Classification of hypersurfaces in homogeneous spaces with specific properties.
problem Classifying hypersurfaces in Riemannian homogeneous spaces with additional assumptions.
method Analyzing hypersurfaces under various conditions in homogeneous spaces CP3. result All extrinsically homogeneous hypersurfaces are classified in all homogeneous CP3 spaces. Classifies homogeneous CR hypersurfaces in low dimensions with maximal symmetry.
problem Classifying CR hypersurfaces with maximal symmetry in low dimensions.
method Introduced modified CR symbols to organize local invariants, classified hypersurfaces through modified symbols, and used Lie group structures.
result Found nine model structures among locally homogeneous 2-nondegenerate hypersurfaces in C4. Eastwood and Ezhov generalized the Cayley surface to the Cayley hypersurface in each dimension, proved some characteristic properties of the Cayley hypersurface and conjectured that a homogeneous hypersurface in affine space satisfying these properties must be the Cayley hypersurface. We will prove this conjecture when…
Classifies homogeneous hypersurfaces in specific 4D geometries.
problem Classifying homogeneous hypersurfaces in 4D Thurston geometries.
method Analyzing subalgebras of Lie algebras and isometry groups.
result Determined all homogeneous hypersurfaces up to ambient isometries.
This paper classifies Möbius homogeneous hypersurfaces in a sphere.
problem Classifying hypersurfaces in a sphere under Möbius transformations.
method Using Möbius transformation group to classify hypersurfaces.
result Möbius homogeneous hypersurfaces are completely classified.
The paper classifies homogeneous hypersurfaces in three 4D Thurston geometries.
problem Classifying homogeneous hypersurfaces in specific 4D geometries.
method Analyzing isometry groups and applying classification techniques.
result Homogeneous hypersurfaces identified in Sol14, Solm,n4 and Nil4. Classifies curvature homogeneous hypersurfaces in S^4 and H^4.
problem Classifying curvature homogeneous hypersurfaces in different space forms.
method Analyzing hypersurfaces in S^4 and H^4, identifying FKM examples and Tsukada's isolate example.
result Identifies an isolated hypersurface with a circle of symmetries and a one-parameter family with no continuous symmetries.
We study the geometry of homogeneous hypersurfaces and their focal sets in complex hyperbolic spaces. In particular, we provide a characterization of the focal set in terms of its second fundamental form and determine the principal curvatures of the homogeneous hypersurfaces together with their multiplicities.
Complete classification of homogeneous real hypersurfaces in complex 3-space.
problem Classifying locally homogeneous real hypersurfaces in C3. method Classification of abstract 5-dimensional real Lie algebras and their representations by algebras of holomorphic vector fields in complex 3-space.
result 47 types of homogeneous hypersurfaces, including 1- or 2-parametric families and single hypersurfaces/families.
We classify the tube domains in C^4 with affinely homogeneous base whose boundary contains a non-degenerate affinely homogeneous hypersurface. It follows that these domains are holomorphically homogeneous and amongst them there are four new examples of unbounded homogeneous domains (that do not have bounded realisation…
The paper classifies hypersurfaces in a specific 4D geometry.
problem Classify homogeneous hypersurfaces in the four-dimensional Thurston geometry mSol04. method Used geometric conditions to classify hypersurfaces with constant principal curvatures.
result Complete classification of homogeneous hypersurfaces in mSol04. We classify totally geodesic and parallel hypersurfaces of four-dimensional non-reductive homogeneous pseudo-Riemannian manifolds.
Study classifies special Hessian rank 2 hypersurfaces in 4D space.
problem Classifying hypersurfaces with constant Hessian rank 2.
method Power series method of equivalence, Lie's classification spirit.
result 34 inequivalent terminal branches, each with a nonempty moduli space.
A Lie hypersurface in the complex hyperbolic space is a homogeneous real hypersurface without focal submanifolds. The set of all Lie hypersurfaces in the complex hyperbolic space is bijective to a closed interval, which gives a deformation of homogeneous hypersurfaces from the ruled minimal one to the horosphere. In th…
We classify all (locally) homogeneous Levi non-degenerate real hypersurfaces in C3 with symmetry algebra of dimension ≥6.
We classify the non-degenerate homogeneous hypersurfaces in real and complex affine four-space whose symmetry group is at least four-dimensional.
Motivated by the physical concept of special geometry two mathematical constructions are studied, which relate real hypersurfaces to tube domains and complex Lagrangean cones respectively. Me\-thods are developed for the classification of homogeneous Riemannian hypersurfaces and for the classification of linear transit…
The paper classifies various types of hypersurfaces in a product space.
problem Classifying hypersurfaces in a specific product space.
method Analyzing hypersurfaces with constant curvatures, product angle functions, and additional conditions.
result Different types of hypersurfaces are classified based on their properties.
Researchers calculate Morse index and nullity for two specific minimal hypersurfaces.
problem Calculating Morse index and nullity for homogeneous minimal hypersurfaces with g=4 or 6. method Analyzing two specific homogeneous minimal hypersurfaces in Sn with g=4. result Obtained irrational eigenvalues in Laplace spectra for the two hypersurfaces.
The paper classifies Hopf hypersurfaces with constant curvatures on complex quadrics.
problem Classifying Hopf hypersurfaces with specific curvature properties.
method Analyzing hypersurfaces on complex quadrics with at most five distinct constant principal curvatures.
result All classified hypersurfaces are open parts of homogeneous examples.
In this paper, extending our previous joint work (Hu et al., Math Nachr 291:343--373, 2018), we initiate the study of Hopf hypersurfaces in the homogeneous NK (nearly Kähler) manifold S3×S3. First, we show that any Hopf hypersurface of the homogeneous NK S3×S3 does …
Hypersurfaces with constant Ricci eigenvalues in real space forms are classified.
problem Classification of curvature homogeneous hypersurfaces in real space forms
method Proving the converse of curvature homogeneity implies constant Ricci eigenvalues
result Hypersurfaces with constant Ricci eigenvalues in real space forms are classified
The image of the Gauss map of any oriented isoparametric hypersurface of the unit standard sphere Sn+1(1) is a minimal Lagrangian submanifold in the complex hyperquadric Qn(C). In this paper we show that the Gauss image of a compact oriented isoparametric hypersurface with g distinct constant princi…
Each hypersurface of a nearly Kähler manifold is naturally equipped with two tensor fields of (1,1)-type, namely the shape operator A and the induced almost contact structure φ. In this paper, we show that, in the homogeneous NK S6 a hypersurface satisfies the condition Aφ+φA=0 if and only if it is …
Researchers classify hypersurfaces in specific symmetric spaces.
problem Classifying homogeneous hypersurfaces in noncompact symmetric spaces.
method Isometric congruence classification of hypersurfaces in SL(3,H)/Sp(3),SO(5,C)/SO(5), and Gr∗(2,Cn+4). result Classification of hypersurfaces up to isometric congruence.
Space-times which allow a slicing into homogeneous spatial hypersurfaces generalize the usual Bianchi models. One knows already that in these models the Bianchi type may change with time. Here we show which of the changes really appear. To this end we characterize the topological space whose points are the 3-dimensiona…
To every Gorenstein algebra A of finite dimension greater than 1 over a field F of characteristic zero, and a projection π on its maximal ideal m with range equal to the annihilator Ann(m) of m, one can associate a certain algebraic hypersurface $S_π\subset{…
We exhibit a family of homogeneous hypersurfaces in affine space, one in each dimension, generalising the Cayley surface.
The study classifies isoparametric and homogeneous hypersurfaces in product spaces.
problem Classifying hypersurfaces in product spaces.
method Exploiting rigidity of constant angle functions and constant principal curvatures.
result Complete classification of isoparametric and homogeneous hypersurfaces in SnimesRm and HnimesRm. New classification of complex hypersurfaces using advanced algebraic methods.
problem Classifying homogeneous C21 hypersurfaces in complex space.
method Power series method of equivalence and Cartan's equivalence method.
result A differential-invariant branching tree for C21 hypersurfaces.
The study classifies Hessian rank 1 hypersurfaces in dimensions 2, 3, and 4.
problem Classifying Hessian rank 1 affinely homogeneous hypersurfaces in specific dimensions.
method Power Series Method of Equivalence, infinitesimal calculations.
result Identified all non-product constant Hessian rank 1 affinely homogeneous hypersurfaces in dimensions 2, 3, and 4.
Hypersurfaces of manifolds of constant nonzero sectional curvature are classificated according their restricted homogeneous holonomy groups.
Study finds maximal symmetry groups for CR structures with specific properties.
problem Determining the maximal dimension of symmetry groups for CR structures.
method Proved the sharp upper bound for the dimension of symmetry groups for homogeneous, 2-nondegenerate CR manifolds.
result The maximal dimension is n2+7 for n≥3. Classifies special hypersurfaces in Gödel spacetimes.
problem Characterizing hypersurfaces in Gödel spacetimes.
method Classification of parallel and totally geodesic hypersurfaces.
result Identified specific types of hypersurfaces in Gödel spacetimes.
Survey on extending rigidity theorems to Riemannian manifolds.
problem Extending classical rigidity theorems to Riemannian manifolds.
method Review and extension of existing rigidity theorems.
result Rigidity results for convex hypersurfaces of homogeneous 3-manifolds.
We introduce analogues of a map due to Rossi and show how they can be used to explicitly determine all covers of certain homogeneous strongly pseudoconvex 3-dimensional hypersurfaces that appear in the classification obtained by E. Cartan in 1932.
We construct a family of balanced signature pseudo-Riemannian manifolds, which arise as hypersurfaces in flat space, that are curvature homogeneous, that are modeled on a symmetric space, and that are not locally homogeneous.
The study classifies isoparametric hypersurfaces in product spaces with constant angle function.
problem Classifying isoparametric hypersurfaces in product spaces with specific curvature conditions.
method Proving constant angle function and using it to classify hypersurfaces.
result Classification of isoparametric and homogeneous hypersurfaces in product spaces.
Study on stability of minimal submanifolds in specific Einstein manifolds.
problem Investigating stability of minimal submanifolds in Einstein manifolds.
method Analyzing homogeneous minimal hypersurfaces in Page space and Sasaki-Einstein manifolds, computing stability operators and indices.
result Determined all homogeneous, minimal hypersurfaces and computed their stability operators and indices.
In this paper, we study biharmonic hypersurfaces in Einstein manifolds. Then, we determine all the biharmonic hypersurfaces in irreducible symmetric spaces of compact type which are regular orbits of commutative Hermann actions of cohomogeneity one.
A Lie hypersurface in the complex hyperbolic space is an orbit of a cohomogeneity one action without singular orbit. In this paper, we classify Ricci soliton Lie hypersurfaces in the complex hyperbolic spaces.
The Gromoll-Meyer sphere is constructed and studied in a homogeneous space.
problem Constructing and studying the Gromoll-Meyer sphere in a specific homogeneous space.
method Constructing a homogeneous isoparametric foliation and transnormal system on the quotient space, analyzing the induced metrics.
result The Gromoll-Meyer sphere has positive Ricci curvature and quasi-positive sectional curvature.
We show that a simply connected Riemannian homogeneous space M which admits a totally geodesic hypersurface F is isometric to either (a) the Riemannian product of a space of constant curvature and a homogeneous space, or (b) the warped product of the Euclidean space and a homogeneous space, or (c) the twisted product o…
Classifies isoparametric hypersurfaces in 3D manifolds.
problem Identifying isoparametric hypersurfaces in specific 3D manifolds.
method By proving constant angle and principal curvatures.
result Established classification of hypersurfaces.
No non-product Hessian rank 1 affine homogeneous hypersurfaces exist in dimensions 5 and above.
problem Identifying non-product Hessian rank 1 affine homogeneous hypersurfaces in higher dimensions.
method Developed a normal form for hypersurfaces under the affine group, up to order ≤ n+5, in any dimension n ≥ 2.
result Non-existence of non-product Hessian rank 1 affine homogeneous hypersurfaces in dimensions 5 and above.
Paper solves Dirichlet problem for p-convex hypersurfaces with curvature constraints.
problem Solving the Dirichlet problem for p-convex hypersurfaces with prescribed curvature. method Proved existence of a graphic hypersurface satisfying the prescribed curvature equation with homogeneous boundary condition, obtained an interior curvature estimate.
result Existence of a graphic hypersurface satisfying the prescribed curvature equation with homogeneous boundary condition.
In this paper, we completely classify homogeneous production functions with an arbitrary number of inputs whose production hypersurfaces are flat. As an immediate consequence, we obtain a complete classification of homogeneous production functions with two inputs whose production surfaces are developable.
A method constructs invariant PDEs on homogeneous manifolds.
problem Finding invariant PDEs on homogeneous manifolds.
method Describes a general method for constructing invariant PDEs by reducing the problem to invariant hypersurfaces under the action of the stability subgroup.
result Describes invariant PDEs for hypersurfaces in Euclidean and conformal spaces.