Compact N-dimensional Manifold Research Articles

Structural stability of diffeomorphisms on two-manifolds

In [8] Robbin proved that the C2-diffeomorphisms on a compact manifold which satisfy Axiom A and the Transversality Condition are structurally stable in the C ~ sense. However, some basic results in the theory of Dynamical Systems, such as the Closing-lemma [7], are only known in the Cl-topology. Therefore, as Robbin observed, it would be of interest to extend his result for C ~ diffeomorphisms. We construct in this paper the analogue of Palis-Smale tubular families [6], for such diffeomorphisms on two dimensional manifolds. These families provide a rather fine geometric structure of the diffeomorphism near the basic sets and the relation between them. From this fact it follows the structural stability of the diffeomorphisms. We will also prove, the stability of diffeomorphisms satisfying Axiom A in a neighborhood of a basic set for two dimensional manifolds and some other cases. We will now set some notation, basic definitions, and the statements of our results. Let Diff ~(M) be the set of C~-diffeomorphisms of M with the C ttopology, where M is a compact n-dimensional manifold without boundary. If feDiffl(M), f2(f) will denote the non-wandering set of f and Per(f) the set of periodic points off. A diffeomorphism f is said to satisfy Axiom A if (i) f2(f) is hyperbolic, namely, the tangent bundle of M over t2 splits in a Df-invariant Whitney sum, TaM=ES~ E", and there exists a riemannian metric ][. II on M such that

On Shikata's distance between differentiable structures

Shikata proved: there is a number e(n) with the following property: If two compact homeomorphic n-dimensional manifolds have a distance less than e (n), then they are diffeomorphic. We improve the known lower bound ∼(n!)−n for e(n) to ∼1/3n−2.

On coerciveness and semiboundedness of general boundary problems

The paper treats coerciveness inequalities (of the form Re(Au, u)≧c ‖u‖ s 2 −λ ‖u‖ 0 2 ,c>0,λ ∈ R) and semiboundedness inequalities (of the form Re (Au, u)≧−λ ‖u‖2) for the general boundary problems associated with an elliptic 2m-order differential operatorA in a compactn-dimensional manifold with boundary. In particular, we study the normal pseudo-differential boundary conditions, for which we determine necessary and sufficient conditions for coerciveness withs=m, and for semiboundedness with ‖u‖ = ‖u‖m, in explicit form.

Finiteness Theorems for Riemannian Manifolds

1. The purpose of this paper is to show that if one puts arbitrary fixed bounds on the size of certain geometrical quantities associated with a riemannian metric, then the set of diffeomorphism classes of compact ndimensional manifolds admitting a metric for which these bounds are satisfied is finite. As an application, we show that the set of diffeomorphism classes of compact n-manifolds, (n , 4) for which some characteristic number is nonzero, and which admit an Einstein metric of nonnegative curvature, is finite. Our main tool is a theorem giving a lower bound for the injectivity radius of the exponential map. The results represent an improved version of a portion of the author's doctoral thesis and he again wishes to thank Professors S. Bochner and J. Simons for their advice and encouragement. A weaker version of part C) of Corollary 3. 3 is due independently to A.

An order relation among topological spaces

In this paper we define a partial ordering on a class of topological spaces as follows('): For two spaces X and Y in the class, we will say X Y which is an onto a-map. (Recall that f is an a-map if there is an open cover ,B of Y such that f-'(13) refines a.) It is easy to check that the relation < is reflexive and transitive. In [7], Kuratowski proved that the closed n-dimensional ball is not _ the n-sphere. More recently, Ulam [8] raised the question whether the 2-dimensional ball is < the 2-dimensional torus. Ulam's question has stimulated interest in the ordering itself, and a result of T. Ganea [5] states, roughly speaking, that if X < Y and Y is a compact n-dimensional manifold and X is an absolute neighborhood retract, then X has the same homotopy type as an n-dimensional manifold. This result then gives Ulam's question a negative answer(2), and suggests studying properties of a space Y which are inherited by a space X satisfying X _ Y. The purpose of this paper is to present certain properties which are inherited, in the above sense. Also, comparison is made between this ordering and the ordering given by dimension (Theorem 1), and (for the class of absolute neighborhood retracts) the ordering given by a-domination for every cover a (Theorem 2). Since the property of possessing a fixed point under every continuous function will be shown to be inherited, another question of Ulam's is answered here [8, Chapter IV, Problem 12].