Geometry and Topology

The following sections are included:

• PREFACE

• CONTENTS

Let γ be a curve associated with a local extension of a space-time. If γ is causal, then the local extension is said to be a causal curve extension. Minkowski space-time and open Robertson-Walker big bang models have causal curve extensions. Complete analytic space-times fail to have endpoint convergent curve extensions. This work yields a new argument for closed cosmological models.

It is shown that the natural Riemannian structure of the tangent sphere bundle of a Riemannian manifold is locally symmetric if and only if either the base manifold is flat or is of dimension 2 and constant curvature 1. The motivation for this result ie the study of the tangent sphere bundle as a contact manifold and the general open question of the classification of locally symmetric contact metric manifolds.

Let M be a closed non-orientable 3-manifold with a Heegaard splitting of genus two. We show that, if every essential Klein bottle (or torus) is separating in M and if M has a non-trivial canonical system of 2-sided tori and Klein bottles, then there is a canonical system such that its intersection with one of the handlebodies in the Heegaard splitting consists of at most two essential disks. We use these results to give a complete list of all the non-simple non-orientable 3-manifolds with a Heegaard splitting of genus two which are not Seifert fibered.

The following sections are included:

• Introduction

• Definitions And Notation

• The Fibered Handle Trading Theorem

• The Factorization Lemma

• Proof Of The Destabilization Theorem

• References

It was conjectured that every mixed foliate CR-submanifold N of a complex hyperbolic space H^m is either a complex submanifold or a totally real submanifold [1,4]. In this article, we prove that the conjecture is true if the holomorphic distribution D of N is of (complex) odd-dimension.

Closed linear subspaces (those defined by polynomial equations) of the sequence space are provided with an echelon form and are then classified and shown to be closed under sum (the most significant result of the paper). Products of closed linear subspaces of the sequence space are proven to be cofinal in products of subspaces of the sequence space. A directed family of closed linear subspaces of the sequence space having a countable cofinal subfamily is seen to cover a closed linear subspace H only when one of its elements contains H. An echelon form for arbitrary subspaces of the sequence space is introduced.

Let M² be an oriented smooth closed surface. A Jordan curve on M² is a continuous map K:[0,1] →, M² (or K: R →M²) such that for each t ∊ (0, 1) (or t∊R) there is an ε>0 such that K is one-to-one on the open #x220A;-neighborhood of t. The orientation on a Jordan curve K on M² will always be the induced orientation by the chosen one on the surface M².

We study the Morse theory of a null geodesic segment with spacelike endmanifolds in an arbitrary Lorentzian manifold.

The paper discusses the uniformity and homogeneity of elastic bodies. After introducing some basic definitions such as configuration space, virtual displacement, forces, global elasticity and several aspects of local elasticity the idea of global uniformity is formulated and compared with the standard definition of (local) uniformity. Then, the existence of the material G-structure for a uniform jet-local elastic body is proved. Observing that homogeneity is equivalent to the flatness of such a G-structure, new explicit conditions for local homogeneity of certain classes of materials are also presented.

The eta invariant of Atiyah et al [1] is an R/Z valued invariant of equivariant unitary bordism which completely detects MU(BG) for spherical space form groups. We use the eta invariant to compute the additive structure of MU(BZ₄).

If M is a smooth compact surface with boundary, with Riemannian metric, then a generic smooth function f has a ‘portrait’ consisting of the level curves of f and the trajectories orthogonal to them. Isomorphisms between portraits are defined in terms of homeomorphisms of M, and we show that each portrait is essentially determined by a linear graph determined by the stable manifolds of f. The classification of portraits is reduced to a combinatorial problem in graph theory. We consider also a ‘generic’ version of the Dirichlet problem, classifying the set of solutions by their portraits.

A procedure amounting to Gâteaux differentiation in the cone of support functions of convex bodies was introduced by Cavallin [1] in 1881 and systematically developed by G. Masotti-Biggigero [2,3] for the derivation of formulas of integral geometry depending on arbitrary functions (see also [4], p.59; [6], p.135). A similar procedure can be used on most integral formulas. As general examples, we treat the differentiation of Stokes's formula and study an invariance property of the integral of the product of the principal curvatures of compact, orientable hypersurfaces in Riemannian spaces.

If X is a metrizable topological space, D(X) stands for the set of all compatible metrics on X. A metric d ∊ D(X) is said to be Heine-Borel if every closed and d-bounded subset of X is compact. For d ∊ D(X) we denote by I(d) the group of all d-isometrics on X. We show that if X is connected and d ∊ D(X) is such that the metric space (X,d) is uniformly locally compact then there exists d* ∊ D(X) such that d* is Heine-Bore1 and I(d) ⊂ I(d*).

Several nonlinear problems from fluid mechanics give rise to operator equations that are special cases of the abstract problem considered here. Nontrivial states branching from the trivial solution of this problem are sought and splitting methods are used to obtain the bifurcation equations. Methods utilizing topological degree are given to solve this system when the reduced bifurcation equations have a gradient structure. The technique is valid even at degenerate critical points of the functional involved and leads to the existence of nontrivial states of the abstract problem. Some additional properties of the solutions are discussed and applications of the theory to fluid mechanics are described.

The purpose of this paper is to show the existence of a density on the punctured unit disk which in a sense behaves more wildly at the origin than one chosen in advance and yet the Picard dimension of which at the origin is an arbitrarily assigned positive integer in advance.

The purpose of this paper is: 1. to give an affirmativs answer to Ulam s-Banach's problem. and 2. to give an example of a homogeneously invertible, Hausdorff topological space that is not normal.

With any r-parameter family of point transformations on a manifold M one can associate three distinct sets of vector fields on M, each set consisting of r elements. The successive imposition of algebraic conditions on these vector fields gives rise to certain differential-geometric structures by means of which the existence of an underlying Lie group may be discerned.

In this paper we introduce a new series of line involutions in the projective space. S₃. of three dimensions. We investigate the various properties of the involutions and determine their characteristics. The characteristics of the involutions are used to classify the new involution and prove their uniqueness (1), (2).

In Ill, Hatcher raised the following question.

Question: For knot exteriors in S³, must the boundary slopes of incompressible, ⁙-incompressible surfaces always be integers?

In this paper we shall give a negative answer to this question, that is:

Theorem. There exist a knot k and an incompressible and ⁙-incompressible surface S in the exterior of k such that the slope of the boundary curves of S is not an integer.

We construct compact convex moduli spaces of harmonic maps between spheres and complex projective spaces.

The purpose of this paper is to illustrate some applications of Topology in General Theory of Relativity by surveying our work in this field. We prove some new results and also point out few open problems.