A characterization of Moebius isoparametric hypersurfaces of the sphere

We consider a proper, umbilic-free immersion of an n-dimensional manifold M in the sphere S ⁿ⁺¹. We show that M is a Moebius isoparametric hypersurface if, and only if, it is a cyclide of Dupin or a Dupin hypersurface with constant Moebius curvature.     

Complete hypersurfaces with constant laguerre scalar curvature in ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

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 R ⁿ are Laguerre form C and Laguerre tensor L. In this paper, 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 ?\(\widetilde L = L - \frac{1}{{n - 1}}tr\left( L \right)\) · Id. If \(\widetilde L = L - \frac{1}{{n - 1}}tr\left( L \right)\), then M is Laguerre equivalent to a Laguerre isotropic hypersurface; and if \({\sup _M}\left\| {\widetilde L} \right\| = \frac{{\sqrt {\left( {n - 1} \right)\left( {n - 2} \right)} R}}{{\left( {n - 1} \right)\left( {n - 2} \right)\left( {n - 3} \right)}},\), M is Laguerre equivalent to the hypersurface ?x: H ¹ × S ^n?2 → R ⁿ .     

Classification of hypersurfaces with parallel Möbius second fundamental form in <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis>+1</Superscript>

Let M ⁿ(n ≥ 2) be an immersed umbilic-free hypersurface in the (n+1)-dimensional unit sphere S ⁿ⁺¹. Then M ⁿ is associated with a so-called Möbius metric g, and a Möbius second fundamental form B which are invariants of M ⁿunder the Möbius transformation group of S ⁿ⁺¹. In this paper, we classify all umbilic-free hypersurfaces with parallel Möbius second fundamental form.     

Dual submanifolds in rational homology spheres

Let Σ be a simply connected rational homology sphere. A pair of disjoint closed submanifolds M_+, M_-? Σ are called dual to each other if the complement Σ-M_+ strongly homotopy retracts onto M_- or vice-versa. In this paper, we are concerned with the basic problem of which integral triples(n; m_+, m-) ∈ N~3 can appear, where n = dimΣ-1 and m_± = codim M_±-1. The problem is motivated by several fundamental aspects in differential geometry.(i) The theory of isoparametric/Dupin hypersurfaces in the unit sphere S~(n+1) initiated by′Elie Cartan, where M_± are the focal manifolds of the isoparametric/Dupin hypersurface M ? S~(n+1), and m± coincide with the multiplicities of principal curvatures of M.(ii) The Grove-Ziller construction of non-negatively curved Riemannian metrics on the Milnor exotic spheres Σ,i.e., total spaces of smooth S~3-bundles over S~4 homeomorphic but not diffeomorphic to S~7, where M_± =P_±×_(SO(4))S~3, P → S~4 the principal SO(4)-bundle of Σ and P_± the singular orbits of a cohomogeneity one SO(4) × SO(3)-action on P which are both of codimension 2.Based on the important result of Grove-Halperin, we provide a surprisingly simple answer, namely, if and only if one of the following holds true:· m_+ = m_-= n;· m_+ = m_-=1/3n ∈ {1, 2, 4, 8};· m_+ = m_-=1/4n ∈ {1, 2};· m_+ = m_-=1/6n ∈ {1, 2};·n/(m_++m_-)= 1 or 2, and for the latter case, m_+ + m_-is odd if min(m_+, m_-)≥2.In addition, if Σ is a homotopy sphere and the ratio n/(m_++m_-)= 2(for simplicity let us assume 2 m_- m_+),we observe that the work of Stolz on the multiplicities of isoparametric hypersurfaces applies almost identically to conclude that, the pair can be realized if and only if, either(m_+, m_-) =(5, 4) or m_+ + m_-+ 1 is divisible by the integer δ(m_-)(see the table on Page 1551), which is equivalent to the existence of(m_--1) linearly independent vector fields on the sphere S~(m_++m_-)by Adams' celebrated work. In contrast, infinitely many counterexamples are given if Σ is a rational homology sphere.     

Certain decompositions of matrices over Abelian rings

A ring R is (weakly) nil clean provided that every element in R is the sum of a (weak) idempotent and a nilpotent. We characterize nil and weakly nil matrix rings over abelian rings. Let R be abelian, and let n ∈ ?. We prove that M ⁿ (R) is nil clean if and only if R/J(R) is Boolean and M _n (J(R)) is nil. Furthermore, we prove that R is weakly nil clean if and only if R is periodic; R/J(R) is ?₃, B or ?₃ ⊕ B where B is a Boolean ring, and that M _n (R) is weakly nil clean if and only if M _n (R) is nil clean for all n ≥ 2.     

Regularity in the Local CR Embedding Problem

We consider a formally integrable, strictly pseudoconvex CR manifold M of hypersurface type, of dimension 2n?1≥7. Local CR, i.e., holomorphic, embeddings of M are known to exist from the works of Kuranishi and Akahori. We address the problem of regularity of the embedding in standard Hölder spaces C ^a (M), a∈R. If the structure of M is of class C ^m , m∈Z, 4≤m≤∞, we construct a local CR embedding near each point of M. This embedding is of class C ^a , for every a, 0≤a<m+(1/2). Our method is based on Henkin’s local homotopy formula for the embedded case, some very precise estimates for the solution operators in it, and a substantial modification of a previous Nash–Moser argument due to the second author.     

Peixoto graph of Morse-Smale diffeomorphisms on manifolds of dimension greater than three

Let M ⁿ be a closed orientable manifold of dimension greater than three and G ₁(M ⁿ ) be the class of orientation-preserving Morse-Smale diffeomorphisms on M ⁿ such that the set of unstable separatrices of every f ∈ G ₁(M ⁿ ) is one-dimensional and does not contain heteroclinic orbits. We show that the Peixoto graph is a complete invariant of topological conjugacy in G ₁(M ⁿ ).     

Rigidity of closed submanifolds in a locally symmetric Riemannian manifold

Let M~n(n ≥ 4) be an oriented closed submanifold with parallel mean curvature in an(n + p)-dimensional locally symmetric Riemannian manifold N~(n+p). We prove that if the sectional curvature of N is positively pinched in [δ, 1], and the Ricci curvature of M satisfies a pinching condition, then M is either a totally umbilical submanifold, or δ = 1, and N is of constant curvature. This result generalizes the geometric rigidity theorem due to Xu and Gu[15].     

Automorphisms of prime order of smooth cubic <Emphasis Type="Italic">n</Emphasis>-folds

In this paper we give an effective criterion as to when a prime number p is the order of an automorphism of a smooth cubic hypersurface of \({\mathbb{P}^{n+1}}\), for a fixed n ≥ 2. We also provide a computational method to classify all such hypersurfaces that admit an automorphism of prime order p. In particular, we show that p < 2 ⁿ⁺¹ and that any such hypersurface admitting an automorphism of order p > 2 ⁿ is isomorphic to the Klein n-fold. We apply our method to compute exhaustive lists of automorphism of prime order of smooth cubic threefolds and fourfolds. Finally, we provide an application to the moduli space of principally polarized abelian varieties.     

An isometrical ℂP<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>-theorem

Let M ⁿ (n ? 3) be a complete Riemannian manifold with sec _M ? 1, and let \(M_i^{n_i }\) (i = 1, 2) be two complete totally geodesic submanifolds in M. We prove that if n₁ + n₂ = n ? 2 and if the distance |M₁M₂| ? π/2, then M _i is isometric to \(\mathbb{S}^{n_i } /\mathbb{Z}_h\), \(\mathbb{C}P^{n_i /2}\), or \(\mathbb{C}P^{n_i /2} /\mathbb{Z}_2 \) with the canonical metric when n _i > 0; and thus, M is isometric to S ⁿ /? _h , ?P^n/2, or ?P^n/2/?₂ except possibly when n = 3 and \(M_1 (or M_2 )\mathop \cong \limits^{iso} \mathbb{S}^1 /\mathbb{Z}_h \) with h ? 2 or n = 4 and \(M_1 (or M_2 )\mathop \cong \limits^{iso} \mathbb{R}P^2 \).     

Linear preservers of row-dense matrices

Let M_m,n be the set of all m × n real matrices. A matrix A ∈ M_m,n is said to be row-dense if there are no zeros between two nonzero entries for every row of this matrix. We find the structure of linear functions T: M_m,n → M_m,n that preserve or strongly preserve row-dense matrices, i.e., T(A) is row-dense whenever A is row-dense or T(A) is row-dense if and only if A is row-dense, respectively. Similarly, a matrix A ∈ M_n,m is called a column-dense matrix if every column of A is a column-dense vector. At the end, the structure of linear preservers (strong linear preservers) of column-dense matrices is found.     

Normal curvature of CR submanifolds of maximal CR dimension of the complex projective space

We prove that there do not exist CR submanifolds Mⁿ of maximal CR dimension of a complex projective space \({\mathbf{P}^{\frac{n+p}{2}}(\mathbf{C})}\) with flat normal connection D of M, when the distinguished normal vector field is parallel with respect to D. If D is lift-flat, then there exists a totally geodesic complex projective subspace \({\mathbf{P}^{\frac{n+1}{2}}(\mathbf{C})}\) of \({\mathbf{P}^{\frac{n+p}{2}}(\mathbf{C})}\) such that M is a real hypersurface of \({\mathbf{P}^{\frac{n+1}{2}}(\mathbf{C})}\).     

A new proof of a theorem of Petersen

Let M be an n-dimensional complete Riemannian manifold with Ricci curvature n- 1. By developing some new techniques, Colding(1996) proved that the following three conditions are equivalent: 1)dGH(M, S~n) → 0; 2) the volume of M Vol(M) → Vol(S~n); 3) the radius of M rad(M) →π. By developing a different technique, Petersen(1999) gave the 4th equivalent condition, namely he proved that the n + 1-th eigenvalue of M, λ_(n+1)(M) → n, is also equivalent to the radius of M, rad(M) →π, and hence the other two.In this paper, we use Colding's techniques to give a new proof of Petersen's theorem. We expect our estimates will have further applications.     

An Intrinsic Rigidity Theorem for Closed Minimal Hypersurfaces in S5 with Constant Nonnegative Scalar Curvature

Let M⁴ be a closed minimal hypersurface in \(\mathbb{S}^5\) with constant nonnegative scalar curvature. Denote by f₃ the sum of the cubes of all principal curvatures, by g the number of distinct principal curvatures. It is proved that if both f₃ and g are constant, then M⁴ is isoparametric. Moreover, the authors give all possible values for squared length of the second fundamental form of M⁴. This result provides another piece of supporting evidence to the Chern conjecture.     

A note on K<Emphasis Type="Italic">ä</Emphasis>hler manifolds with almost nonnegative bisectional curvature

In this note, we prove the following result. There is a positive constant ε(n, Λ) such that if M ⁿ is a simply connected compact Kähler manifold with sectional curvature bounded from above by Λ, diameter bounded from above by 1, and with holomorphic bisectional curvature H ≥ ?ε(n, Λ), then M ⁿ is diffeomorphic to the product M ₁ × ? × M _k , where each M _i is either a complex projective space or an irreducible Kähler–Hermitian symmetric space of rank ≥ 2. This resolves a conjecture of Fang under the additional upper bound restrictions on sectional curvature and diameter.     

A tournament approach to pattern avoiding matrices

We consider the following Turán-type problem: given a fixed tournament H, what is the least integer t = t(n,H) so that adding t edges to any n-vertex tournament, results in a digraph containing a copy of H. Similarly, what is the least integer t = t(T _n ,H) so that adding t edges to the n-vertex transitive tournament, results in a digraph containing a copy of H. Besides proving several results on these problems, our main contributions are the following:

• Pach and Tardos conjectured that if M is an acyclic 0/1 matrix, then any n × n matrix with n(log n) ^O(1) entries equal to 1 contains the pattern M. We show that this conjecture is equivalent to the assertion that t(T _n ,H) = n(log n) ^O(1) if and only if H belongs to a certain (natural) family of tournaments.
• We propose an approach for determining if t(n,H) = n(log n) ^O(1). This approach combines expansion in sparse graphs, together with certain structural characterizations of H-free tournaments. Our result opens the door for using structural graph theoretic tools in order to settle the Pach–Tardos conjecture.

     

Adapted complex tubes on the symplectization of pseudo-Hermitian manifolds

Let (M, θ) be a pseudo-Hermitian space of real dimension 2n + 1, that is M is a CR-manifold of dimension 2n + 1 and θ is a contact form on M giving the Levi distribution \({HT(M) \subset TM}\). Let \({M^\theta \subset T^* M}\) be the canonical symplectization of (M, θ) and let M be identified with the zero section of M ^θ . Then M ^θ is a manifold of real dimension 2(n + 1) which admits a canonical foliation by surfaces parametrized by \({\mathbb{C} \ni t+i\sigma\mapsto \phi^{\theta}_{p}(t+i\sigma)=\sigma\theta_{g_t(p)}}\), where \({p \in M}\) is arbitrary and g _t is the flow generated by the Reeb vector field associated to the contact form θ. Let J be an (integrable) complex structure defined in a neighbourhood U of M in M ^θ . We say that the pair (U, J) is an adapted complex tube on M ^θ if all the parametrizations \({\phi^{\theta}_{p}(t+i\sigma)}\) defined above are holomorphic on \({(\phi^{\theta}_{p})^{-1}(U)}\). In this paper we prove that if (U, J) is an adapted complex tube on M ^θ , then the real function E on \({M^\theta\subset T^*M}\) defined by the condition \({\alpha=E (\alpha)\theta_{\pi(\alpha)}}\), for each \({\alpha \in M^\theta}\), is a canonical defining function for M which satisfies the homogeneous Monge–Ampère equation (dd ^c E)ⁿ⁺¹ = 0. We also prove that if M and θ are real analytic then the symplectization M ^θ admits an unique maximal adapted complex tube.     

A-priori upper bounds for the set covering problem

A matrix M∈R^n×n is said to be a column sufficient matrix if the solution set of LCP(M,q) is convex for every q∈Rⁿ. In a recent article, Qin et al. (Optim. Lett. 3:265–276, 2009) studied the concept of column sufficiency property in Euclidean Jordan algebras. In this paper, we make a further study of this concept and prove numerous results relating column sufficiency with the Z and Lypaunov-like properties. We also study this property for some special linear transformations.     

Conjugacy classes in Möbius groups

Let \({\mathbb H^{n+1}}\) denote the n + 1-dimensional (real) hyperbolic space. Let \({\mathbb {S}^{n}}\) denote the conformal boundary of the hyperbolic space. The group of conformal diffeomorphisms of \({\mathbb {S}^{n}}\) is denoted by M(n). Let M _o (n) be its identity component which consists of all orientation-preserving elements in M(n). The conjugacy classification of isometries in M _o (n) depends on the conjugacy of T and T ^?1 in M _o (n). For an element T in M(n), T and T ^?1 are conjugate in M(n), but they may not be conjugate in M _o (n). In the literature, T is called real if T is conjugate in M _o (n) to T ^?1. In this paper we classify real elements in M _o (n). Let T be an element in M _o (n). Corresponding to T there is an associated element T _o in SO(n + 1). If the complex conjugate eigenvalues of T _o are given by \({\{e^{i\theta_j}, e^{-i\theta_j}\}, 0 < \theta_j \leq \pi, j=1,\ldots,k}\) , then {θ₁, . . . , θ _k } are called the rotation angles of T. If the rotation angles of T are distinct from each-other, then T is called a regular element. After classifying the real elements in M _o (n) we have parametrized the conjugacy classes of regular elements in M _o (n). In the parametrization, when T is not conjugate to T ^?1 , we have enlarged the group and have considered the conjugacy class of T in M(n). We prove that each such conjugacy class can be induced with a fibration structure.     

Coherence relative to a weak torsion class

Let R be a ring. A subclass T of left R-modules is called a weak torsion class if it is closed under homomorphic images and extensions. Let T be a weak torsion class of left R-modules and n a positive integer. Then a left R-module M is called T-finitely generated if there exists a finitely generated submodule N such that M/N ∈ T; a left R-module A is called (T,n)-presented if there exists an exact sequence of left R-modules

$$0 \to {K_{n - 1}} \to {F_{n - 1}} \to \cdots \to {F_1} \to {F_0} \to M \to 0$$

such that F₀,..., F_n?1 are finitely generated free and K_n?1 is T-finitely generated; a left R-module M is called (T,n)-injective, if Ext ⁿ _R (A,M) = 0 for each (T, n+1)-presented left R-module A; a right R-module M is called (T,n)-flat, if Tor ^R _n (M,A) = 0 for each (T, n+1)-presented left R-module A. A ring R is called (T,n)-coherent, if every (T, n+1)-presented module is (n + 1)-presented. Some characterizations and properties of these modules and rings are given.