Method finds all cross caps formally isometric to a given one.
problem Identifying cross caps that are formally isometric to a given one.
method Finding cross caps with matching Taylor expansions of first fundamental forms.
result A countable family of intrinsic invariants recognizes formal isometry classes completely.
In four dimensions one can use the chiral part of the spin connection as the main object that encodes geometry. The metric is then recovered algebraically from the curvature of this connection. We address the question of how isometries can be identified in this "pure connection" formalism. We show that isometries are r…
For a closed Kähler manifold with a Hamiltonian action of a connected compact Lie group by holomorphic isometries, we construct a formal Frobenius manifold structure on the equivariant cohomology by exploiting a natural DGBV algebra structure on the Cartan model.
Researchers found a new 8D Taub-NUT-like metric using harmonic superspace.
problem Finding a new 8-dimensional Taub-NUT-like manifold.
method Used harmonic superspace formalism to derive the metric.
result Derived a new 8D Taub-NUT-like metric.
Abstract: Proves Tutte's sequence connection to complex space forms.
problem Relating algebra of isometry invariant valuations to combinatorics.
method Proves Fu's power series conjecture.
result Fu's power series conjecture is proven, linking algebra to combinatorics.
We construct N=2 supersymmetric nonlinear sigma models whose target spaces are tangent as well as cotangent bundles over the quadric surface Q^{n-2} = SO(n)/[SO(n-2)\times U(1)]. We use the projective superspace framework, which is an off-shell formalism of N=2 supersymmetry.
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.
Let N be the space of Gaussian distribution functions over R, regarded as a 2-dimensional statistical manifold parameterized by the mean μ and the deviation σ. In this paper we show that the tangent bundle of N, endowed with its natural Kähler structure, is the Siegel-Jacobi space…
Reconstruct spacetime from order and number of points.
problem Reconstruct spacetime from chronological relations and i.i.d. samples.
method Relaxing hypotheses of Gromov reconstruction theorem, using random adjacency matrices and chronological relations.
result Spacetime can be recovered by only knowing 'order' and 'number' of its points.
Developed new Crofton formulas for pseudo-Riemannian spaces.
problem Computing volumes and curvature integrals in pseudo-Riemannian space forms.
method Introduced Crofton formulas using distributions and Alesker's Radon transform.
result Explicit Crofton formulas for all isometry-invariant valuations on pseudo-Riemannian spaces.
In Thurston's notes, he gives two different definitions of the Gromov norm (also called simplicial volume) of a manifold and states that they are equal but does not prove it. Gromov proves it in the special case of hyperbolic manifolds as a consequence of his proof that simplicial volume is proportional to volume. We g…
Lean 4 library formalizes mathematical finance, verifying over 200 theorems.
problem Formal verification of complex financial mathematics.
method Lean 4 proof assistant, Mathlib, BrownianMotion package, formal verification of over 200 theorems.
result Formal verification yields certified unification of known financial results.
Lean 4 library formalizes mathematical finance, verifying over 200 theorems.
problem Formal verification of complex financial mathematics.
method Lean 4 proof assistant, Mathlib, and BrownianMotion package.
result Formal verification yields certified unification of known results.
For singular metrics, there is no Quillen metric formalism on cohomology determinant. In this paper, we develop an admissible theory, with which the arithmetic Deligne-Riemann-Roch isometry can be established for singular metrics. As an application, we first study Weil-Petersson metrics and Takhtajan-Zograf metrics on …
We study n-ary commutative superalgebras and L∞-algebras that possess a skew-symmetric invariant form, using the derived bracket formalism. This class of superalgebras includes for instance Lie algebras and their n-ary generalizations, commutative associative and Jordan algebras with an invariant form. We…
The study classifies Heintze groups up to isometry and quasi-isometry in low dimensions.
problem Classifying Heintze groups up to isometry and quasi-isometry in low dimensions.
method Analyzing quasi-isometries and isometries of Heintze groups, applying existing tools to groups of dimension 4 and 5.
result Complete classification of simply connected solvable groups in dimension 4 and groups of polynomial growth in dimension 5 up to isometry.
This paper bridges Kahler geometry and quantum mechanics in lognormal statistical models.
problem Evolution of spectral curves in Siegel Jacobi space through Schrodinger equation.
method Kahler geometry induced on lognormal statistical manifold, Dombrowski's construction.
result Time-dependent Schrodinger equation with varying energy.
Isometries of cones embed in those of asymptotic shrinking Ricci solitons.
problem Characterizing isometries of asymptotically conical shrinking Ricci solitons.
method Analyzing the embedding of isometries from the cone's cross-section into the entire shrinker.
result Isometries of the cone's cross-section embed in the entire shrinker.
Characterizes and proves isometries in Kähler potential spaces.
problem Understanding isometries in Kähler potential spaces.
method Characterization and proof of local isometries.
result Existence and uniqueness of isometries proved.
Differential structure on partial isometries over Grassmannian constructed.
problem No specific problem stated; abstract focuses on method and result.
method Construction of differential structure on partial isometries over restricted Grassmannian.
result Set of partial isometries over restricted Grassmannian becomes a Banach Lie groupoid.
Lifts isometries in orbit spaces for compact groups.
problem Isometries in orbit spaces of compact groups.
method Equivariant isometry of original Euclidean space.
result Simple formula for connected component of isometry group.
Recently, a metric construction for the Calabi-Yau 3-folds from a four-dimensional hyperkahler space by adding a complex line bundle was proposed. We extend the construction by adding a U(1) factor to the holomorphic (3,0)-form, and obtain the explicit formalism for a generic hyperkahler base. We find that a discrete c…
Sharp stability of isometries on Heisenberg group proven.
problem Quantitative stability of isometries on the Heisenberg group.
method Proving quasi-isometries close to isometries with specific closeness orders.
result Quasi-isometries of Heisenberg group are close to isometries with specific closeness orders.
Study reveals structure of isometry group for specific manifolds.
problem Understanding the isometry group of non-compact, homogeneous manifolds.
method Analyzes non-compact, homogeneous manifolds with immortal Ricci flows.
result Establishes structure result for isometry group.
Study on holomorphic isometries between complex domains, revealing geometric properties.
problem Characterizing holomorphic isometries between bounded symmetric domains.
method Analyzing holomorphic isometries between complex unit ball and other bounded symmetric domains, using classical results for complex-analytic subvarieties of Stein manifolds.
result Images of holomorphic isometries have specific geometric properties, including intersections with affine-linear subspaces.
Quantum isometry groups exist for certain metric spaces.
problem Existence of quantum isometry groups for specific metric spaces.
method Proved existence for geodesic metrics and uniformly distributed measure spaces.
result Quantum isometry groups are classical (commutative) for Riemannian manifolds.
Study finds all isometries for specific Lie groups.
problem Identifying isometry groups in nonunimodular Lie groups.
method Examined left-invariant Riemannian metrics on Lie groups of dimension four.
result Determined full group of isometries for each metric.
Explicit isometry groups found for nearly Kähler manifolds.
problem Understanding the symmetries of nearly Kähler manifolds.
method Alternative, less algebraic approach to find isometry groups.
result Explicit expression for isometry groups of six-dimensional nearly Kähler manifolds.
Compact Lie groups have compact isometry groups with pseudo-Riemannian metrics.
problem Characterizing isometry groups of compact Lie groups with pseudo-Riemannian metrics.
method Analyzing left-invariant pseudo-Riemannian metrics on compact Lie groups.
result Isometry groups of compact Lie groups are compact.
Proves linear extension of isometries in smooth 2D Banach spaces.
problem Linear extension of isometries in absolutely smooth 2D Banach spaces.
method Analyzes isometries between unit spheres of smooth Banach spaces.
result Any isometry extends to a linear isometry of Banach spaces.
Proper holomorphic isometries between Bergman domains are biholomorphisms.
problem Characterizing isometries between Bergman domains.
method New method from Information Geometry.
result Proper holomorphic local isometries are biholomorphisms.
Every element of PU(2,1) can be decomposed into at most 4 special elliptic isometries.
problem Understanding the length of elements in PU(2,1) relative to special elliptic isometries.
method Generalizing the involution length of the complex hyperbolic plane, calculating the α-length of PU(2,1) and describing decompositions of isometries. result Every element of PU(2,1) can be decomposed into at most 4 special elliptic isometries.
Computes Weyl group of Kähler toric manifold isometries.
problem Computing the Weyl group of Kähler toric manifold isometries.
method Analyzes the group of holomorphic isometries of a Kähler toric manifold with real analytic Kähler metric.
result Computed the Weyl group of the group of holomorphic isometries.
Maps preserve distances in non-positively curved spaces.
problem Understanding continuous maps between boundaries of Hadamard manifolds.
method Circumcenter extension maps, cross ratio preservation, visibility conditions.
result Circumcenter extension maps are rough isometries under certain conditions.
Study of isometries in spacetimes without observer horizons.
problem Understanding the symmetries of spacetimes without specific boundaries.
method Analysis of isometry groups in causal spacetimes without observer horizons.
result The group of time orientation-preserving isometries acts properly on the spacetime.
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.
Characterizes self-isometries of Riemannian metrics on compact manifolds.
problem Understanding isometries of Riemannian metrics.
method Characterization of self-isometries and proof of isometry conditions.
result Two Riemannian metric spaces are isometric if and only if their manifolds are diffeomorphic.
We study the rigidity of complete, embedded constant mean curvature surfaces in R^3. Among other things, we prove that when such a surface has finite genus, then intrinsic isometries of the surface extend to isometries of R^3 or its isometry group contains an index two subgroup of isometries that extend.
For real hyperbolic spaces, the dynamics of individual isometries and the geometry of the limit set of nonelementary discrete isometry groups have been studied in great detail. Most of the results were generalised to discrete isometry groups of simply connected Riemannian manifolds of pinched negative curvature. For sy…
The study defines conditions for Finsler spacetime structures in (α,β)-metrics and identifies their isometries.
problem Conditions for Finsler spacetime structures in (α,β)-metrics. method Established necessary and sufficient conditions for Finsler spacetime structures.
result Identified (α,β)-Finsler spacetimes and determined the relation between isometries of (α,β)-metrics and the underlying pseudo-Riemannian metric. The paper defines quasi-isometry for almost contact metric manifolds and explores its properties.
problem Understanding quasi-isometry in almost contact metric manifolds.
method Definition and study of quasi-isometry for almost contact metric manifolds.
result Established a relation between scalar curvature and quasi-isometric constants.
Maximal isometries define canonical connections in sub-Riemannian spaces.
problem Characterizing sub-Riemannian spaces with maximal isometry groups.
method Study of sub-Riemannian spaces with maximal isometry groups and canonical connections.
result Canonical connections generalize Levi-Civita connections and have more invariants.
In this paper we develop a complete theory of factorization for isometries of hyperbolic 4-space. Of special interest is the case where a pair of isometries is linked, that is, when a pair of isometries can be expressed each as compositions of two involutions, one of which is common to both isometries. Here we develop …
Study describes isometry groups of specific Lie groups.
problem Understanding isometry groups of compact Lie groups.
method Analyzing bi-invariant metrics on SO(4) and U(n).
result Full groups of isometries identified for specific Lie groups.
The study shows finite measure-preserving isometry groups for certain metric measure spaces.
problem Understanding the structure of isometry groups in metric measure spaces.
method Analyzing synthetic negative Ricci curvature and Bakry-Émery Ricci curvature.
result The measure-preserving isometry group is finite for compact metric measure spaces with specific curvature conditions.
Study groups of piecewise isometries in tessellations of Euclidean space.
problem Understanding the structure of groups formed by cutting and gluing tessellations.
method Proving structure results about groups of piecewise isometries of tessellations, including elementary amenability.
result Groups of piecewise isometries of tessellations are elementary amenable.
Using an extension to isometries of the associated Sasaki structure, we establish a Lie transformation group structure for the set of isometries of a pseudo-Finsler conical metric.
New research shows uncountably many quasi-isometry classes of groups of type FP.
problem Identifying distinct quasi-isometry classes of groups of type FP. method Constructing uncountable families of groups and proving quasi-isometry classes.
result Uncountably many quasi-isometry classes of groups of type FP.