相关论文: Generalized Curvatures of Curves in $\mathbb{R}^n$

In the first part we use Gromov's K--area to define the K--area homology which stabilizes into singular homology on the category of pairs of compact smooth manifolds. The second part treats the questions of certain curvature gaps. For…

An integral geometric curvature is defined as the index expectation K(x) = E[i(x)] if a probability measure m is given on vector fields on a Riemannian manifold or on a finite simple graph. Such curvatures are local, satisfy Gauss-Bonnet…

This work introduces the framed curvature flow, a generalization of both the curve shortening flow and the vortex filament equation. Here, the magnitude of the velocity vector is still determined by the curvature, but its direction is given…

We discuss notions of Gauss curvature and mean curvature for polyhedral surfaces. The discretizations are guided by the principle of preserving integral relations for curvatures, like the Gauss/Bonnet theorem and the mean-curvature force…

We consider a regular smooth curve in $\mathbb{E}^n$ such that its coordinates' components are the fundamental solutions of the differential equation $ y^{(n)} (x) - y(x) = 0 ,$ $x \in \mathbb{R} $ of order $n$. We show that the total first…

It is proved that smooth closed curves of given length minimizing the principal eigenvalue of the Schr\"odinger operator $-\frac{d^2}{ds^2}+\kappa^2$ exist. Here $s$ denotes the arclength and $\kappa$ the curvature. These minimizers are…

We prove that every immersed $C^2$-curve $\gamma$ in $\mathbb R^n$, $n\geqslant 3$ with curvature $k_{\gamma}$ can be $C^1$-approximated by immersed $C^2$-curves having prescribed curvature $k>k_{\gamma}$. The approximating curves satisfy a…

Given a singular Riemannian foliation on a compact Riemannian manifold, we study the mean curvature flow equation with a regular leaf as initial datum. We prove that if the leaves are compact and the mean curvature vector field is basic,…

We derive a generalized deviation equation -- analogous to the well-known geodesic deviation equation -- for test bodies in General Relativity. Our result encompasses and generalizes previous extensions of the standard geodesic deviation…

The basic idea of quantum complexity geometry is to endow the space of unitary matrices with a metric, engineered to make complex operators far from the origin, and simple operators near. By restricting our attention to a finite subgroup of…

In this paper, we give a generalization of Fenchel's theorem for closed curves as frontals in Euclidean space $\mathbb{R}^n$. We prove that, for a non-co-orientable closed frontal in $\mathbb{R}^n$, its total absolute curvature is greater…

A sequence of generalizations of Cartan's conservation of torsion theorem is given for n-dimensional differentiable manifolds having a general linear connection.

We study generalized matrix models corresponding to n-point Virasoro conformal blocks on Riemann surfaces with arbitrary genus g. Upon AGT correspondence, these describe four dimensional N=2 SU(2)^{n+3g-3} gauge theories with generalized…

We prove a linear bound on the average total curvature of the central path of linear programming theory in terms on the number of independent variables of the primal problem, and independent on the number of constraints.

Based on two classical notions of curvature for curves in general metric spaces, namely the Menger and Haantjes curvatures, we introduce new definitions of sectional, Ricci and scalar curvature for networks and their higher dimensional…

In this note, we discuss the mean curvature flow of graphs of maps between Riemannian manifolds. Special emphasis will be placed on estimates of the flow as a non-linear parabolic system of differential equations. Several global existence…

We study the center problem for the class $\mathcal E_\Gamma$ of Abel differential equations $\frac{dv}{dt}=a_1 v^2+a_2 v^3$, $a_1,a_2\in L^\infty ([0,T])$, such that images of Lipschitz paths $\tilde A:=\bigl(\int_0^\cdot a_1(s)ds,…

We establish existence and uniqueness results for the modified binormal curvature flow equation that generalizes the binormal curvature flow equation for a curve in $\R^3.$ In this generalization, the velocity of the curve is still directed…

Curvature tensors of higher-spin gauge theories have been known for some time. In the past, they were postulated using a generalization of the symmetry properties of the Riemann tensor (curl on each index of a totally symmetric rank-$n$…

According to the classification scheme of the generalized random matrix ensembles, we present various kinds of concrete examples of the generalized ensemble, and derive their joint density functions in an unified way by one simple formula…