The paper explores conditions for constructing and extending infinitesimal isometries on special sub-Riemannian manifolds.
problem Finding conditions for infinitesimal isometries on special sub-Riemannian manifolds.
method Introducing $\is^*$-regular and $\is$-regular points to construct and extend infinitesimal isometries.
result Conditions on special sub-Riemannian manifolds allow for the construction and extension of infinitesimal isometries.
The paper proves properties of strain tensors on surfaces with changing Gauss curvature.
problem Regularity of solutions to strain tensor equations on surfaces with variable Gauss curvature.
method Proof of regularity, density property, and matching property.
result Established matching property and density of smooth infinitesimal isometries.
A Jacobi field on a Riemannian manifold M is defined along a geodesic. We generalize this notion to an arbitrary smooth curve, and call it an infinitesimal isometry along the curve. We give two approaches to this: 1) compute the complete prolongation of the Killing equation and then restrict to the curve, and 2) comput…
Maps between certain Lipschitz manifolds are isometries if they preserve volume.
problem Volume preservation and isometry conditions for Lipschitz manifolds.
method Volume-preserving 1-Lipschitz maps from integral currents onto infinitesimally Euclidean Lipschitz manifolds.
result Volume-preserving maps are isometries under given conditions.
New concept of effective isometries for compliant shells.
problem Inadequate classification of isometric deformations for compliant shells.
method Introduce effective isometric deformations defined by first-order isometry in a small scale separation parameter.
result Effective isometries are solutions to a quasilinear second-order PDE.
Research explores Lorentzian distances on a specific geometric plane.
problem Investigating Lorentzian structures on a 2D geometric plane.
method Analyzes sectional curvature, attainable sets, and Lorentzian length maximizers.
result Describes distance properties and spheres in the context of Lorentzian geometry.
Let g be a real finite-dimensional Lie algebra equipped with a symmetric bilinear form ⟨⋅,⋅⟩. We assume that ⟨⋅,⋅⟩ is nil-invariant. This means that every nilpotent operator in the smallest algebraic Lie subalgebra of endomomorphims containing the adjoint repres…
Research explores Lie algebras in Riemannian manifolds.
problem Understanding Lie algebras in Riemannian manifolds.
method Analyzes the Lie algebra of infinitesimal isometries.
result Identifies two commutative ideals in the Lie algebra.
Let W be a compact hyperbolic n-manifold with totally geodesic boundary. We prove that if n>3 then the holonomy representation of pi_1 (W) into the isometry group of hyperbolic n-space is infinitesimally rigid.
Study Lie 2-group actions on Riemannian groupoids, proving existence and developing geometric Killing vector fields.
problem Understanding isometric actions of Lie 2-groups on Riemannian groupoids.
method Exhibit properties, prove existence, construct bi-invariant metrics, provide infinitesimal description.
result Existence of 2-equivariant Slice Theorem and Equivariant Tubular Neighborhood Theorem.
The study explores maximal symmetry in Ricci solitons on Lie groups.
problem Maximal symmetry in left-invariant Riemannian metrics and Ricci solitons.
method Analysis of left-invariant metrics and Ricci solitons on Lie groups, using tools from previous work on Einstein metrics.
result Expanding homogeneous Ricci solitons have maximal isometry algebras but not always maximal isometry groups.
Developed a new concept of isometric surfaces in isotropic space.
problem Missing notion of isometric deformations in isotropic space.
method Using Gauss' Theorema Egregium as a condition, developed a new concept.
result Natural analogues of Euclidean space isometries found in isotropic space.
New framework discovers non-affine continuous symmetries in neural networks.
problem Lack of efficient methods for detecting non-affine continuous symmetries in neural networks.
method Computational framework for discovering infinitesimal generators of multi-parameter group actions.
result Framework can discover non-affine continuous symmetries in neural networks.
Researchers extend parametrization of Margulis spacetimes using strip deformations.
problem Parametrize Margulis spacetimes with decorated horoballs.
method Use strip deformations to parametrize complete finite-area hyperbolic surfaces with spikes decorated with horoballs.
result Generalized parametrization of Margulis spacetimes with photons.
We study the Lie algebra of infinitesimal isometries on compact Sasakian and K--contact manifolds. On a Sasakian manifold which is not a space form or 3--Sasakian, every Killing vector field is an infinitesimal automorphism of the Sasakian structure. For a manifold with K--contact structure, we prove that there exists …
We establish a link between the holomorphic derivatives of Thurston's hyperbolic gluing equations on an ideally triangulated finite volume hyperbolic 3-manifold and the cohomology of the sheaf of infinitesimal isometries. Moreover, we provide a geometric reformulation of the non-abelian Reidemeister torsion correspondi…
The moduli space NK of infinitesimal deformations of a nearly Kähler structure on a compact 6-dimensional manifold is described by a certain eigenspace of the Laplace operator acting on co-closed primitive (1,1) forms. Using the Hermitian Laplace operator and some representation theory, we compute the space NK on all 6…
A new construction of naturally reductive spaces is presented. This construction gives a large amount of new families of naturally reductive spaces. First the infinitesimal models of the new naturally reductive spaces are constructed. A concrete transitive group of isometries is given for the new spaces and also the na…
A torsion-free G_2 structure admitting an infinitesimal isometry is shown to give rise to a 4-manifold equipped with a complex symplectic structure and a 1-parameter family of functions and 2-forms linked by second order equations. Reversing the process in various special cases leads to the construction of explicit met…
In this work we investigate solvable and nilpotent Lie groups with special metrics. The metrics of interest are left-invariant Einstein and algebraic Ricci soliton metrics. Our main result shows that the existence of a such a metric is intrinsic to the underlying Lie algebra. More precisely, we show how one may determi…
Let V be a finite dimensional vector space over a field F of characteristic different from 2, and let Q be a nondegenerate, symmetric, bilinear form on V. Let Cℓ(V,Q) be the Clifford algebra determined by V and Q. The bilinear form Q extends in a natural way to a nondegenerate, symmetric, bilinear fo…
Theory of symmetric rigidity in hyperbolic geometry.
problem Symmetric rigidity in hyperbolic frameworks.
method Gain graphs and orbit rigidity matrix, matroidal sparsity conditions.
result Characterization of infinitesimal rigidity for Gamma-symmetric frameworks.
Shells resist three out of six possible loads if simply connected.
problem Understanding the load resistance of shells.
method Formal mathematical analysis of shell strains and deflections.
result The space of strains is three-dimensional for simply-connected shells.
A three-dimensional Riemannian manifold has locally 6, 4, 3, 2, 1 or none independent Killing vectors. We present an explicit algorithm for computing dimension of the infinitesimal isometry algebra. It branches according to the values of curvature invariants. These are relative differential invariants computed via curv…
Study of motion control systems on Lie groups with specific geometric constraints.
problem Controlling motion systems on Lie groups with geometric constraints.
method Analysis of control systems on Lie groups, focusing on infinitesimal roto-translations and geodesics.
result Explicit geodesics found for the sub-Riemannian structure on the Lie group.
Study on symmetries of quaternionic Kähler manifolds with S^1-symmetry.
problem Understanding symmetries in quaternionic Kähler manifolds.
method Associate central extensions of Lie algebras to infinitesimal automorphisms and study one-loop deformed metrics.
result One-loop deformed metrics increase cohomogeneity by at most one.
The paper studies hybrid connections on Hessian manifolds and their properties.
problem Investigating hybrid connections on Hessian manifolds.
method Defining and analyzing hybrid connections as incompressible affine connections projective to a flat connection D. result The difference abla−D is determined by the logarithmic differential of a Hessian potential function. We describe a family of 4-dimensional hyperbolic orbifolds, constructed by deforming an infinite volume orbifold obtained from the ideal, hyperbolic 24-cell by removing two walls. This family provides an infinite number of infinitesimally rigid, infinite covolume, geometrically finite discrete subgroups of the isometry…
We investigate the local deformation space of 3-dimensional cone-manifold structures of constant curvature κ∈{−1,0,1} and cone-angles ≤π. Under this assumption on the cone-angles the singular locus will be a trivalent graph. In the hyperbolic and the spherical case our main result is a vanishing theorem fo…
Apart from global topological problems an affine homogeneous space is locally described by its curvature, its torsion and a slightly less tangible object called its connection in a given base point. Using this description of the local geometry of an affine homogeneous space we construct an algebraic variety $\mathfrak{…
Having developed a description of indefinite extrinsic symmetric spaces by corresponding infinitesimal objects in the preceding paper we now study the classification problem for these algebraic objects. In most cases the transvection group of an indefinite extrinsic symmetric space is not semisimple, which makes the cl…
We study the neutral Kähler metric on the space of time-like lines in Lorentzian E13, which we identify with the total space of the tangent bundle to the hyperbolic plane. We find all of the infinitesimal isometries of this metric, as well as the geodesics, and interpret them in terms of the Lorentzian metr…
The construction (by Kapranov) of the space of infinitesimal paths on a manifold is extended to include higher dimensional infinitesimal objects, encoding contractions of infinitesimal loops. This full infinitesimal groupoid is shown to have the algebra of polyvector fields as its non-linear cohomology.
Study on bending knots and energy changes in 3D space.
problem Understanding energy changes in knots under small deformations.
method Analyzes infinitesimal bending of knots and energy changes using Willmore and Mobius energies.
result Changes in energy under small deformations of knots have been quantified.
In this paper, we establish an infinitesimal equivariant index formula in the noncommutative geometry framework using Greiner's approach to heat kernel asymptotics. An infinitesimal equivariant index formula for odd dimensional manifolds is also given. We define infinitesimal equivariant eta cochains, prove their regul…
Cauchy used infinitesimals in differential geometry and integral geometry.
problem Applying infinitesimals in differential and integral geometry.
method Using infinitesimals as numbers in differential and integral geometry.
result Valid application of infinitesimals in geometric probability, differential geometry, elasticity, and Dirac delta functions.
Metric spaces with certain curvature properties are universally infinitesimally Hilbertian.
problem Analyzing the infinitesimal geometry of metric spaces with curvature bounds.
method Proving a metric space with a Gromov-Hausdorff tangent splitting property is universally infinitesimally Hilbertian.
result Metric spaces with curvature bounds are universally infinitesimally Hilbertian.
Weiss and, independently, Mazzeo and Montcouquiol recently proved that a 3--dimensional hyperbolic cone-manifold (possibly with vertices) with all cone angles less than 2π is infinitesimally rigid. On the other hand, Casson provided 1998 an example of an infinitesimally flexible cone-manifold with some of the cone an…
In this paper, we prove that infinitesimal equivariant Chern-Connes characters are well-defined. We decompose an equivariant index as a pairing of infinitesimal equivariant Chern-Connes characters with the Chern character of an idempotent matrix. We compute the limit of infinitesimal equivariant Chern- Connes character…
Paper generalizes connections between Lie groups and affine connections.
problem Exploring properties of infinitesimal groups and affine connections.
method Introducing second-order infinitesimal groups and using them to define Lie brackets and connections.
result Generalized correspondence between symmetric and non-symmetric affine connections.
Ancient formula connects volume forms and infinitesimal square volumes in manifolds.
problem Relating volume forms and infinitesimal square volumes in Riemannian manifolds.
method Uses Heron's formula to link these concepts.
result Established a connection between volume forms and infinitesimal square volumes.
New examples of rigid Lie foliations with dense leaves found.
problem Infinitesimal rigidity of Lie foliations with dense leaves.
method Construction of specific Lie foliations.
result First examples of infinitesimally rigid Riemannian foliations with dense leaves.
Smooth manifolds have been always understood intuitively as spaces with an affine geometry on the infinitesimal scale. In Synthetic Differential Geometry this can be made precise by showing that a smooth manifold carries a natural structure of an infinitesimally affine space. This structure is comprised of two pieces o…
Lie algebroids are like infinitesimal Lie groupoids.
problem None explicitly stated in the abstract.
method Overview article.
result Lie algebroids are infinitesimal counterparts of Lie groupoids.
New classification of hypersurfaces with conformal variations.
problem Classifying hypersurfaces with conformal infinitesimal variations.
method Analyzing hypersurfaces in conformal geometry, extending previous work by Cartan and Sbrana.
result The class of hypersurfaces with conformal infinitesimal variations is larger than previously known.
Using standard analysis only, we present an extension ∙R of the real field containing nilpotent infinitesimals. On the one hand we want to present a very simple setting to formalize infinitesimal methods in Differential Geometry, Analysis and Physics. On the other hand we want to show that these infinitesim…
We study infinitesimal conformal deformations of a triangulated surface in Euclidean space and investigate the change in its extrinsic geometry. A deformation of vertices is conformal if it preserves length cross-ratios. On one hand, conformal deformations generalize deformations preserving edge lengths. On the other h…
We study infinitesimal Einstein deformations on compact flat manifolds and on product manifolds. Moreover, we prove refinements of results by Koiso and Bourguignon which yield obstructions on the existence of infinitesimal Einstein deformations under certain curvature conditions.