This paper connects Salem numbers and totally real fields to Thurston's construction of pseudo-Anosov maps.
problem Understanding which algebraic units can be stretch factors of pseudo-Anosov maps.
method Using Thurston's construction, the paper shows that every Salem number and every totally real field can be represented as stretch factors of pseudo-Anosov maps.
result Every Salem number and every totally real field can be represented as stretch factors of pseudo-Anosov maps arising from Thurston's construction.
We define a `Higgs field' for a four-dimensional spinc-manifold to be a smooth section of its positive half-spinor bundle, transverse to the zero section, and defined only up to a positive functional factor. This is intended to be a generalization of almost complex structures on real four-manifolds, each of which ma…
In this paper we give a characterization of real hypersurfaces in noncompact complex two-plane Grassmannian SU2,m/S(U2Um), m≥2 with Reeb vector field ξ belonging to the maximal quaternionic subbundle Q. Then it becomes a tube over a totally real totally geodesic HHn, m=2n, in …
Study on PMC surfaces in complex space forms, linking biconservative and totally real properties.
problem Characterizing PMC surfaces in complex space forms and their properties.
method Analyzing interactions between PMC, totally real, and biconservative properties; proving rigidity and reduction codimension results.
result PMC surfaces in non-flat complex space forms are biconservative if and only if totally real.
The paper extends arithmetic Chern-Simons invariants to real quadratic fields and calculates mod 2 Dijkgraaf-Witten invariants.
problem Calculating arithmetic Dijkgraaf-Witten invariants for real quadratic number fields.
method Using modified étale cohomology groups and fundamental groups, explicit formulas are derived for real quadratic fields.
result Explicit formulas for mod 2 arithmetic Dijkgraaf-Witten invariants for real quadratic fields are provided.
Algorithm finds real-analytic Legendrian representatives for every link type.
problem Finding explicit expressions for Legendrian representatives and Bateman fields.
method Algorithm based on trigonometric polynomials and solving linear equations.
result No compact subset of R^3 can contain an electromagnetic knot indefinitely.
Defines flag structures on real 3-manifolds and proves null curvature models.
problem Characterizing and classifying real 3-manifolds with specific geometric structures.
method Introduces flag structures, constructs adapted connections, and defines invariants.
result Null curvature models are given by totally real submanifolds in flag space.
Totally umbilical hypersurfaces in Spin^c manifolds with special spinors have constant mean curvature.
problem Characterizing totally umbilical hypersurfaces in Spin^c manifolds with specific spinor fields.
method Proving constant mean curvature for hypersurfaces carrying parallel, real or imaginary Killing spinors.
result Results extend to Spin^c case, generalizing O. Kowalski's theorem.
We present a new equation with respect to a unit vector field on Riemannian manifold Mn such that its solution defines a totally geodesic submanifold in the unit tangent bundle with Sasaki metric and apply it to some classes of unit vector fields. We introduce a class of covariantly normal unit vector fields and pro…
We classify real hypersurfaces in complex space forms with constant principal curvatures and whose Hopf vector field has two nontrivial projections onto the principal curvature spaces. In complex projective spaces such real hypersurfaces do not exist. In complex hyperbolic spaces these are holomorphically congruent to …
We prove that a ``bouillabaisse'' surface (translation surface which has two transverse parabolic elements) has totally real trace field. As a corollary, non trivial Veech groups which have no parabolic elements do exist. The proof follows Veech's viewpoint on Thurston's construction of pseudo-Anosov diffeomorphisms.
The paper classifies A-principal Hopf hypersurfaces in complex quadrics.
problem Characterizing A-principal Hopf hypersurfaces in complex quadrics. method Analyzing the singularities of the unit normal vector field and using geometric properties.
result A A-principal Hopf hypersurface in Qm is an open part of a tube around a totally geodesic Qm+1 in Qm. We give a complete list of those left invariant unit vector fields on three-dimensional Lie groups with the left-invariant metric that generate a totally geodesic submanifold in the unit tangent bundle of a group with the Sasaki metric. As a result, each class of three-dimensional Lie groups admits the totally geodesic…
The purpose of this note is to establish the following theorem: Let N be a Kahler manifold, L be a compact oriented immersed minimal Lagrangian submanifold in N and V be a holomorphic vector field in a neighbourhood of L in N. Let div(V) be the (complex) divergence of V. Then the integral of div(V) over L is 0. Vice ve…
New Fuchsian groups found with special embedding properties.
problem Finding new Fuchsian groups with specific embedding properties.
method Using period domains and properties of complex hyperbolic surfaces.
result First cocompact nonarithmetic Fuchsian groups with modular embedding not commensurable with triangle groups.
It is shown that the sum of class numbers of orders in totally complex quartic fields with no real quadratic subfield obeys an asymptotic law similar to the prime numbers, as the bound on the regulators tends to infinity. Here only orders which are maximal at a given set of primes containing an even number of elements …
We obtain bounds on the least dimension of an affine space that can contain an n-dimensional submanifold without any pairs of parallel or intersecting tangent lines at distinct points. This problem is closely related to the generalized vector field problem, non-singular bilinear maps, and the immersion problem for re…
Following the previous work of Nikulin and Agol, Belolipetsky, Storm, and Whyte it is known that there exist only finitely many (totally real) number fields that can serve as fields of definition of arithmetic hyperbolic reflection groups. We prove a new bound on the degree nk of these fields in dimension 3: nk d…
Let Γ be a nonelementary discrete subgroup of SU(n,1) or Sp(n,1). We show that if the trace field of Γ is contained in R, Γ preserves a totally geodesic submanifold of constant negative sectional curvature. Furthermore if Γ is irreducible, Γ is a Zariski dense irreducible discrete subgroup of SO(n,1…
We give a full geometrical description of local totally geodesic unit vector field on Riemannian 2-manifold, considering the field as a local imbedding of the manifold into its unit tangent bundle with the Sasaki metric.
Using the Arthur-Selberg trace formula we express the index of a Dirac operator on an arithmetic quotient over a totally real field with at least two real embeddings as the integral over the index form plus a sum of orbital integrals. For the Euler operator these orbital integrals are shown to vanish for products of ra…
We study curvature restrictions of Levi-flat real hypersurfaces in complex projective planes, whose existence is in question. We focus on its totally real Ricci curvature, the Ricci curvature of the real hypersurface in the direction of the Reeb vector field, and show that it cannot be greater than -4 along a Levi-flat…
Constructs pseudo-Anosov homeomorphisms using positive twists.
problem Creating explicit examples of pseudo-Anosov maps with specific properties.
method Using positive half-twists to construct pseudo-Anosov homeomorphisms on n-times punctured spheres and surfaces with higher genus.
result Produces examples of pseudo-Anosov maps with non-trivial trace fields and unit circle Galois conjugates.
We prove that the Hopf vector field is a unique one among geodesic covariantly normal unit vector fields on spheres such that the submanifold generated by the field is totally geodesic in the unit tangent bundle with Sasaki metric. As application, we give a new proof of stability (instability) of the Hopf vector field …
By attaching a Lie algebra of germs of analytic vector fields to every point of a (real or complex) analytic variety V we construct the Nagano foliation of the variety. We prove that the Nagano foliation of V is a stratification. The treatment of the subject is totally coordinate free but relies on the Oka-Cartan-Serre…
We show that for each genus there are only finitely many algebraically primitive Teichmueller curves C, such that i) C lies in the hyperelliptic locus and ii) C is generated by an abelian differential with two zeros of order g-1. We prove moreover that for these Teichmueller curves the trace field of the affine group i…
In the paper we investigate submanifolds in a tangent bundle endowed with g-natural metric G, defined by a vector field on a base manifold. We give a sufficient condition for a vector field on M to defined totally geodesic submanifold in (TM,G). The parallel vector field is discussed in more detail.
Study on totally real submanifolds in (LCS)n-manifolds.
problem Characterizing properties of totally real submanifolds in (LCS)n-manifolds. method Analysis of submanifolds with respect to Levi-Civita and quarter symmetric metric connections.
result Scalar curvature of C-totally real submanifolds is same for both connections.
New classifications of totally real surfaces in nearly Kähler C⁴.
problem Classifying totally real surfaces in nearly Kähler C⁴.
method Investigation under various assumptions, including extrinsic homogeneity, minimality, total umbilicity, and Codazzi-like properties.
result Multiple new examples of totally real surfaces, including parallel and non-parallel Codazzi-like cases.
We consider a unit normal vector field of (local) hyperfoliation on a given Riemannian manifold as a submanifold in the unit tangent bundle with Sasaki metric. We give an explicit expression of the second fundamental form for this submanifold and a rather simple condition its totally geodesic property in the case of a …
We prove that a Riemannian product of type M x R (where R denotes the Euclidean line) admits totally umbilical hypersurfaces if and only if M has locally the structure of a warped product and we give a complete description of the totally umbilical hypersurfaces in this case. Moreover, we give a necessary and sufficient…
The study defines fields of definition for triangle groups as Fuchsian groups.
problem Characterizing the fields of definition for triangle groups as Fuchsian groups.
method Analyzing the trace field and properties of compact hyperbolic triangle groups.
result Exactly eleven compact hyperbolic triangle groups are conjugate to subgroups of \(\mathrm{PSL}_2(K)\) where \(K\) is a specific field.
P.Lecomte has proposed to take into account the covariant derivatives used to build ordering prescriptions for the naturality of transformation properties and has conjectured that there exists an natural ordering prescription for differential operators of any orders between density bundles which in addition is invarian…
Study foliations on Riemannian manifolds with specific vector fields, focusing on geometric properties.
problem Investigate foliations transverse to closed conformal vector fields on Riemannian manifolds.
method Analyze conditions for totally geodesic leaves and geometric constraints on foliations.
result Characterize totally geodesic foliations and classify minimal and constant mean curvature foliations.
This paper provides an explicit formula for complex structures in embeddings of manifolds.
problem Finding explicit formulas for complex structures in embeddings of manifolds.
method Recursive expression and fiberwise Taylor expansion of the canonical complex structure.
result Evidence of canonical vanishing of integrability equations in general settings.
In an earlier paper we discussed soldered forms, multivector fields and Riemannian metrics. In particular, we showed that a Riemannian submanifold is totally geodesic iff the metric is soldered to the submanifold. In the present note we discuss general, soldered tensor fields. In particular, we prove that the almost co…
We show the non-vanishing of cohomology groups of sufficiently small congruence lattices in SL(1,D), where D is a quaternion division algebras defined over a number field E contained inside a solvable extension of a totally real number field. As a corollary, we obtain new examples of compact, arithmetic, hyperbol…
Review of image compressive sensing algorithms for beginners.
problem Efficiently processing images with limited data.
method Comprehensive review of Total variation methods and other algorithms.
result Standardized comparison of algorithms for compressive sensing applications.
The paper characterizes Clairaut conformal submersions on Ricci solitons.
problem Characterizing Clairaut conformal submersions on Ricci solitons.
method Calculating scalar and Ricci tensors, providing necessary conditions for fibres and base manifolds to be Ricci solitons and Einstein, and solving Poisson equations.
result Necessary and sufficient conditions for Clairaut conformal submersions to be harmonic.
Challenge to separate Earth's magnetic field from vehicle's magnetic field for accurate navigation.
problem Separate Earth's magnetic field from vehicle's magnetic field for accurate magnetic navigation.
method Use machine learning (ML) and integrate physics of magnetic navigation (SciML) to remove aircraft magnetic field from total magnetic field.
result A model can be constructed to effectively remove aircraft magnetic field from the dataset.
Study on totally real flat minimal surfaces in quaternionic projective space.
problem Characterizing the moduli space of totally real flat minimal immersions in HP^3.
method Analyzing the moduli space of linearly full totally real flat minimal immersions from C into HP^3.
result The moduli space has three components, each a 6-dimensional manifold.
Study shows convergence for mean curvature flow on almost minimal totally real submanifolds.
problem Mean curvature flow of totally real submanifolds.
method Quantitative analysis of almost minimal submanifolds.
result Established convergence result for mean curvature flow.
We show that totally real elliptic Lefschetz fibrations that admit a real section are classified by their "real loci" which is nothing but an S1-valued Morse function on the real part of the total space. We assign to each such real locus a certain combinatorial object that we call a \emph{necklace diagram}. On the o…
New inequality shows all special submanifolds in light cone are totally umbilical spheres.
problem Characterizing submanifolds with parallel mean curvature in Lorentz-Minkowski spacetime.
method Established an integral inequality and used it to derive a rigidity result.
result All compact submanifolds with parallel mean curvature in light cone are totally umbilical spheres.
The paper proves properties of Brauer equivalent number fields and applies them to geometric spaces.
problem Understanding properties of Brauer equivalent number fields.
method Analyzes Brauer groups and arithmetic locally symmetric spaces.
result Constructs incommensurable spaces with identical geodesic surfaces.
New method for mesh denoising using TGV of normal vector field.
problem Improving mesh quality by removing noise.
method Proposes a novel TGV formulation for normal vector fields on triangular meshes.
result New method outperforms existing techniques in mesh denoising experiments.
The following results are proved: Theorem 1. A totally real semiparallel submanifold of constant curvature with parallel f-structure in the normal bundle of a Kähler manifold N is flat or a totally geodesic submanifold of N. Theorem 2. A totally real minimal semiparallel submanifold M with parallel f-structure in the n…
The paper studies energy functionals on spinor bundles and their critical points.
problem Critical points of energy functionals on spinor bundles.
method Analyzes energy functionals on universal spinor bundles over spin manifolds, modifies for arbitrary dimensions, and applies to solve ODEs.
result Critical points are pairs of Ricci-flat metrics and real parallel spinor fields for n=3 or 7.