Related papers: Some geometric properties of Riemann's non-differe…

In the infinite dimensional Heisenberg group, we construct a left invariant weak Riemannian metric that gives a degenerate geodesic distance. The same construction yields a degenerate sub-Riemannian distance. We show how the standard notion…

We extend the Besicovitch-Federer projection theorem to transversal families of mappings. As an application we show that on a certain class of Riemann surfaces with constant negative curvature and with boundary, there exist natural…

This work revolves around properties and applications of functions whose nonlocal gradient, or more precisely, finite-horizon fractional gradient, vanishes. Surprisingly, in contrast to the classical local theory, we show that this class…

In this work we study differential geometry in $N$ dimensional Riemann curved spaces using Pfaff derivatives. Avoiding the classical partial derivative the Pfaff derivatives are constructed in a more sophisticated way and make evaluations…

We analyse second order (in Riemann curvature) geometric flows (un-normalised) on locally homogeneous three manifolds and look for specific features through the solutions (analytic whereever possible, otherwise numerical) of the evolution…

Certain aspects of nonrelativistic diffeomorphisms in 2+1 dimensions are investigated. These include a nonrelativistic limit of some relativistic actions in 3 dimensions, the Seiberg-Witten map, a modification of the viscosity tensor in…

One field of fluid dynamics concerns the search for variational principles. So far, the Hamiltonian view and Riemannian geometry has been applied to find geodesics for hydrodynamic systems. Compared to Riemannian geometry sub-Riemannian…

Hausdorff relation, topologically identifying points in a given space, belongs to elementary tools of modern mathematics. We show that if subtle enough mathematical methods are used to analyze this relation, the conclusions may be…

We prove a general existence theorem for nonlinear partial differential systems of any order in one complex variable. A special case of first order contains a well-known theorem of Nijenhuis and Woolf concerning local existence of…

We discover a fundamental exterior differential system of Riemannian geometry; indeed, an intrinsic and invariant global system of differential forms of degree $n$ associated to any given oriented Riemannian manifold $M$ of dimension $n+1$.…

We prove that the Hausdorff dimension of the graph of a prevalent continuous function is 2. We also indicate how our results can be extended to the space of continuous functions on $[0,1]^d$ for $d \in \mathbb{N}$ and use this to obtain…

We introduce fractional flat space, described by a continuous geometry with constant non-integer Hausdorff and spectral dimensions. This is the analogue of Euclidean space, but with anomalous scaling and diffusion properties. The basic tool…

We define the tangential derivative, a notion of directional derivative which is invariant under diffeomorphisms. In particular this derivative is invariant under changes of chart and is thus well-defined for functions defined on a…

We consider the three dimensional Heisenberg nilflows. Under a full measure set Diophantine condition on the generator of the flow we construct Bufetov functionals which are asymptotic to ergodic integrals for sufficiently smooth functions,…

We solve the following problem for $n=2:$ Is any n-dimensional Finsler manifold $(M, F)$ with a function $f$ which is nonconstant and smooth on $M$ satisfying $ \dfrac{\partial g^{ij}}{\partial y^k}\dfrac{\partial f}{\partial x^i}=0, $ a…

The aim of this article is to present a comparative review of Riemannian and Finsler geometry. The structures of cut and conjugate loci on Riemannian manifolds have been discussed by many geometers including H. Busemann, M. Berger and W.…

For a complete noncompact Riemannian manifold with nonnegative Ricci curvature, we show that bounded biharmonic functions are constant and the space consists of biharmonic functions with polynomial growth of a fixed rate is finite…

We develop a geometric formulation of fluid dynamics, valid on arbitrary Riemannian manifolds, that regards the momentum-flux and stress tensors as 1-form valued 2-forms, and their divergence as a covariant exterior derivative. We review…

We discuss hypersurface motions in Riemannian manifolds whose normal velocity is a function of the induced hypersurface volume element and derive a second order partial differential equation for the corresponding time function $\tau(x)$ at…

Within a framework of noncommutative geometry, we develop an analogue of (pseudo) Riemannian geometry on finite and discrete sets. On a finite set, there is a counterpart of the continuum metric tensor with a simple geometric…