ISOPARAMETRIC HYPERSURFACES AND COMPLEX STRUCTURES

The main purpose of this note is to construct almost complex or complex structures on certain isoparametric hypersurfaces in unit spheres.As a consequence,complex structures on S^(1)×S^(7)×S^(6),and on S^(10)×S^(3)×S(2)with vanishing first Chern class,are built.

On the Chern conjecture for isoparametric hypersurfaces

For a closed hypersurface Mn?Sn+1(1)with constant mean curvature and constant non-negative scalar curvature,we show that if tr(Ak)are constants for k=3,...,n-1 and the shape operator A,then M is isoparametric.The result generalizes the theorem of de Almeida and Brito(1990)for n=3 to any dimension n,strongly supporting the Chern conjecture.

Isoparametric Hypersurfaces Induced by Navigation in Lorentz Finsler Geometry

Using a navigation process with the datum(F,V),in which F is a Finsler metric and the smooth tangent vector field V satisfies F(−V(x))>1 everywhere,a Lorentz Finsler metric F˜can be induced.Isoparametric functions and isoparametric hypersurfaces with or without involving a smooth measure can be defined for F˜.When the vector field V in the navigation datum is homothetic,we prove the local correspondences between isoparametric functions and isoparametric hypersurfaces before and after this navigation process.Using these correspondences,we provide some examples of isoparametric functions and isoparametric hypersurfaces on a Funk space of Lorentz Randers type.

CLASSIFICATIONS OF DUPIN HYPERSURFACES IN LIE SPHERE GEOMETRY

This is a survey of local and global classification results concerning Dupin hypersurfaces in S^(n)(or R^(n))that have been obtained in the context of Lie sphere geometry.The emphasis is on results that relate Dupin hypersurfaces to isoparametric hypersurfaces in spheres.Along with these classification results,many important concepts from Lie sphere geometry,such as curvature spheres,Lie curvatures,and Legendre lifts of submanifolds of S^(n)(or R^(n)),are described in detail.The paper also contains several important constructions of Dupin hypersurfaces with certain special properties.

Mobius Isoparametric Hypersurfaces in S^(n+1) with Two Distinct Principal Curvatures

A hypersurface x: M→S^(n+1) without umbilic point is called a Mbius isoparametric hypersurface if its Mbius form Φ=-ρ^(-2)∑_i(ei(H)+∑_j(h_(ij)-Hδ_(ij))e_j(logρ))θ_i vanishes and its Mbius shape operator S=ρ^(-1)(S-Hid) has constant eigenvalues. Here {e_i} is a local orthonormal basis for I=dx·dx with dual basis {θ_i}, II=∑_(ij)h_(ij)θ_iθ_J is the second fundamental form, H=1/n∑_i h_(ij), ρ~2=n/(n-1)(‖II‖~2-nH^2) and S is the shape operator of x. It is clear that any conformal image of a (Euclidean) isoparametric hypersurface in S^(n+1) is a Mbius isoparametric hypersurface, but the converse is not true. In this paper we classify all Mbius isoparametric hypersurfaces in S^(n+1) with two distinct principal curvatures up to Mbius transformations. By using a theorem of Thorbergsson [1] we also show that the number of distinct principal curvatures of a compact Mbius isoparametric hypersurface embedded in S^(n+1) can take only the values 2, 3, 4, 6.

On the Blaschke isoparametric hypersurfaces in the unit sphere with three distinct Blaschke eigenvalues

An immersed umbilic-free submanifold in the unit sphere is called Blaschke isoparametric if its Mobius form vanishes identically and all of its Blaschke eigenvalues are constant. In this paper, we give a complete classification for all Blaschke isoparametric hypersurfaces with three distinct Blaschke eigenvalues.

Isoparametric hypersurfaces in Funk spaces

Funk metrics are a kind of important Finsler metrics with constant negative flag curvature. In this paper, it is proved that any isoparametric hypersurface in Funk spaces has at most two distinct principal curvatures. Moreover, a complete classification of isoparametric families in a Funk space is given.

On Ricci tensor of focal submanifolds of isoparametric hypersurfaces

A-manifolds and/3-manifolds, introduced by Gray （1978）, are two significant classes of Einstein-like Riemannian manifolds. A Riemannian manifold is Ricci parallel if and only if it is simultaneously an A-manifold and a B-manifold. The present paper proves that both focal submanifolds of each isoparametric hypersurface in unit spheres with g = 4 distinct principal curvatures are A-manifolds. As for the focal submanifolds with g = 6, m = 1 or 2, only one is an A-manifold, and neither is a B-manifold.

Isoparametric hypersurfaces in Finsler space forms

In this paper, we study isoparametric hypersurfaces in Finsler space forms by investigating focal points, tubes and parallel hypersurfaces of submanifolds. We prove that the focal submanifolds of isoparametric hypersurfaces are anisotropic-minimal and obtain a general Cartan-type formula in a Finsler space form with vanishing reversible torsion, from which we give some classifications on the number of distinct principal curvatures or their multiplicities.

On Mbius Form and Mbius Isoparametric Hypersurfaces

An umbilic-free hypersurface in the unit sphere is called MSbius isoparametric if it satisfies two conditions, namely, it has vanishing MSbius form and has constant MSbius principal curvatures. In this paper, under the condition of having constant MSbius principal curvatures, we show that the hypersurface is of vanishing MSbius form if and only if its MSbius form is parallel with respect to the Levi-Civita connection of its MSbius metric. Moreover, typical examples are constructed to show that the condition of having constant MSbius principal curvatures and that of having vanishing MSbius form are independent of each other.

Laguerre Isoparametric Hypersurfaces in R^n with Two Distinct Non-zero Principal Curvatures

An umbilical free oriented hypersurfacex：M→Rnwith non-zero principal curvatures is called a Laguerre isoparametric hypersurface if its Laguerre form C=i Ciωi=iρ1（Ei（logρ）（r ri）Ei（r））ωi vanishes and Laguerre shape operator S=ρ1（S 1 rid）has constant eigenvalues.Hereρ=i（r ri）2,r=r1＋r2＋···＋rn 1n 1is the mean curvature radius andSis the shape operator ofx.{Ei}is a local basis for Laguerre metric g=ρ2III with dual basis{ωi}and III is the third fundamental form ofx.In this paper,we classify all Laguerre isoparametric hypersurfaces in Rn（n〉3）with two distinct non-zero principal curvatures up to Laguerre transformations.

On Laguerre Form and Laguerre Isoparametric Hypersurfaces

Let x ： M^n-1→ R^nbe an umbilical free hypersurface with non-zero principal curvatures.M is called Laguerre isoparametric if it satisfies two conditions, namely, it has vanishing Laguerre form and has constant Lauerre principal curvatures. In this paper, under the condition of having constant Laguerre principal curvatures, we show that the hypersurface is of vanishing Laguerre form if and only if its Laguerre form is parallel with respect to the Levi-Civita connection of its Laguerre metric.

Moment maps and isoparametric hypersurfaces of OT-FKM type

Associated with a Clifford system on R^(2 l),a Spin(m+1)action is induced on R^(2 l).An isoparametric hypersurface N in S^(2 l-1)of OT-FKM(Ozeki,Takeuchi,Ferns,Karcher and Miinzner)type is invariant under this action,and so is the Cartan-Munzner polynomial F(x).This action is extended to a Hamiltonian action on C^(2 l).We give a new description of F(x)by the moment mapμ:C2 l→t^(*),where t≌o(m+1)is the Lie algebra of Spin(m+1).It also induces a Hamiltonian action on CP^(2 l-1).We consider the Gauss map g of N into the complex hyperquadric Q_(2 l-2)(C)■CP^(2 l-1),and show that g(N)lies in the zero level set of the moment map restricted to Q_(2 l-2)(C).

On hypersurfaces of H^(2)×H^(2)

In this paper,we study hypersurfaces of H^(2)×H^(2).We first classify the hypersurfaces with constant principal curvatures and constant product angle functions.Then we classify homogeneous hypersurfaces and isoparametric hypersurfaces,respectively.Finally,we classify the hypersurfaces with at most two distinct constant principal curvatures,as well as those with three distinct constant principal curvatures under some additional conditions.

Topology and Curvature of Isoparametric Families in Spheres

An isoparametric family in the unit sphere consists of parallel isoparametric hypersurfaces and their two focal submanifolds.The present paper has two parts.The first part investigates topology of the isoparametric families,namely the homotopy,homeomorphism,or diffeomorphism types,parallelizability,as well as the Lusternik-Schnirelmann category.This part extends substantially the results of Wang(J Differ Geom 27:55-66,1988).The second part is concerned with their curvatures;more precisely,we determine when they have non-negative sectional curvatures or positive Ricci curvatures with the induced metric.

Mbius Homogeneous Hypersurfaces with Three Distinct Principal Curvatures in S^(n+1)

Let x ： M^n→S^n＋1 be an immersed hypersurface in the （n ＋1）-dimensional sphere S^n＋1. If, for any points p,q ∈ M^n, there exists a Mobius transformation Ф ： S^n＋l →S^n＋1 such that Ф o x（M^n） = x（M^n） and Ф o x（p） = x（q）, then the hypersurface is called a Mobius homogeneous hypersurface. In this paper, the Mobius homogeneous hypersurfaces with three distinct principal curvatures are classified completely up to a Mobius transformation.

Finiteness results for equifocal hypersurfaces in compact symmetric spaces

We prove that there are only finitely many diffeomorphism types of curvature-adapted equifocal hypersurfaces in a simply connected compact symmetric space.Moreover,if the symmetric space is of rank one,the result can be strengthened by dropping the condition curvature-adapted.

Willmore Submanifolds in the Unit Sphere via Isoparametric Functions

In this paper,we show that both focal submanifolds of each isoparametric hypersurface in the sphere with six distinct principal curvatures are Willmore,hence all focal submanifolds of isoparametric hypersurfaces in the sphere are Willmore.

Complete Hypersurfaces with Constant Laguerre Scalar Curvature in R^n

Let x ： M^n-1→ R^n be an umbilical free hypersurface with non-zero principal curvatures. Two basic invariants of M under the Laguerre transformation group of Rn are Laguerre form C and Laguerre tensor L. In this paper, we prove the following theorem： Let M be an （n-1）-dimensional （n 〉 3） complete hypersurface with vanishing Laguerre form and with constant Laguerre scalar curvature R in R^n, denote the trace-free Laguerre tensor by L = L -1/n-1tr（L）· Id. If supM ｜｜L｜｜=0,then M is Laguerre equivalent to a Laguerre isotropic hypersurface; and if supM ｜｜L^-｜｜≡√（n-1）（n-2）R/ （n-1）（n-2）（n-3） , M isLaguerre equivalent to the hypersurface x^- ： H^1× S^n-2 → R^n.

An Intrinsic Rigidity Theorem for Closed Minimal Hypersurfaces in S^5 with Constant Nonnegative Scalar Curvature

Let M^4 be a closed minimal hypersurface in S^5 with constant nonnegative scalar curvature. Denote by f_3 the sum of the cubes of all principal curvatures, by g the number of distinct principal curvatures. It is proved that if both f_3 and g are constant,then M^4 is isoparametric. Moreover, the authors give all possible values for squared length of the second fundamental form of M^4. This result provides another piece of supporting evidence to the Chern conjecture.