6-dimensional manifolds with effective T4-actions

Suppose the four dimensional torus T⁴ acts effectively on a 6-manifold M so that the orbit space M¹ is a closed 2-disk, and there exist no exceptional orbits, and the isotropy groups span T⁴. Then the fundamental group of M is a finite abelian group with at most two generators. In this paper, we obtain a homology classification of manifolds of this type under an additional hypothesis that one of the two generators is trivial. We then use this result to obtain a complete classification of simply connected 6-manifolds supporting effective T⁴-actions.     

Strongly Jónsson and strongly HS modules

Let R be a commutative ring with identity and let M be an infinite unitary R-module. Then M is a Jónsson module provided every proper R-submodule of M has smaller cardinality than M. In this note, we strengthen this condition and call an R-module M (which may be finite) strongly Jónsson provided distinct R-submodules of M have distinct cardinalities. We present a classification of these modules, and then we study a sort of dual notion. Specifically, we consider modules M   for which M/N$M/N$ and M/K$M/K$ have distinct cardinalities for distinct R-submodules N and K of M; we call such modules strongly HS (see the introduction for etymology). We conclude the paper with a classification of the strongly HS modules over an arbitrary commutative ring.     

Special conformal groups of a Riemannian manifold and Lie point symmetries of the nonlinear Poisson equation

We obtain a complete group classification of the Lie point symmetries of nonlinear Poisson equations on generic (pseudo) Riemannian manifolds M. Using this result we study their Noether symmetries and establish the respective conservation laws. It is shown that the projection of the Lie point symmetries on M are special subgroups of the conformal group of M. In particular, if the scalar curvature of M vanishes, the projection on M of the Lie point symmetry group of the Poisson equation with critical nonlinearity is the conformal group of the manifold. We illustrate our results by applying them to the Thurston geometries.     

On real hypersurfaces with η-parallel curvature tensor in complex space forms

The purpose of this paper is to give a complete classification of real hypersurfces M in complex space forms M _n(c), c≠0 in terms of an η-parallel curvature tensor and a certain commutative condition defined on the distribution T ₀={X∈T _x M| X⊥ξ} of M in M _n(c).     

Classes of general H-matrices

Let M(A) denote the comparison matrix of a square H-matrix A, that is, M(A) is an M-matrix. H-matrices such that their comparison matrices are nonsingular are well studied in the literature. In this paper, we study characterizations of H-matrices with either singular or nonsingular comparison matrices. The spectral radius of the Jacobi matrix of M(A) and the generalized diagonal dominance property are used in the characterizations. Finally, a classification of the set of general H-matrices is obtained.     

Ruled surfaces of finite type in 3- dimensional minkowski space

We show that a ruled surface M in E₁ ³ is of finite type if and only if M is minimal, or M is a part of a circular cylinder, or M is a part of a hyperbolic cylinder, or M is an isoparametric surface with null rules. We also give a complete classification of isoparametric ruled surfaces with null rules.     

p-q-convergence of sequences of measurable functions

We introduce new types of convergence of sequences of measurable functions stronger than convergence in measure for each pair of positive real numbers p, q and we obtain a classification of convergences in measure. Also in the space M of sequences of measurable functions converging in measure to zero, we introduce in a natural way an equivalence relation ∼, and in the quotient space M=M/∼ a metric, under which M turns to be a complete metric space.     

Diffeomorphism classification of finite group actions on disks

In this paper we discuss the diffeomorphism classification of finite group actions on disks. We answer the question when an action on a space M can be extended to an action on a disk such that the action is free away from M. Let the singular set consist of the points with nontrivial isotropy group. We show (under some dimension assumptions) that disks with diffeomorphic neighborhoods of the singular set can be imbedded into each other. As a consequence we find a classification of group actions on disks in terms of the neighborhood of the singular set and an element in the Whitehead group of G.     

The Residually Weakly Primitive Geometries of M 22

We announce the classification of all firm and residually connected geometries satisfying the conditions (I P)₂ and (2T)₁ and on which the Mathieu group M ₂₂ acts flag-transitively and residually weakly primitively. The complete list of geometries is available as a supplement to this note [6].     

A note on the {\rho}-Nitsche conjecture

Let (M, g) be a complete, simply connected Riemannian manifold of dimension 3 without conjugate points. We show that if M is asymptotically harmonic of constant h = 0, then M is a flat manifold. This theorem shows that any asymptotically harmonic manifold in dimension 3 is a symmetric space, thus completing the classification of asymptotically harmonic manifolds in dimension 3.     

On the classification of nonsingular 2×2×2×2 hypercubes

As a first step in the classification of nonsingular 2×2×2×2 hypercubes up to equivalence, we resolve the case where the base field is finite and the hypercubes can be written as a product of two 2×2×2 hypercubes. (Nonsingular hypercubes were introduced by D. Knuth in the context of semifields. Where semifields are related to hypercubes of dimension 3, this paper considers the next case, i.e., hypercubes of dimension 4.) We define the notion of ij-rank (with 1 ≤ i < j ≤ 4) and prove that a hypercube is the product of two 2×2×2 hypercubes if and only if its 12-rank is at most 2. We derive a ‘standard form’ for nonsingular 2×2×2×2 hypercubes of 12-rank less than 4 as a first step in the classification of such hypercubes up to equivalence. Our main result states that the equivalence class of a nonsingular 2×2×2×2 hypercube M of 12-rank 2 depends only on the value of an invariant δ ₀(M) which derives in a natural way from the Cayley hyperdeterminant det₀ M and another polynomial invariant det M of degree 4. As a corollary we prove that the number of equivalence classes is (q + 1)/2 or q/2 depending on whether the order q of the field is odd or even.     

On positive quaternionic Kähler manifolds with certain symmetry rank

Let M be a positive quaternionic Kähler manifold of real dimension 4m. In this paper we show that if the symmetry rank of M is greater than or equal to [m/2] + 3, then M is isometric to HP ^m or Gr2(C ^m+2). This is sharp and optimal, and will complete the classification result of positive quaternionic Kähler manifolds equipped with symmetry. The main idea is to use the connectedness theorem for quaternionic Kähler manifolds with a group action and the induction arguments on the dimension of the manifold.     

Transparent Pairs

Let M be a closed orientable Riemannian surface. Consider an SO(3)-connection A and a Higgs field $\Phi:M\to \mathfrak{so}(3)$ . The pair (A,??) naturally induces a cocycle over the geodesic flow of M. We classify (up to gauge transformations) cohomologically trivial pairs (A,??) with finite Fourier series in terms of a suitable B?cklund transformation. In particular, if M is negatively curved we obtain a full classification of SO(3)-transparent pairs.     

Classification of gradient-like flows on dimensions two and three

We consider here gradient-like flows of classC ^r ( ^r ≥1) on a closed manifoldM of classC ^r+1 and dimension two or three. We study the classification of these flows by the relation of topological equivalence. In this sense, the flows which are more relevant are the polar flows (only one source and only one sink).     

Ruled Weingarten Hypersurfaces in Hyperbolic Space {{\mathbb{H}}^{n+1}}

In this work we give a local classification of connected ruled Weingarten hypersurfaces M ⁿ , n ≥ 3 in the hyperbolic space ${{\mathbb{H}}^{n+1} \subset {\mathbb{L}}^{n+2}}$ .     

Minimal Submanifolds in Sp+3 with Constant Scalar Curvature

Let M be a compact, minimal 3-dimensional submanifold with constant scalar curvature R immersed in the standard sphere S^3+p. In codimension 1, we know from the work that has been done on Chern’s conjecture that M is isoparametric and R = 3D0, R = 3D3 or R = 3D6. In this paper we extend this result from codimension one to compact submanifolds with a flat normal bundle and give a complete classification.     

M-classification of regular semigroups

On any regular semigroup S, the least group congruence σ, the greatest idempotent pure congruence τ and the least band congruence β are used to give the M -classification of regular semigroups as follows. These congruences generate a sublattice Λ of the congruence lattice C(S) of S. We consider the triples (Λ, K, T), where K and T are the restrictions of the K- and T-relations on {C(S) to Λ. Such triples are characterized abstractly and form the objects of a category M whose morphisms are surjective T-preserving homomorphisms subject to a mild condition. The class of regular semigroups is made into a category M whose morphisms are fairly restricted homomorphisms. The main result of the paper is the existence of a representative functor from M to M. Several properties of the classification of regular semigroups induced by this functor are established.     

Locally conformal Kähler structures on compact nilmanifolds with left-invariant complex structures

Let (M, g, J) be a compact Hermitian manifold and \(\Omega\) the fundamental 2-form of (g, J). A Hermitian manifold (M, g, J) is called a locally conformal Kähler manifold if there exists a closed 1-form α such that \(d\Omega=\alpha \wedge \Omega\) . The purpose of this paper is to give a completely classification of locally conformal Kähler nilmanifolds with left-invariant complex structures.     

Similarity of multiplication operators on the Sobolev disk algebra

In this paper, we give a similarity classification for the multiplication operator M _g on the Sobolev disk algebra $R(\mathbb{D})$ with g analytic on the closure of the unit disk $\mathbb{D}$ .     

Homogeneous structures and rigidity of isoparametric submanifolds in Hilbert space

We study isoparametric submanifolds of rank at least two in a separable Hilbert space, which are known to be homogeneous by the main result in [E. Heintze and X. Liu, Ann. of Math. (2), 149 (1999), 149?C181], and with such a submanifold M and a point x in M we associate a canonical homogeneous structure ?? _x (a certain bilinear map defined on a subspace of T _x M × T _x M). We prove that ?? _x , together with the second fundamental form ?? _x , encodes all the information about M, and we deduce from this the rigidity result that M is completely determined by ?? _x and (????) _x , thereby making such submanifolds accessible to classification. As an essential step, we show that the one-parameter groups of isometries constructed in [E. Heintze and X. Liu, Ann. of Math. (2), 149 (1999), 149?C181] to prove their homogeneity induce smooth and hence everywhere defined Killing fields, implying the continuity of ?? (this result also seems to close a gap in [U. Christ, J. Differential Geom., 62 (2002), 1?C15]). Here an important tool is the introduction of affine root systems of isoparametric submanifolds.