We show that if S is the maximal ideal space of certain ultrapowers of C_0(L) spaces then: C_0(S, [Copf ]) allows polar decompositions while C_0(S, ℝ) does not, which answers a question of Greim and Rajalopagan [4]. Also, S is almost homogeneous but not transitive, which answers a question of Wood [9].

Let X be a complex infinite dimensional Banach space. We use σ_a(T) andσ_{ea}(T) , respectively, to denote the approximate point spectrum and the essential approximate point spectrum of a bounded operator T onX . Also, \pi _a(T) denotes the set <$>{\rm{iso} σ_a(T)\backslash σ_{ea}(T)}<$>. An operator T onX obeys the a-Browder's theorem provided that<$>σ_{ea}(T) =σ_a(T\,)\backslash π_a(T)<$> . We investigate connections between the Browder's theorems, the spectral mapping theorem and spectral continuity.

In this paper, we show new results on slant submanifolds of an almost contact metric manifold. We study and characterize slant submanifolds of K-contact and Sasakian manifolds. We also study the special class of three-dimensional slant submanifolds. We give several examples of slant submanifolds.

1991 Mathematics Subject Classification 53C15, 53C40.

For a C1 function f:ℝ^n →ℝ\;(n \ge 2), we consider the least numberk of distinct critical points that f must possess when restricted to the sphere S=\{x\in ℝ^n: \Vert x\Vert =1\}. Clearly k \ge 2 (for f attains its absolute minimum and maximum onS ), and a result of Lusternik and Schnirelmann establishes thatk=n if f is even. Here we prove that k=n if, for a given orthonormal system (e_i), \max\limits_{S \cap V_i}\,f<\min\limits_{S \cap V_i^\bot}\,f, for all i=1, …n-1, where V_i is the subspace spanned by e_1, …, e_i andV_i^\bot its orthogonal complement. It is shown that this criterion is satisfied by suitably restricted perturbations of quadratic forms having n distinct eigenvalues.

In this paper, we prove that if M^2 is a complete maximal spacelike surface of an anti-de Sitter space {\bf H}^{4}_{2}(c) with constant scalar curvature, then S=0, S={-10c\over 11}, S={-4c\over 3} or S=-2c, where S is the squared norm of the second fundamental form of M^{2}. Also

(1) S=0 if and only if M^2 is the totally geodesic surface {\bf H}^2(c);

(2) S={-4c\over 3} if and only if M^2 is the hyperbolic Veronese surface;

(3) S=-2c if and only if M^2 is the hyperbolic cylinder of the totally geodesic

surface {\bf H}^{3}_{1}(c) of {\bf H}^{4}_{2}(c).

1991 Mathematics Subject Classifaction 53C40, 53C42.

In its original formulation Lang's theorem referred to a semilinear map on an n-dimensional vector space over the algebraic closure of GF(p): it fixes the vectors of a copy ofV(n, p^h) . In other words, every semilinear map defined over a finite field is equivalent by change of coordinates to a map induced by a field automorphism. We provide an elementary proof of the theorem independent of the theory of algebraic groups and, as a by-product of our investigation, obtain a convenient normal form for semilinear maps. We apply our theorem to classical groups and to projective geometry. In the latter application we uncover three simple yet surprising results.

We prove that if n \geq 2 then A_nd cannot be a maximal left ideal of then -th Weyl algebra A_n over ℂ for almost every operator d of degree 2 of A_n.

An identity property defined for a pair of 2-complexes (Y,X) first arose in 1993 within a strategy for constructing a counterexample of infinite type to Whitehead's Asphericity Conjecture. In this note we make use of the theory of pictures to characterize a more general right N-identity property, where N < \pi _1Y. We also define combinatorial asphericity (CA) for the pair (Y,X) and determine a test for (CA) in the case that Y is obtained from X by the addition of a single 2-cell. This test can be used to determine an explicit generating set for \pi _2Y.