This paper constructs real algebraic maps that are topologically special generic maps.
problem Constructing smooth maps in differential topology and real algebraic geometry.
method Constructs real algebraic maps that are topologically special generic maps.
result Real algebraic maps are topologically special generic maps.
New class of maps restricts manifolds strongly in algebraic topology.
problem Restricting manifolds in algebraic topology.
method Proposed a class of generalized special generic maps.
result Extended fundamental results on structures and algebraic topological properties.
The paper examines rational homology spheres that admit special generic maps into Euclidean spaces.
problem Whether rational homology n-spheres admit special generic maps into Rp for p<n. method Stein factorization technique to derive a necessary homological condition.
result New results on the (non-)existence of special generic maps for specific rational homology spheres.
Standard special generic maps solve a homotopy sphere problem.
problem Determining integers p for which homotopy spheres admit special generic maps into Euclidean spaces.
method Using Stein factorization, we introduce standard special generic maps and study their properties.
result Standard special generic maps give a filtration of the group of homotopy spheres related to the Gromoll filtration.
Smooth maps show Gromoll filtration for spheres.
problem Understanding Gromoll filtration for special generic maps.
method Analyzing smooth maps with definite folds and Gromoll filtration.
result Gromoll filtration equals p for special generic maps.
The paper establishes a correspondence between special Kähler manifolds and their deformations.
problem Mapping between affine and projective special Kähler manifolds.
method Formulation of a correspondence and application to r-maps in string theory.
result One-parameter deformations of projective special Kähler manifolds correspond to perturbative α'-corrections.
Defines special Kaehler structures on groups and their c-map properties.
problem Defining special Kaehler structures on groups.
method Intrinsic definition and defining equations without special cone computations.
result The c-map of special Kaehler structures on groups results in quaternionic Kaehler structures.
New method for special generic maps into 3D space.
problem Understanding special generic maps into 3D space.
method Constructing pseudo quotient maps onto 2D polyhedra.
result Explicit construction of special generic maps into 3D space.
The paper generalizes Reeb spaces for special generic maps and lifts smooth functions.
problem Constructing lifts of smooth maps, especially Morse functions.
method Defining and generalizing quotient maps onto Reeb spaces of special generic maps and constructing lifts.
result Lifts of Morse functions can be constructed using the generalized maps.
A smooth map between smooth manifolds is called a special generic map if it has only definite fold points as its singularities. In this paper, we give conditions for a special generic map into the 3-dimensional Euclidean space to be factored as the composition of an embedding and a projection for certain dimensions.
Maps with boundary definite fold points restrict manifold structure.
problem Restricting the global structure of manifolds with boundary.
method Introducing boundary special generic maps and deriving differential-topological restrictions.
result New results on non-singular extensions of special generic maps.
The paper justifies two questions on special maps on subgroup of reals.
problem Justifying two questions on special maps on subgroup of reals.
method Different points of view and discussion of two versions of Anderson's Involution Conjecture.
result Discussion of two versions of Anderson's Involution Conjecture.
The study classifies special manifolds leading to quaternionic Kähler manifolds.
problem Classifying special projective real manifolds and their quaternionic Kähler counterparts.
method Classification through projective special real manifolds and supergravity maps.
result Found a series of complete quaternionic Kähler manifolds of co-homogeneity one.
The study shows manifolds with special generic maps also have nice multisections.
problem Characterizing manifolds with special generic maps and multisections.
method Analyzing manifolds with special generic maps and their properties, and showing how these maps restrict the differentiable structures of spheres and manifolds.
result Manifolds admitting special generic maps also admit nice generalized multisections.
The paper studies geodesic mappings in special Riemannian manifolds.
problem Investigating geodesic mappings in specific Riemannian manifolds.
method Proof of geodesic mappings properties and new results on geodesic mappings of various Riemannian manifolds.
result New results on geodesic mappings of quasi Einstein, Ricci recurrent, and Ricci symmetric manifolds.
Study gluing and deformations of special Lagrangian submanifolds.
problem Deforming special Lagrangian submanifolds in Calabi-Yau manifolds.
method Proves well-defined gluing map and local diffeomorphism for deformations.
result Gluing map defines local diffeomorphism between deformations.
The study restricts manifolds with certain explicit SGL maps and constructs them.
problem Restrictions on manifolds admitting specific SGL maps.
method Generalization of Morse functions and canonical projections to construct SGL maps.
result Manifolds admitting certain explicit SGL maps are strongly topologically restricted.
We give an intrinsic definition of (affine very) special real manifolds and realise any such manifold M as a domain in affine space equipped with a metric which is the Hessian of a cubic polynomial. We prove that the tangent bundle N=TM carries a canonical structure of (affine) special Kähler manifold. This gives a…
Study geodesic mappings and concircular fields in pseudo-Riemannian manifolds.
problem Characterize geodesic mappings and concircular fields in pseudo-Riemannian manifolds.
method Analyze geodesic mappings and concircular fields on Vn(K)-spaces, proving properties of the set of solutions. result The set of solutions of geodesic mappings on Vn(K)-spaces forms a special Jordan algebra, and the set of solutions generated by consircular fields is an ideal of this algebra. Under the action of the c-map, special Kahler manifolds are mapped into a class of quaternion-Kahler spaces. We explicitly construct the corresponding Swann bundle or hyperkahler cone, and determine the hyperkahler potential in terms of the prepotential of the special Kahler geometry.
Tropical curves match to special Lagrangian shapes.
problem Connecting tropical geometry to special Lagrangian shapes.
method Gluing construction that matches tropical local models to Lagrangian shapes.
result Locally planar tropical curves can be realized as special Lagrangian limits.
Authors compute Weingarten map and curvatures for SL(n, R).
problem Computing curvatures of the special linear group.
method Elementary calculations to derive Weingarten map and curvatures.
result Explicit computation of curvatures at identity of SL(n, R).
We determine all complete projective special real surfaces. By the supergravity r-map, they give rise to complete projective special Kähler manifolds of dimension 6, which are distinguished by the image of their scalar curvature function. By the supergravity c-map, the latter manifolds define in turn complete quaternio…
Abstract: Generalizes supergravity c-map to quaternionic manifolds.
problem Construct quaternionic manifolds from hypercomplex manifolds.
method Construct conical hypercomplex manifolds and associate quaternionic manifolds.
result Quaternionic manifolds can be associated to special complex manifolds.
Construct special Lagrangian submanifolds in complex projective space.
problem Finding special Lagrangian submanifolds in complex projective space.
method Moment map technique and classification of cohomogeneity one actions.
result Examples of special Lagrangian submanifolds constructed.
Proves algebraicity of maps to hyperbolic varieties using specialization techniques.
problem Analytic maps to hyperbolic varieties over complex numbers.
method Transcendental specialization technique.
result Proves equivalence of algebraicity and Borel hyperbolicity for certain maps.
The paper uses singularity theory to find normal forms for sub-Riemannian exponential maps.
problem Finding normal forms near critical points of sub-Riemannian exponential maps.
method Singularity theory applied to sub-Riemannian structures.
result Normal forms for sub-Riemannian exponential maps in specific cases.
Method constructs special Lagrangian submanifolds using symmetries.
problem Constructing special Lagrangian submanifolds in Calabi-Yau manifolds.
method Using generalized perpendicular symmetries and moment maps of Lie group actions.
result Constructs non-flat Calabi-Yau manifolds with Stenzel metrics.
Constructs harmonic maps between special geometric shapes.
problem Creating harmonic maps between specific types of geometric shapes.
method Equivariant harmonic maps constructed between cohomogeneity one manifolds.
result Developed a method to construct harmonic maps.
New homogeneous special Lagrangian submanifolds discovered in nearly Kähler CP3.
problem Exploring special Lagrangian submanifolds in nearly Kähler CP3.
method Intrinsically and extrinsically using moving frame and moment-type maps.
result Classification of totally geodesic special Lagrangian submanifolds and homogeneity of special Lagrangians.
Study special complexified Kähler forms in mirror symmetry.
problem Extend Kähler class concepts to complexified Kähler classes.
method Provide moment map interpretation and prove existence results.
result Existence of special complexified Kähler forms in toric cases.
Study examines conditions for quotient maps of foliated manifolds to be locally trivial.
problem Conditions for quotient maps of foliated manifolds to be locally trivial.
method Analyzes necessary and sufficient conditions for a quotient map to be a locally trivial fibration.
result Necessary and sufficient conditions for the map to be a locally trivial fibration are presented.
Generalized are the investigated in other works of the author transports along paths in fibre bundles to transports along arbitrary maps in them. Their structure and some properties are studied. Special attention is paid to the linear case and the case when the map's domain is a Cartesian product of two sets. Also cons…
In the present paper we prove Liouville-type theorems: non-existence theorems for some complete Riemannian almost product manifolds and special mappings of complete Riemannian manifolds which generalize similar results for compact manifolds.
This is the sixth in a series of papers constructing examples of special Lagrangian m-folds in C^m. We present a construction of special Lagrangian cones in C^3 involving two commuting o.d.e.s, motivated by the first two papers of the series. Then we generalize it to a construction of non-conical special Lagrangian 3-f…
Isothermic nets created from special maps for smooth surfaces.
problem Creating discrete curvature lines on surfaces.
method Special discrete holomorphic maps and lifted-folding.
result Isothermic nets with spherical parameter lines constructed efficiently.
Derives stress-energy tensor for polyharmonic maps.
problem Characterizing polyharmonic maps between Riemannian manifolds.
method Derives stress-energy tensor and uses it to characterize polyharmonic maps.
result Characterizes polyharmonic maps, focusing on triharmonic maps.
New PDEs for k-harmonic maps link to calibrated fibrations.
problem Understanding k-harmonic maps and their relation to calibrated fibrations. method Analyzing two special classes of k-harmonic maps between Riemannian manifolds. result Explicit noncompact examples of the second class of maps.
Investigates harmonic self-maps' stability on cohomogeneity one manifolds.
problem Stability of harmonic self-maps on cohomogeneity one manifolds.
method Systematic study of Jacobi equation for harmonic self-maps.
result Explicit solutions for specific cases show identity map's stability.
Study of associative submanifolds in Berger space SO(5)/SO(3).
problem Characterizing and classifying associative submanifolds in Berger space.
method Geometric correspondence with pseudo-holomorphic curves, analysis of special Gauss maps.
result Existence of infinitely many topological types of compact associative 3-folds.
New method characterizes surface quadrilateral layouts as special immersions.
problem Characterize surface quadrilateral layouts mathematically.
method Characterizes quadrilateral layouts as special immersions of a cut representation of the surface into the Euclidean plane.
result Mathematically describes and generalizes integer grid maps.
Formanek and Procesi have demonstrated that Aut(F_n) is not linear for n >2. Their technique is to construct nonlinear groups of a special form, which we call FP-groups, and then to embed a special type of automorphism group, which we call a poison group, in Aut(F_n), from which they build an FP-group. We first prove t…
Study of deformed Hermitian Yang-Mills equations with variable Kähler metrics.
problem Solving special Lagrangian type equations with variable metrics.
method Introducing extended gauge group to couple moment maps and scalar curvature.
result Solutions satisfy a mixture of K-stability and Bridgeland-type stability.
Hessian geometry explains special geometries in N=2 theories.
problem Explaining special geometries in N=2 theories. method Formulating N=2 theories in terms of Hessian structures. result Hessian geometry relates special geometries of vector multiplets to hypermultiplet geometries.
Quantum map counts BPS states in special theories.
problem Computing protected spin characters in class S theories.
method Geometric approach from 5D supersymmetric Yang-Mills theory.
result Explicit computation of protected spin characters in various examples.
The study finds conditions for a special vector field in singularities.
problem Existence of Milnor vector field for new singularities.
method Introducing sufficient conditions for the existence of the Milnor vector field.
result Conditions guarantee the existence of the Milnor vector field for new classes of singularities.
Classifies finite orbits of mapping class group actions on surface group representations.
problem Finite orbits of mapping class group actions on surface group representations.
method Classification based on surface genus and representation properties.
result Finite orbits correspond to homomorphisms with finite image for genus at least two, and to finite or special dihedral representations for genus one.
Solved standard problems for special mappings in complex 4-space.
problem Standard problems for holomorphic Segre preserving mappings.
method Analytical solutions for non-trivial real-formal hypersurfaces.
result Solved specific problems in complex 4-space.