Study on conjugate connections and statistical structures on specific manifolds.
problem Understanding connections and structures on almost Norden manifolds.
method Investigation of conjugate connections with respect to Norden metrics and almost complex structure.
result Obtained conjugate connections of Levi-Civita connections induced by Norden metrics.
Characterizes connections on multivariate normal distributions.
problem Characterizing connections on statistical manifold of multivariate normal distributions.
method Analyzes statistical manifold (N,gF,ablaA,ablaA∗) of multivariate normal distributions. result The Amari-Chentsov connection ablaA is characterized by conjugate symmetry. The paper characterizes metallic pseudo-Riemannian manifolds using conjugate connections and tensor structures.
problem Characterizing metallic pseudo-Riemannian manifolds.
method Using conjugate connections and tensor structures, the paper derives new characterizations and conditions for these manifolds.
result A necessary and sufficient condition for a non-integrable metallic pseudo-Riemannian manifold to be a quasi metallic pseudo-Riemannian manifold is derived.
Study on conjugate points in a C∗-algebra's Grassmann manifold.
problem Characterizing conjugate points in a C∗-algebra's Grassmann manifold. method Analyzes connections and geodesics in the Riemannian metric induced by the Killing form.
result Points that are tangent conjugate in the classical setting may not be conjugate in a C∗-algebra's Grassmann manifold. Properties of pairs of product conjugate connections are stated with a special view towards the integrability of the given almost product structure. We define the analogous in product geometry of the structural and the virtual tensors from the Hermitian geometry and express the product conjugate connections in terms of…
The paper explores connections and curvature tensors on specific geometric manifolds.
problem Investigating properties of connections and curvature tensors on almost anti-Hermitian manifolds.
method Introduced three types of conjugate connections and proved a Klein group result.
result Derived a necessary and sufficient condition for an anti-Kähler structure.
The quotients Y=X/conj by the complex conjugation conjX→X for complex rational and Enriques surfaces X defined over R are shown to be diffeomorphic to connected sums of $\barCP2$, whenever Y are simply connected.
New connections found on zero-mean multivariate normal distributions.
problem Characterizing statistical connections on zero-mean multivariate normal distributions.
method Investigating invariant conjugate symmetric statistical connections on the submanifold of zero-mean multivariate normal distributions.
result Invariant connections on zero-mean multivariate normal distributions are not uniquely characterized by invariance under the general linear group action.
Study local limits of subgroups in SL3(R).
problem Understanding subgroups of SL3(R) in a specific topology. method Conjugation and Chabauty topology analysis.
result Described local limits of all connected subgroups.
Study cyclic group actions on specific high-dimensional manifolds.
problem Classify smooth actions of cyclic groups on certain high-dimensional manifolds.
method Analyzes smooth orientation-preserving actions of Z/m on (n−1)-connected 2n-manifolds. result Classifications up to smooth conjugation for specific cases of n and m. Improved method for numerical conformal mappings on complex domains.
problem Accurate and efficient computation of conformal mappings on multiply connected domains.
method Generalization and refinement of the conjugate function method using high-order finite element methods.
result Achieved accurate and efficient construction of boundary values for multiply connected domains.
Condition for Anosov flows on non-compact manifolds without conjugate points.
problem Conditions for Anosov flows on non-compact manifolds.
method Formulated a condition for complete, connected and non-compact Riemannian manifolds.
result No conjugate points in geodesic flow if Anosov.
Given two points of a Generalized Robertson-Walker spacetime, the existence, multiplicity and causal character of geodesic connecting them is characterized. Conjugate points of such geodesics are related to conjugate points of geodesics on the fiber, and Morse-type relations are obtained. Applications to bidimensional …
In this article, we show the existence of conjugations on many simply-connected spin 6-manifolds with free integral cohomology. In a certain class the only condition on X^6 to admit a conjugation with fixed point set M^3 is the obvious one: the existence of a degree-halving ring isomorphism between the Z_2-cohomologies…
Abelian subgroups in certain spaces are simple.
problem Characterizing Abelian subgroups in specific geometric spaces.
method Analyzing the asymptotic norm of Abelian subgroups.
result Each Abelian subgroup is isomorphic to Zk. Support theorem proved for X-ray transform on certain non-compact manifolds.
problem Support theorem for X-ray transform on non-compact manifolds with conjugate points.
method Use of plane covers and support theorem for simple manifolds by Krishnan.
result Support theorem proved for simply connected 2-step nilpotent Lie groups and some non-homogeneous 3D manifolds.
We give a positive answer to the Chavel's conjecture [J. Diff. Geom. 4 (1970), 13-20]: a simply connected rank one normal homogeneous space is symmetric if any pair of conjugate points are isotropic. It implies that all simply connected rank one normal homogeneous space with the property that the isotropy action is var…
The study examines Hopfian properties of conjugation quandles and their underlying groups.
problem Understanding the relationship between Hopfian properties of conjugation quandles and their underlying groups.
method Examined Hopfian and residual finiteness properties of conjugation quandles of specific groups.
result Conjugation quandles of Baumslag-Solitar groups are infinitely generated and not necessarily Hopfian.
The goals of this article are twofold : 1) to compute the conjugate locus of a geodesic that lies in the center of a simply connected, 2-step nilpotent Lie group with a left invariant metric 2) compare the isometry types of two such nilpotent Lie groups whose conjugate loci for central geodesics are "the same" in a sui…
Quotients Y=X/conj by the complex conjugation conjX→X for complex surfaces X defined over R tend to be completely decomposable when they are simply connected, i.e., split into connected sums $\#_n CP^2\#_m\barCP^2$ if w2(Y)=0, or into #n(S2×S2) if w2(Y)=0. The author proves this prope…
The study describes good involutions in quandles and Alexander quandles.
problem Characterizing and enumerating good involutions in quandles and Alexander quandles.
method Completely describing good involutions of free and subquandles of twisted conjugation quandles of groups, including Alexander quandles.
result Explicit mappings for good involutions of linear quandles up to order 23.
The paper introduces statistical and geometric structures on anti-commutable pre-Leibniz algebroids.
problem Generalizing differential geometric structures to algebroids.
method Introducing statistical, conjugate connection, and Hessian structures on anti-commutable pre-Leibniz algebroids.
result Statistical and conjugate connection structures are equivalent for admissible connections.
Let G be a compact, simply connected Lie group. If C1,C2 are two G-conjugacy classes, then the set of elements in G that can be written as products g=g1g2 of elements gi∈Ci is invariant under conjugation, and its image under the quotient map $G\to G/\operatorname{Ad}(G…
Groups of contact transformations are uniformly simple if supporting open books are flexible.
problem Understanding the algebraic and geometric properties of contact transformation groups.
method Exploring the connection between contact transformations, open book decompositions, and flexible Weinstein manifolds.
result The connected component of the identity in the automorphism group of a closed contact manifold is a uniformly simple group.
Study of Dirac operator with chiral boundary conditions on spin manifolds.
problem Reconstructing metrics and connections from boundary data.
method Defining boundary conjugation map and showing its symbolic determination.
result Reconstruction of Riemannian manifolds and spin structures from boundary data.
The study examines continuous mean curvature functions on manifolds without conjugate points.
problem Understanding properties of manifolds with specific curvature functions.
method Analyzing simply connected Riemannian manifolds with continuous horospherical mean curvature functions.
result Compact rank one manifolds without conjugate points are locally symmetric spaces of negative curvature.
Classifies good involutions in conjugation subquandles and racks.
problem Classifying quandles with good involutions for applications in surface-knot theory.
method Study of subquandles of conjugation quandles, including core quandles; analysis of good involutions of faithful racks.
result Sharp bounds on the number of good involutions of racks in these families.
Quandle coloring detects causality in spacetime links.
problem Detecting causality in spacetime links using quandle colorings.
method Conjugation quandle of dihedral group D5 combined with Alexander-Conway polynomial.
result The conjugation quandle over D5 distinguishes causally related and unrelated events.
We prove that any non-simply connected planar domain can be properly and minimally embedded in H^2 x R. The examples that we produce are vertical bi-graphs, and they are obtained from the conjugate surface of a Jenkins-Serrin graph.
The paper proposes a new method for updating neural network parameters using gradient conjugate priors.
problem Learning probability distributions of observed data using neural networks.
method Gradient conjugate prior (GCP) update for neural networks, connecting to log-likelihood maximization.
result The method differs from classical Bayesian updates, leading to different limiting behavior of prior parameters.
Paper shows mapping classes are largely determined by their finite quotient actions.
problem Understanding the equivalence of mapping classes based on their finite quotient actions.
method Analyzes procongruent conjugacy classes and their dependence on finite quotients.
result Procongruent conjugacy classes are largely determined by their finite quotient actions.
It is known that the bundle of Dirac spinors is produced as a direct sum of two bundles - the bundle of chiral spinors and its Hermitian conjugate bundle. In this paper some aspects of metric connections for chiral and Dirac spinors are resumed and their relation is studied.
The paper studies conjugating complex representations into real ones.
problem Understanding representations of surface groups into complex Lie groups.
method Analyzes representations of finitely generated groups into PGL(k,C) and determines conjugacy conditions. result Identifies representations in the larger variety that are conjugate in PGL(k,C) to a representation in PGL(k,R). We compute the equivariant K-theory KG∗(G) for a simply connected Lie group G (acting on itself by conjugation). We prove that KG∗(G) is isomorphic to the algebra of Grothendieck differentials on the representation ring. We also study a special example of a non-simply connected Lie group G, namely PSU(3),…
We prove a formula for the conjugation action on the knot Floer complex of the connected sum of two knots. Using the formula we construct a homomorphism from the smooth concordance group to an abelian group consisting of chain complexes with homotopy automorphisms, modulo an equivalence relation. Using our connected su…
The paper studies four Hermitian structures on twistor spaces.
problem Understanding the geometry of Hermitian surfaces via twistor spaces.
method Defining four almost Hermitian structures on twistor spaces using canonical connections.
result Relations between twistor space geometry and Hermitian surface geometry.
The topology of the orbit space, Y, for the action of the complex conjugation on a complex surface, X, defined over reals, is studied. I give a criterion for blow-up stable triviality of Y (which implies vanishing of its Seiberg-Witten invariants). The main result concerns the double planes branched along the com…
Geodesic flows on specific manifolds are structurally stable.
problem Stability of geodesic flows on compact manifolds without conjugate points.
method Analyzing the C∞ compact manifold (M,g) with quasi-convex universal covering and divergent geodesic rays. result Proved the C1-stability conjecture for geodesic flows of compact manifolds. The paper extends a method for numerical conformal mappings to surfaces using Laplace-Beltrami equations.
problem Computing conformal mappings between Riemannian surfaces.
method Adapting the conjugate function method to Riemannian surfaces using hp-adaptive finite element methods. result Highly accurate numerical computations of conformal mappings on surfaces, including complex geometries.
Let (M,g) be a complete, simply connected Riemannian manifold of dimension 3 without conjugate points. We show that M is a flat manifold, provided M is asymptotically harmonic of constant h=0.
The paper connects geodesic flows and limit sets on visibility manifolds.
problem Understanding dynamics and ergodic properties on non-compact visibility manifolds.
method Analyzing geodesic flows and Patterson-Sullivan measures on visibility manifolds without conjugate points.
result The positivity of the Patterson-Sullivan measure of the Myrberg limit set is equivalent to the conservativity of the geodesic flow.
Let (M,g) be a complete, simply connected Riemannian manifold of dimension 3 without conjugate points. We show that M is a hyperbolic manifold of constant sectional curvature, provided M is asymptotically harmonic of constant h > 0.
Holonomy for Lie subalgebroids defined via bisections.
problem Defining holonomy for Lie subalgebroids.
method Minimal integration and conjugation by bisections.
result Holonomy action for Lie subalgebroids extends to singular cases.
We give an explicit construction of any simply-connected superconformal surface φ:M2→R4 in Euclidean space in terms of a pair of conjugate minimal surfaces g,h:M2→R4. That φ is superconformal means that its ellipse of curvature is a circle at any point. We characterize the pairs (g,h) of …
A well-known Lemma in Riemannian geometry by Klingenberg says that if x0 is a minimum point of the distance function d(p,⋅) to p in the cut locus Cp of p, then either there is a minimal geodesic from p to x0 along which they are conjugate, or there is a geodesic loop at p that smoothly goes throu…
New manifolds found without interior conjugate points.
problem Existence of interior conjugate points in hyperbolic manifolds.
method Construction of non-trapping asymptotically hyperbolic manifolds.
result Found manifolds without interior conjugate points.
The paper classifies equivariant gerbe connections on Lie groups.
problem Classifying equivariant gerbe connections on Lie groups.
method Using Meinrenken's G-equivariant bundle gerbe connections and differential quotient stacks.
result Isomorphism classes of G-equivariant gerbe connections are classified by degree three differential equivariant cohomology.
The paper disproves a conjecture about isomorphic subgroups in finite groups.
problem The existence of non-isomorphic subgroups that are isomorphic in extensions of finite groups.
method Constructing extensions of finite groups to show non-isomorphic pre-images of subgroups.
result Subgroups of finite groups that are isomorphic in extensions are not conjugate.