Einstein-Weyl structures on lightlike hypersurfaces

We study Weyl structures on lightlike hypersurfaces endowed with a conformal structure of certain type and specific screen distribution: the Weyl screen structures. We investigate various differential geometric properties of Einstein-Weyl screen structures on lightlike hypersurfaces and show that, for ambient Lorentzian space ? ₁ ⁿ⁺² and a totally umbilical screen foliation, there is a strong interplay with the induced (Riemannian) Weyl-structure on the leaves.     

Curvature and conjugate points in Lorentz symmetric spaces

We give some relations between conjugate points and curvature in a locally symmetric Lorentzian manifold. In the compact case, we show that the sectional curvature of timelike planes is non positive, and the lightlike sectional curvature of null planes is non negative. We also compute the lightlike conjugate loci of Cahen–Wallach manifolds, which are an important family of symmetric Lorentzian spaces.     

Pseudoconvexity in Lorentzian doubly warped products

A Lorentzian manifold M is said to be null (resp. causally) pseudoconvex if, given any compact set K in M, there exists a compact set K' in M such that any null (resp. causal) geodesic segment with both endpoints in K lies in K'. Various implications of causal and null pseudoconvexity on the geodesic structure of a Lorentzian manifold have been studied in several recent papers by Beem and Parker, Beem and Ehrlich, and Low. We provide sufficient conditions for a Lorentzian doubly warped product manifold to be null pseudoconvex. These conditions are not necessary and provide new examples of non-globally hyperbolic spacetimes which are null pseudoconvex.     

A characterization of totally <Emphasis Type="Italic">η</Emphasis>-umbilical real hypersurfaces and ruled real hypersurfaces of a complex space form

We give a characterization of totally η-umbilical real hypersurfaces and ruled real hypersurfaces of a complex space form in terms of totally umbilical condition for the holomorphic distribution on real hypersurfaces. We prove that if the shape operator A of a real hypersurface M of a complex space form M ⁿ (c), c ≠ 0, n ⩾ 3, satisfies g(AX, Y) = ag(X, Y) for any X, Y ∈ T ₀(x), a being a function, where T ₀ is the holomorphic distribution on M, then M is a totally η-umbilical real hypersurface or locally congruent to a ruled real hypersurface. This condition for the shape operator is a generalization of the notion of η-umbilical real hypersurfaces.     

On semi-parallel lightlike hypersurfaces of indefinite Kenmotsu manifolds

We study semi-parallel lightlike hypersurfaces of an indefinite Kenmotsu manifold, tangent to the structure vector field. Some Theorems on parallel and semi-parallel vector field, geodesibility of lightlike hypersurfaces are obtained. The geometrical configuration of such lightlike hypersurfaces is established. We prove that, in totally contact umbilical lightlike hypersurfaces of an indefinite Kenmotsu manifold which has constant ${\overline{\phi}}$ -holomorphic sectional curvature c, tangent to the structure vector field and such that its distribution is parallel, the parallelism and semi-parallelism notions are equivalent.     

Biharmonic hypersurfaces in a Riemannian manifold with non-positive Ricci curvature

In this article, we show that, for a biharmonic hypersurface (M, g) of a Riemannian manifold (N, h) of non-positive Ricci curvature, if ò_M|H|² v_g < ￥{\int_M\vert H\vert^2 v_g<\infty}, where H is the mean curvature of (M, g) in (N, h), then (M, g) is minimal in (N, h). Thus, for a counter example (M, g) in the case of hypersurfaces to the generalized Chen’s conjecture (cf. Sect. 1), it holds that ò_M|H|² v_g=￥{\int_M\vert H\vert^2 v_g=\infty} .     

Semiclassical <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> Estimates of Quasimodes on Curved Hypersurfaces

Let M be a compact manifold of dimension n, P=P(h) a semiclassical pseudodifferential operator on M, and u=u(h) an L ² normalized family of functions such that P(h)u(h) is O(h) in L ²(M) as h↓0. Let H⊂M be a compact submanifold of M. In a previous article, the second-named author proved estimates on the L ^p norms, p≥2, of u restricted to H, under the assumption that the u are semiclassically localized and under some natural structural assumptions about the principal symbol of P. These estimates are of the form Ch ^−δ(n,k,p) where k=dim H (except for a logarithmic divergence in the case k=n−2, p=2). When H is a hypersurface, i.e., k=n−1, we have δ(n,n−1, 2)=1/4, which is sharp when M is the round n-sphere and H is an equator.     

A Report on Canonical Null Curves and Screen Distributions for Lightlike Geometry

The general theory of lightlike submanifolds makes use of a non-degenerate screen distribution which is not unique and, therefore, the induced objects (starting from null curves) depend on the choice of a screen, which creates a problem. The purpose of this paper is to report on the existence of a canonical representation of null curves of Lorentzian manifolds and the choice of a canonical or a good screen for large classes of lightlike hypersurfaces of semi-Riemannian manifolds. We also prove a new theorem on the existence of an integrable canonical screen, subject to a geometric condition, and supported by a physical application.      

Remark on the anisotropic prescribed mean curvature equation on arbitrary domains

In this article we consider the Dirichlet problem for hypersurfaces of aniso- tropic prescribed mean curvature H = H(x, u, N) depending on ${x \in \varOmega \subset \mathbb {R}^n}In this article we consider the Dirichlet problem for hypersurfaces of aniso- tropic prescribed mean curvature H = H(x, u, N) depending on x ? \varOmega ì \mathbb Rⁿ{x \in \varOmega \subset \mathbb {R}^n}, the height u of the hypersurface M = graph u over \varOmega{\varOmega} and the unit normal N to M at (x, u). We give a condition relating H and the mean curvature of ?\varOmega{\partial \varOmega} that guarantees the existence of smooth solutions even for not necessarily convex domains.     

The localization of 1-cohomology of transitive Lie algebroids

For a transitive Lie algebroid A on a connected manifold M and its representation on a vector bundle F, we define a morphism of cohomology groups rk: Hk(A,F) → Hk(Lx,Fx), called the localization map, where Lx is the adjoint algebra at x ∈ M. The main result in this paper is that if M is simply connected, or H (LX,FX) is trivial, then T is injective. This means that the Lie algebroid 1-cohomology is totally determined by the 1-cohomology of its adjoint Lie algebra in the above two cases.     

Maximality of entropy in finite von Neumann algebras

It is shown that the entropy function H(N ₁,…,N _k ) on finite dimensional von Neumann subalgebras of a finite von Neumann algebra attains its maximal possible value H(⋁ℓ=1k N _ℓ) if and only if there exists a maximal abelian subalgebra A of ⋁ℓ=1k N _ℓ such that A=⋁ℓ=1k(A∩N _ℓ). Oblatum 24-IV-1997 & 6-V-1997     

Geometry of lightlike hypersurfaces of an indefinite Sasakian manifold

In this paper, we study the geometry of lightlike hypersurfaces of an indefinite Sasakian manifold. The main result is to prove three characterization theorems for such a lightlike hypersurface. In addition to these main theorems, we study the geometry of totally geodesic lightlike hypersurfaces of an indefinite Sasakian manifold.     

Einstein-like metrics on three-dimensional homogeneous Lorentzian manifolds

We completely classify three-dimensional homogeneous Lorentzian manifolds, equipped with Einstein-like metrics. Similarly to the Riemannian case (E. Abbena et al., Simon Stevin Quart J Pure Appl Math 66:173–182, 1992), if (M, g) is a three-dimensional homogeneous Lorentzian manifold, the Ricci tensor of (M, g) being cyclic-parallel (respectively, a Codazzi tensor) is related to natural reductivity (respectively, symmetry) of (M, g). However, some exceptional examples arise. The author is supported by funds of MURST, GNSAGA and the University of Lecce.     

On convergence of sections of sequences in Banach spaces

An elementary proof of the (known) fact that each element of the Banach spaceℓ _w ^p (X) of weakly absolutelyp-summable sequences (if 1≤p<∞) in the Banach spaceX is the norm limit of its sections if and only if each element ofℓ _w ^p (X) is a norm null sequence inX, is given. Little modification to this proof leads to a similar result for a family of Orlicz sequence spaces. Some applications to spaces of compact operators on Banach sequence spaces are considered.     

Preduals of semigroup algebras

For a locally compact group G, the measure convolution algebra M(G) carries a natural coproduct. In previous work, we showed that the canonical predual C ₀(G) of M(G) is the unique predual which makes both the product and the coproduct on M(G) weak^*-continuous. Given a discrete semigroup S, the convolution algebra ℓ ¹(S) also carries a coproduct. In this paper we examine preduals for ℓ ¹(S) making both the product and the coproduct weak^*-continuous. Under certain conditions on S, we show that ℓ ¹(S) has a unique such predual. Such S include the free semigroup on finitely many generators. In general, however, this need not be the case even for quite simple semigroups and we construct uncountably many such preduals on ℓ ¹(S) when S is either ℤ₊×ℤ or (ℕ,⋅).     

On the first pontrjagin class of homotopy complex projective spaces

Let M ²ⁿ be a closed smooth manifold homotopy equivalent to the complex projective space ℂP(n). It is known that the first Pontrjagin class p ₁(M) of M ²ⁿ has the form (n+1+24α(M))u ² for some integer α(M) where u is a generator of H ²(M; ℤ). We prove that α(M) is even when n is even but not divisible by 64.     

Lorentzian similarity manifolds

An (m+2)-dimensional Lorentzian similarity manifold M is an affine flat manifold locally modeled on (G,ℝ ^m+2), where G = ℝ ^m+2 ⋊ (O(m+1, 1)×ℝ⁺). M is also a conformally flat Lorentzian manifold because G is isomorphic to the stabilizer of the Lorentzian group PO(m+2, 2) of the Lorentz model S ^m+1,1. We discuss the properties of compact Lorentzian similarity manifolds using developing maps and holonomy representations.     

Generalized difference sequence spaces on seminormed space defined by Orlicz functions

In this paper we define the sequence space ℓ _M (Δ ^m , p, q, s) on a seminormed complex linear space by using an Orlicz function. We study its different algebraic and topological properties like solidness, symmetricity, monotonicity, convergence free etc. We prove some inclusion relations involving ℓ _M (Δ ^m , p, q, s).      

Differentiable and analytic families of continuous martingales in manifolds with connection

Summary.  We prove that the derivative of a differentiable family X _t (a) of continuous martingales in a manifold M is a martingale in the tangent space for the complete lift of the connection in M, provided that the derivative is bicontinuous in t and a. We consider a filtered probability space (Ω,(ℱ _t )_{0≤ t ≤1}, ℙ) such that all the real martingales have a continuous version, and a manifold M endowed with an analytic connection and such that the complexification of M has strong convex geometry. We prove that, given an analytic family a↦L(a) of random variable with values in M and such that L(0)≡x ₀∈M, there exists an analytic family a↦X(a) of continuous martingales such that X ₁(a)=L(a). For this, we investigate the convexity of the tangent spaces T ^{( n )} M, and we prove that any continuous martingale in any manifold can be uniformly approximated by a discrete martingale up to a stopping time T such that ℙ(T<1) is arbitrarily small. We use this construction of families of martingales in complex analytic manifolds to prove that every ℱ₁-measurable random variable with values in a compact convex set V with convex geometry in a manifold with a C ¹ connection is reachable by a V-valued martingale. Received: 14 March 1996/In revised form: 12 November 1996     

Screen conformal lightlike submanifolds of semi-Riemannian manifolds

Since the induced objects on a lightlike submanifold depend on its screen distribution which, in general, is not unique and hence we can not use the classical submanifold theory on a lightlike submanifold in the usual way. Therefore, in present paper, we study screen conformal lightlike submanifolds of a semi-Riemannian manifold, which are essential for the existence of unique screen distribution. We obtain a characterization theorem for the existence of screen conformal lightlike submanifolds of a semi-Riemannian manifold. We prove that if the differential operator D^s is a metric Otsuki connection on transversal lightlike bundle for a screen conformal lightlike submanifold then semi-Riemannian manifold is a semi-Euclidean space. We also obtain some characterization theorems for a screen conformal totally umbilical lightlike submanifold of a semi-Riemannian space form. Further, we obtain a necessary and sufficient condition for a screen conformal lightlike submanifold of constant curvature to be a semi-Euclidean space. Finally, we prove that for an irrotational screen conformal lightlike submanifold of a semi-Riemannian space form, the induced Ricci tensor is symmetric and the null sectional curvature vanishes.