in the three-dimensional Minkowski space we introduce locally natural principal parameters and prove that such a surface is determined uniquely up to motion by a special invariant function, which satisfies a…
Some aspects of the connection between differential geometry and multidimensional soliton equations are discussed.
In this work we construct and analyze exact solutions describing Ricci flows and nonholonomic deformations of four dimensional (4D) Taub-NUT spacetimes. It is outlined a new geometric techniques of constructing Ricci flow solutions. Some…
The paper advances the hypothesis that the multi-field is a determinable, that is, a physical object characterized by indeterminate values with respect to some properties. The multi-field is a realist interpretation of the wave function in…
Functional bases of second-order differential invariants of the Euclid, Poincar\'e, Galilei, conformal, and projective algebras are constructed. The results obtained allow us to describe new classes of nonlinear many-dimensional invariant…
We study noncommutative Ricci flow in a finite dimensional representation of a noncommutative torus. It is shown that the flow exists and converges to the flat metric. We also consider the evolution of entropy and a definition of scalar…
The article is devoted to approximate, global and along curves differentiability of functions over non-archimedean infinite fields with non-trivial valuations. Fields with zero and non-zero characteristics are considered. Spaces of…
An abstract theory of ultradifferentiable sheafs is developed. Moreover, various applications to the theory of linear partial differential equations, differential geometry and, in particular, CR geometry are discussed.
A general formalism to solve nonlinear differential equations is given. Solutions are found and reduced to those of second order nonlinear differential equations in one variable. The approach is uniformized in the geometry and solves…
We review the main aspects of Ricci flows as they arise in physics and mathematics. In field theory they describe the renormalization group equations of the target space metric of two dimensional sigma models to lowest order in the…
This is a survey article on recent progress of comparison geometry and geometric analysis on Finsler manifolds of weighted Ricci curvature bounded below. Our purpose is two-fold: Give a concise and geometric review on the birth of weighted…
We give a natural way to identify between two scales, potentially arbitrarily far apart, in a non-compact Ricci-flat manifold with Euclidean volume growth when a tangent cone at infinity has smooth cross section. The identification map is…
Integrable discrete scalar equations defined on a~two or a three dimensional lattice can be rewritten as difference systems in bond variables or in face variables respectively. Both the difference systems in bond variables and the…
We hypothesise that properties of space could underly some patterns observed in nature. We explore the possibility that the observed variety of matter particles and the pattern of their properties arise due to the non-orientability of the…
We find an infinite number of noncommutative geometries which posses a differential structure. They generalize the two dimensional noncommutative plane, and have infinite dimensional representations. Upon applying generalized coherent…
We obtain integral formulas for a metric-affine space equipped with two complementary orthogonal distributions. The integrand depends on the Ricci and mixed scalar curvatures and invariants of the second fundamental forms and integrability…
An example of an infinite dimensional and separable Banach space is given, that is not isomorphic to a subspace of l1 with no infinite equilateral sets.
Differentials on Riemann surfaces correspond to translation surfaces with conical singularities, and affine transformations acting on them preserve the orders of these singularities. This viewpoint allows the moduli spaces of differentials…