In this paper we study the r-stability of closed spacelike hypersurfaces with constant r-th mean curvature in conformally stationary spacetimes of constant sectional curvature. In this setting, we obtain a characterization of r-stability through the analysis of the first eigenvalue of an operator naturally attached to the r-th mean curvature. As an application, we treat the case in which the spacetime is the de Sitter space.

We introduce the notion of generalized Tanaka-Webster connection for hypersurfaces in complex two-plane Grassmannians G₂(C^m+2) and give a non-existence theorem for Hopf hypersurfaces in G₂(C^m+2) with parallel shape operator in this connection.

Let S be a Riemann surface of type (p, n) with 3p + n > 4 and n ≥ 1. Let a be a puncture of S. We show that for any Dehn twist t_c along a simple closed geodesic c on S, there exists a sequence {f_m} of pseudo-Anosov maps of S such that for sufficiently large integers m, the products f_m $\circ$ t_c^k are pseudo-Anosov for all integers k. As a corollary, we prove that for a multi-twist M₂ on $\tilde{S}$ along two disjoint simple closed geodesics, there are infinitely many pseudo-Anosov maps of S that are isotopic to M₂ as a is filled in.

An inequality proved by Frank, Steinmetz and Weissenborn relates the frequency of poles of a function meromorphic in the plane to the frequency of zeros of a linear differential polynomial in that function with small coefficients. A version of this inequality is established in terms of the frequency of distinct zeros of the linear differential polynomial.

In a previous work, the authors gave a definition of `front bundles'. Using this, we give a realization theorem for wave fronts in space forms, like as in the fundamental theorem of surface theory. As an application, we investigate the behavior of principal singular curvatures along A₂-singularities of hypersurfaces with non-negative sectional curvature in Euclidean space.

Let e $\in$ {-1, +1}. Let a,b $\in$ Z be such that x⁶ + ax⁴ + bx² + e is irreducible in Z[x]. The cubic field C = Q (α), where α³ + aα² + bα + e = 0, is said to lift to the sextic field K = Q(θ), where θ⁶ + aθ⁴ + bθ² + e = 0. The field K is called the lift of C. If {1, α, α²} is an integral basis for C (so that C is monogenic), we investigate conditions on a and b so that {1, θ, θ², θ³, θ⁴, θ⁵} is an integral basis for the lift K of C (so that K is monogenic). As the sextic field K contains a cubic subfield (namely C), there are eight possibilities for the Galois group of K. For five of these Galois groups, we show that infinitely many monogenic sextic fields can be obtained in this way, and for the remaining three Galois groups, we show that only finitely many monogenic fields can arise in this way, when e $\in$ {-1, +1}.

We study $\mathscr{D}$-homothetic deformations of almost α-Kenmotsu structures. We characterize almost contact metric manifolds which are CR-integrable almost α-Kenmotsu manifolds, through the existence of a canonical linear connection, invariant under $\mathscr{D}$-homothetic deformations. If the canonical connection associated to the structure (φ, ξ, η, g) has parallel torsion and curvature, then the local geometry is completely determined by the dimension of the manifold and the spectrum of the operator h′ defined by 2αh′ = ($\mathscr{L}$_ξφ) $\circ$ φ. In particular, the manifold is locally equivalent to a Lie group endowed with a left invariant almost α-Kenmotsu structure. In the case of almost α-Kenmotsu (κ, μ)′-spaces, this classification gives rise to a scalar invariant depending on the real numbers κ and α.

Let G_k be a split reductive group over a field k corresponding to a compact Lie group G. Let G_k be a nontrivial G_k-torsor over a field k. In this paper we study the Chow ring of G_k. For example when (G, p) = (G₂, 2), we have the isomorphism CH^*(G_k)₍₂₎ $\cong$ Z₍₂₎.

In this paper we generalize the classical finite dimensional selection theorem due to Michael [12, theorem 1.2] to the case where the target space is only a Hausdorff uniform space. This also generalizes the zero-dimensional selection theorem of Fakhoury-Gieler [7, 8]. The proof of this generalization utilizes an elegant construction due to Ageev.

We classify parallel real hypersurfaces in a complex space form for the generalized Tanaka-Webster connection.

We study circular trajectories for Sasakian magnetic fields on geodesic spheres, horospheres and tubes around totally geodesic complex hypersurfaces in a complex hyperbolic space. Investigating their extrinsic shapes in the ambient complex hyperbolic space, we give conditions for them to be bounded and to be closed. By use of information on lengths of circles in complex space forms, we give expressions of lengths of circular trajectories on those real hypersurfaces and show that their length spectrum is a discrete subset of a real line.

When does a topological group G have a Hausdorff compactification bG with a remainder belonging to a given class of spaces? In this paper, we mainly improve some results of A. V. Arhangel'skiĭ and C. Liu's. Let G be a non-locally compact topological group and bG be a compactification of G. The following facts are established: (1) If bG$\backslash$G has locally a k-space with a point-countable k-network and π-character of bG$\backslash$G is countable, then G and bG are separable and metrizable; (2) If bG$\backslash$G has locally a δθ-base, then G and bG are separable and metrizable; (3) If bG$\backslash$G has locally a quasi-G_δ-diagonal, then G and bG are separable and metrizable. Finally, we give a partial answer for a question, which was posed by C. Liu in [16].

In this paper we construct torus invariant special Lagrangian submanifolds in the canonical bundle K_M of the toric positive Kähler Einstein manifold M. We construct a Ricci-flat metric on K_M using the Calabi ansatz to show that K_M is a Calabi-Yau manifold. Then, using moment map techniques developed in [6], we construct special Lagrangian submanifolds in K_M.