Study of invariant surfaces in isotropic and pseudo-isotropic geometries.
problem Prescribed curvature for invariant surfaces in singular metrics.
method Analysis of one-parameter subgroups of isotropic rigid motions, computation of fundamental forms and curvatures.
result Generalization of revolution and helicoidal surfaces to singular metrics.
Study on spherical Finsler metrics with isotropic curvature rigidity.
problem Characterizing and understanding spherically symmetric Finsler metrics with isotropic E-curvature. method Provided the correct formula for mean Berwald curvature, established differential equations for projective and dual flatness, and derived a rigidity result.
result Rigidity result on spherically symmetric Finsler metrics with isotropic E-curvature. The study of Laguerre isotropic hypersurfaces with rigidity and isoparametric properties.
problem Characterizing and understanding Laguerre isotropic hypersurfaces.
method Analyzing hypersurfaces with zero Laguerre form and constant eigenvalues of the Laguerre tensor.
result For L-isotropic hypersurfaces, if they are also L-isoparametric, the constant λ must be zero. This paper explores non-periodic folding of Spidron units, revealing nonlinear dynamics.
problem Understanding the kinematics and nonlinear phenomena of Spidron units.
method Analysis of single unit cell kinematics and recursive construction of multiple cells.
result Non-periodic folding restricts isotropic folding as the number of unit cells increases.
Equations for minimal surfaces from rigid motions in high dimensions.
problem Finding minimal surfaces from rigid motions in RN. method Derives equations for minimal surfaces using rigid motions in RN. result Equations for minimal surfaces in RN. The paper proves rigidity results for non-positively curved homogeneous Finsler metrics.
problem Rigidity of non-positively curved homogeneous Finsler metrics.
method Analyzes and proves rigidity results for specific Finsler metrics with non-positive flag curvature.
result Homogeneous Finsler spaces with non-positive flag curvature and isotropic S-curvature are either Riemannian or locally Minkowskian.
Neural network estimates rigid motion in stroke imaging to improve image quality.
problem Rigid patient motion during C-arm CBCT imaging reduces image quality.
method Neural network trained to regress reprojection error based on image information.
result Neural network outperforms entropy-based method in motion estimation.
The paper studies how points and lines can move while preserving incidences.
problem Understanding how point-line configurations can move while maintaining their geometric relationships.
method Developed a projective rigidity matrix to analyze the infinitesimal motions and dependencies of point-line configurations.
result The symmetry-adapted projective rigidity matrix provides a more detailed analysis of symmetric configurations and their motions.
The abstract proves properties of Berwald spaces with non-zero flag curvature.
problem Characterizing Berwald spaces with non-zero flag curvature.
method Analyzes properties of Berwald manifolds with non-zero flag curvature, proving extensions of previous theorems.
result Every Berwald manifold with non-zero flag curvature is Riemannian.
The paper derives Gauss-Bonnet theorems for deformed connections in affine and rigid motions groups.
problem Computing curvature and geodesic curvature for surfaces and curves in affine and rigid motions groups.
method Defined deformed Schouten-Van Kampen connections, computed Gaussian curvature limits, and signed geodesic curvature.
result Derived Gauss-Bonnet theorems for deformed connections in affine and rigid motions groups.
In this note we find a generic defining function of projective motion in the 6-dimensional rigid h-space.
Paper proves every homogeneous Landsberg surface is either Riemannian or locally Minkowskian.
problem Characterizing homogeneous Landsberg surfaces.
method Proved isotropic flag curvature and used it to prove rigidity.
result Every homogeneous Landsberg surface is either Riemannian or locally Minkowskian.
Study travel time tomography for transversely isotropic media using modified pseudodifferential calculus.
problem Travel time tomography problem for transversely isotropic media.
method Modified scattering pseudodifferential calculus to solve the tomography problem.
result Construction and use of modified pseudodifferential calculus to solve the tomography problem.
Study timelike surfaces with parallel mean curvature in Minkowski 4-space.
problem Existence and uniqueness of timelike surfaces with parallel mean curvature.
method Introduce canonical parameters and prove existence and uniqueness theorem.
result Each timelike surface with parallel mean curvature is determined by three geometric functions.
The paper examines isotropic cosmological space-times with changing sectional curvature.
problem Cosmological space-times with changing sectional curvature.
method Analysis of a family of geometrically well-behaved cosmological space-times foliated by isotropic hypersurfaces.
result Only space-time isometries ensure the rigidity properties of isotropic cosmological space-times.
A Steiner type formula for continuous translation invariant Minkowski valuations is established. In combination with a recent result on the symmetry of rigid motion invariant homogeneous bivaluations, this new Steiner type formula is used to obtain a family of Brunn-Minkowski type inequalities for rigid motion intertwi…
The paper calculates curvature limits and proves Gauss-Bonnet theorems in affine and Minkowski groups.
problem Computing curvature limits in affine and Minkowski groups.
method Analyzing Euclidean C2-smooth surfaces and curves in affine and Minkowski groups. result Gauss-Bonnet theorems in affine and Minkowski groups are proven.
Paper provides closed-form time derivatives for rigid body systems.
problem Need for time derivatives of equations of motion in robotics.
method Lie group formulation for rigid body systems to derive closed-form derivatives up to second-order.
result Closed-form equations provide direct insight into system dynamics.
Unified shape spaces with preserved invariances and regular metrics.
problem Combining shape invariances from Kendall's spaces with regular metrics.
method Defined a Sobolev-type operator to achieve the desired geometry, preserving invariances and regularity.
result Achieved a new landmark shape space with regular metrics and preserved invariances.
Geometrically interpolates rigid body motions with initial and terminal twists.
problem Finding spatial trajectories between prescribed initial and terminal poses.
method Derives solutions for k-IV-TIP and k-BV-TIP for k=1,...,4.
result Automatic cubic interpolation identical to minimum acceleration curve when twists are zero.
Study cohomological equation for robotic screw motions on SE(3).
problem Understanding obstruction phenomena in robotic rigid-body motion.
method Combining Fourier analysis and Peter-Weyl theory, reduce to finite-dimensional linear transport systems.
result Explicit screw motion illustrates resonance conditions and finite-dimensional obstructions.
Study proves rigidity of harmonic maps from 2-torus to complex projective space.
problem Rigidity of isotropic harmonic maps from a 2-torus to a complex projective space.
method Proves rigidity through holomorphic embeddings and complete linear systems.
result Ensures rigidity of harmonic bands in condensed matter physics.
Study of rigid body displacements in a projective space over dual numbers with geometric interpretations.
problem Understanding rigid body displacements in a novel geometric space.
method Projective differential geometry over the ring of dual numbers.
result Existence of non-straight curves with multiple osculating tangents.
In this work, we are interested in the differential geometry of curves in the simply isotropic and pseudo-isotropic 3-spaces, which are examples of Cayley-Klein geometries whose absolute figure is given by a plane at infinity and a degenerate quadric. Motivated by the success of rotation minimizing (RM) frames in Eucli…
Paper studies how discrete space curves with constant torsion deform to model linkage motions.
problem Modeling and understanding the motion of discrete space curves with constant torsion.
method Using semi-discrete mKdV equations to describe the motion of discrete space curves.
result The motion of discrete space curves is governed by semi-discrete mKdV equations.
Let (Mn,g), n≥4, be a compact simply-connected Riemannian manifold with nonnegative isotropic curvature. Given 0<l≤L, we prove that there exists $\eps = \eps (l,L,n)$ satisfying the following: If the scalar curvature s of g satisfies l≤s≤L and the Einstein tensor satisfies $$ | Ric - \fr…
Construct intrinsic Langevin dynamics for rigid inclusions on curved surfaces.
problem Stochastic dynamics of rigid inclusions on curved surfaces.
method Cartan's method of moving frames, Hamiltonian equations, intrinsic Langevin equations, Fokker-Planck equation.
result Extracted overdamped equations for accurate simulations of diffusion processes.
We study the motion of a charge on a conformally flat Riemannian torus in the presence of magnetic field. We prove that for any non-zero magnetic field there always exist orbits of this motion which have conjugate points. We conjecture that the restriction of conformal flatness of the metric is not essential for this r…
New method for curve comparison using iterated integrals and moving frames.
problem Comparing curves robustly to noise and transformations.
method Moving frame method paired with log-signature transform.
result Algorithmic construction of invariants for curve equivalence under rigid motions.
Constructs surfaces that can be tiled by a finite set of rigid motion congruence classes of tiles.
problem Creating surfaces that can be tiled by a finite set of rigid motion congruence classes of tiles.
method Constructs examples with various topologies and describes all monotilings by finite edge prototiles.
result Describes all monotilings by finite edge prototiles with three or less edges.
On the one hand, we prove that the Clifford torus in C2 is unstable for Lagrangian mean curvature flow under arbitrarily small Hamiltonian perturbations, even though it is Hamiltonian F-stable and locally area minimising under Hamiltonian variations. On the other hand, we show that the Clifford torus is r…
New equations for rigid body motion on infinite-dimensional spaces of operators.
problem Integrating rigid body dynamics on infinite-dimensional spaces of operators.
method Introducing pseudo-Riemannian metrics and adapting classical integrability theory.
result Existence of geodesics and integrals of motion for the rigid body equations.
Using the basic Lie symmetry method, we find the most general Lie point symmetries group of the ∇u=f(u) Poisson's equation, which has a subalgebra isomorphic to the 3−dimensional special Euclidean group SE(3) or group of rigid motions of R3. Looking the adjoint representation of ${\rm SE}(3)…
We show that timelike maximal cylinders in $\RR^{1 + 2}$ always develop singularities in finite time and that, infinitesimally at a generic singularity, their time slices are evolved by a rigid motion or a self-similar motion. We also prove a mild generalization in non-flat backgrounds.
In this paper, we continue studying the 6-dimensional pseudo-Riemannian space V^6(g_{ij}) with signature [++--], which admits projective motions, i. e. continuous transformation groups preserving geodesics. In particular, we determine a necessary and sufficient condition that the 6-dimensional rigid h-spaces have const…
We prove that Ricci flows with almost maximal extinction time must be nearly round, provided that they have positive isotropic curvature when crossed with R2. As an application, we show that positively curved metrics on S3 and RP3 with almost maximal width must be nearly round.
We propose a unified computational framework for the problem of deformation and rigidity of submanifolds in a homogeneous space under geometric constraint. A notion of 1-rigidity of a submanifold under admissible deformations is introduced. It measures how a deformation deviates from a one parameter family of motions u…
We study the boundary rigidity problem with partial data consisting of determining locally the Riemannian metric of a Riemannian manifold with boundary from the distance function measured at pairs of points near a fixed point on the boundary. We show that one can recover uniquely and in a stable way a conformal factor …
Study proves rigid spectral properties of planets with metric discontinuities.
problem Establishing spectral rigidity for spherically symmetric planets with discontinuities.
method Novel trace formula applied to two wave types in spherically symmetric manifolds with boundary and interior interfaces.
result Spectral rigidity of spherically symmetric planets with discontinuities is proven.
Paper derives and applies a parallel transport equation on Lie groups.
problem Efficiently solving parallel transport on Lie groups with left-invariant metrics.
method Derives a parallel transport equation in Lie algebra, applies it to SE(3), and compares to existing methods.
result Stable and efficient parallel transport implementation on Lie groups.
New approach for obstacle avoidance in robotics using learned representations.
problem Challenges in sensor-based motion planning for new and dynamic environments.
method Proposes a new obstacle representation using PointNet architecture trained jointly with policies for obstacle avoidance.
result Significant improvements in accuracy and efficiency compared to state of the art.
Paper shows surfaces with same SRNF but different shapes.
problem SRNF degeneracy in shape space.
method Introduced Square Root Normal Field (SRNF) to represent and compare surfaces.
result Examples of surfaces with same SRNF but different shapes.
Extended Regge complex for linearized Riemann-Cartan geometry and cohomology.
problem Cohomology of the Regge complex in three dimensions.
method Constructing a discrete version of linearized Riemann-Cartan geometry on any triangulation.
result The cohomology of the Regge complex is isomorphic to the infinitesimal-rigid-body-motion-valued de~Rham cohomology.
The study examines spacelike hypersurfaces in Minkowski space with constant σn−1 curvature.
problem Characterizing spacelike hypersurfaces with constant σn−1 curvature in Minkowski space. method Analyzing hypersurfaces with bounded principal curvatures and proving properties of their convexity.
result Hypersurfaces with constant σn−1 curvature in Minkowski space are either convex or can be split into a product form. New surface area measures defined for ball-convex bodies, leading to entropy and inequalities.
problem Defining and analyzing surface area measures for ball-convex bodies.
method Introducing Lp relative surface areas, proving invariance and inequalities, and using geometric interpretations. result Established inequalities and a new notion of entropy for ball-convex bodies.
A description of continuous rigid motion compatible Minkowski valuations is established. As an application, we present a Brunn-Minkowski type inequality for intrinsic volumes of these valuations.
Maps between positively curved manifolds with non-increasing area are rigid.
problem Understanding maps between manifolds with positive curvature and non-increasing area.
method Exploring the graphical mean curvature flow and using Brendle's sphere theorem.
result Maps between certain positively curved manifolds are homotopy trivial, Riemannian submersion, local isometry, or isometric immersion.
Suppose curves are moving by curvature in a plane, but one embeds the plane in R3 and looks at the plane from an angle. Then circles shrinking to a round point would appear to be ellipses shrinking to an ``elliptical point,'' and the surface energy would appear to be anisotropic as would the mobility. The result of …