arXiv research

In this paper we study geodesic mappings of $n$-dimensional surfaces of revolution. From the general theory of geodesic mappings of equidistant spaces we specialize to surfaces of revolution and apply the obtained formulas to the case of rotational ellipsoids. We prove that such $n$-dimensional ellipsoids admit non tri…

The study classifies singularities of spherical orthotomic curves.

The generalization of the n-dimensional cube, an n-dimensional chain, the exterior derivative and the integral of a differential n-form on it are introduced and investigated. The analogue of Stokes theorem for the differential space is given.

The paper characterizes special curves and generalizes rectifying-type curves in n-dimensional space.

Study characterizes involutes and evolutes of curves in n-dimensional space.

We extend the edge version of the classical Menger's Theorem for undirected graphs to $n$-dimensional simplicial complexes with chains over the field $\mathbb{F}_2$. The classical Menger's Theorem states that two different vertices in an undirected graph can be connected by $k$ pairwise edge-disjoint paths if, and only…

The paper constructs graph models for n-dimensional manifolds.

New Coxeter groups yield n-dimensional Sierpiński boundaries.

This paper constructs wild knots from beaded necklaces using a Schottky group.

The paper studies scalar curvature of self-shrinkers and proves curvature bounds.

The paper proves an equivariant Kastler-Kalau-Walze theorem for various spin manifolds.

The paper generalizes fractals to higher dimensions using affine transformations.

A method to calculate generalized curvatures of curves in n-dimensional space.

Research shows surfaces close to planes in Hausdorff distance.

We show that an n-dimensional compactum X embeds in R^m, where m>3(n+1)/2, if and only if X x X - Δadmits an equivariant map to S^{m-1}. In particular, X embeds in R^{2n}, n>3, iff the top power of the (twisted) Euler class of the factor-exchanging involution on X x X - Δis trivial. Assuming that X quasi-embeds in R^{2…

The study characterizes rectifying curves in n-dimensional space.

We study the Abreu's equation in n-dimensional polytopes and derive interior estimates of solutions under the assumption of the uniform K-stability.

We study sequences of integral current spaces $(X_j,d_j,T_j)$ such that the integral current structure $T_j$ has weight $1$ and no boundary and, all $(X_j,d_j)$ are closed Alexandrov spaces with curvature uniformly bounded from below and diameter uniformly bounded from above. We prove that for such sequences either the…

Paper tackles tensor completion for 3D or higher exponential signals.

Developed concentrated liquidity in n-dimensional AMM with polar coordinates in Rust.

We present two classical conjectures concerning the characterization of manifolds: the Bing Borsuk Conjecture asserts that every $n$-dimensional homogeneous ANR is a topological $n$-manifold, whereas the Busemann Conjecture asserts that every $n$-dimensional $G$-space is a topological $n$-manifold. The key object in bo…

The n-dimensional torus is uniquely characterized by specific harmonic forms.

In this paper we parametrize the symmetry group of the n-dimensional Berwald-Moor metric. Some properties of this Lie group are studied, and its corresponding Lie algebra is computed.

We give a short proof of the systolic inequality for the n-dimensional torus. The proof uses minimal hypersurfaces. It is based on the Schoen-Yau proof that an n-dimensional torus admits no metric of positive scalar curvature.

A manifold is T-embedded into an affine space if its tangent spaces at distinct points are disjoint. We prove that an n-dimensional disc cannot be T-embedded into 2n-dimensional space.

This paper classifies complete self-shrinkers in R^(n+1) with nonnegative constant scalar curvature.

Generalized Cauchy's surface area formula to arbitrary submanifolds in R^n.

Smooth knots can be embedded into a specific Menger continuum.

Characterizes C-projective vector fields on Randers spaces.

We introduce the equation of n-dimensional totally geodesic submanifolds of a manifold E as a submanifold of the second order jet space of n-dimensional submanifolds of E. Next we study the geometry of n-Grassmannian equivalent connections, that is linear connections without torsion admitting the same equation of n-dim…

Let $x:M\to S^{n+p}$ be an $n$-dimensional submanifold in an $(n+p)$-dimensional unit sphere $S^{n+p}$, $x:M\to S^{n+p}$ is called a Willmore submanifold to the following Willmore functional: $\int_M(S-nH^2)^{\frac{n}{2}}dv,$ where $S=\sum\limits_{α,i,j}(h^α_{ij})^2$ is the square of the length of the second fundam…

Researchers show Hodge numbers modulo m can be achieved by smooth projective varieties.

Study on symmetry defects of complete intersections in complex space.

We describe the fundamental groups of ordered and unordered $k-$point sets in the n-dimensional complex space $C^n$ generating an affine subspace of fixed dimension.

In this paper we solve the problem of finding integrals of equations determining the Killing tensors on an $n$ -dimensional differentiable manifold $M$ endowed with an equiaffine $SL(n, R)$ -structure and discuss possible applications of obtained results in Riemannian geometry.

Cannon, Floyd, and Parry have studied subdivisions of the 2-sphere extensively, especially those corresponding to 3-manifolds, in an attempt to prove Cannon's conjecture. There has been a recent interest in generalizing some of their tools, such as extremal length, to higher dimensions. We define finite subdivision rul…

An $n$-dimensional Hartogs domain $D_F$ with strongly pseudoconvex boundary can be equipped with a natural \K metric $g_F$. In this paper we prove that if $g_F$ is an extremal \K metric then $(D_F, g_F)$ is biholomorphically isometric to the $n$-dimensional complex hyperbolic space.

It is well known that the space of oriented lines of Euclidean space has a natural symplectic structure. Moreover, given an immersed, oriented hypersurface S the set of oriented lines that cross S orthogonally is a Lagrangian submanifold. Conversely, if \bar{S} an n-dimensional family of oriented lines is Lagrangian, t…

Study improves Reilly's inequality for $p$-Laplacian on submanifolds.

Reduces gradient Ricci solitons to ODEs for easier analysis.

The number of functionally independent scalar invariants of arbitrary order of a generic pseudo--Riemannian metric on an $n$--dimensional manifold is determined.

Author simplifies and generalizes p-adic integer action construction.

We show that any $n$-dimensional Fano manifold $X$ admitting Kähler-Einstein metrics satisfies that the anti-canonical volume is less than or equal to the value $(n+1)^n$. Moreover, the equality holds if and only if $X$ is isomorphic to the $n$-dimensional projective space.

We show some characterizations of hyperspheres in the $(n+1)$-dimensional Euclidean space ${\Bbb E}^{n+1}$ with intrinsic and extrinsic properties such as the $n$-dimensional area of the sections cut off by hyperplanes, the $(n+1)$-dimensional volume of regions between parallel hyperplanes, and the $n$-dimensional surf…

The paper generalizes curvature bounds for submanifolds with singularities.

We derive a dimensionally-reduced limit theory for an $n$-dimensional nonlinear elastic body that is slender along $k$ dimensions. The starting point is to view an elastic body as an $n$-dimensional Riemannian manifold together with a not necessarily isometric $W^{1,2}$-immersion in $n$-dimensional Euclidean space. The…

Equal diagonal energies proven on Liouville surfaces.

Let M be a complete non-compact connected Riemannian n-dimensional manifold. We first prove that, for any fixed point p in M, the radial Ricci curvature of M at p is bounded from below by the radial curvature function of some non-compact n-dimensional model. Moreover, we then prove, without the pointed Gromov-Hausdorff…