Sphere Theorem Research Articles

A converse to the P. A. Smith theorem for nonunitary homology spheres

If p is an odd prime and F is the fixed point set of a smooth Zp action on Sn or Dn, then F is a smooth manifold with a unitary structure. Conversely; most Zp homology disks or spheres with unitary structures are fixed point sets of smooth Zp actions on Dn or Sn for suitable n. The results of this paper show that an arbitrary oriented mod p homology disk or sphere is the fixed point set of a smooth Zp action on some Z[l/2]-homology disk or sphere. This result is in general the best possible.

Inequalities relating sectional curvatures of a submanifold to the size of its second fundamental form and applications to pinching theorems for submanifolds

The Gauss curvature equation is used to prove inequalities relating the sectional curvatures of a submanifold with the corresponding sectional curvature of the ambient manifold and the size of the second fundamental form. These inequalities are then used to show that if a manifold M ¯ \overline M is δ \delta -pinched for some δ > 1 4 \delta > \tfrac {1}{4} , then any submanifold M M of M ¯ \overline M that has small enough second fundamental form is δ M {\delta _M} -pinched for some δ M > 1 4 {\delta _M} > \tfrac {1}{4} . It then follows from the sphere theorem that the universal covering manifold of M M is a sphere. Some related results are also given.

An Algebraic Periodicity Theorem for Spheres

An Algebraic Periodicity Theorem for Spheres

An algebraic periodicity theorem for spheres

A periodicity theorem is given for spheres over a field of finite level generalizing results of Jouanolou. An extension of this result gives families of smooth affine varities with isomorphic K K -groups.

A Sphere Theorem for Manifolds of Positive Ricci Curvature

A Sphere Theorem for Manifolds of Positive Ricci Curvature

A sphere theorem for manifolds of positive Ricci curvature

Instead of injectivity radius, the contractibility radius is estimated for a class of complete manifolds such that Ri c M ⩾ 1 , K M ⩾ − κ 2 {\text {Ri}}{{\text {c}}_M} \geqslant 1,{K_M} \geqslant - {\kappa ^2} and the volume of M ⩾ M \geqslant the volume of the ( π − ε ) (\pi - \varepsilon ) -ball on the unit m m -sphere, m = dim M m = {\text {dim }}M . Then for a suitable choice of ε = ε ( m , k ) \varepsilon = \varepsilon (m,k) every M M belonging to this class is homeomorphic to S m {S^m} .

Embedded Minimal Surfaces, Exotic Spheres, and Manifolds with Positive Ricci Curvature

Let N be a three dimensional Riemannian manifold. Let E be a closed embedded surface in N. Then it is a question of basic interest to see whether one can deform : in its isotopy class to some canonical embedded surface. From the point of view of geometry, a natural canonical surface will be the extremal surface of some functional defined on the space of embedded surfaces. The simplest functional is the area functional. The extremal surface of the area functional is called the minimal surface. Such minimal surfaces were used extensively by Meeks-Yau [MY21 in studying group actions on three dimensional manifolds. In [MY2], the theory of minimal surfaces was used to simplify and strengthen the classical Dehn's lemma, loop theorem and the sphere theorem. In the setting there, one minimizes area among all immersed surfaces and proves that the extremal object is embedded. In this paper, we minimize area among all embedded surfaces isotopic to a fixed embedded surface. In the category of these surfaces, we prove a general existence theorem (Theorem 1). A particular consequence of this theorem is that for irreducible manifolds an embedded incompressible surface is isotopic to an embedded incompressible surface with minimal area. We also prove that there exists an embedded sphere of least area enclosing a fake cell, provided the complementary volume is not a standard ball, and provided there exists no embedded one-sided RP2. By making use of the last result, and a cutting and pasting argument, we are able to settle a well-known problem in the theory of three dimensional manifolds. We prove that the covering space of any irreducible orientable three dimensional manifold is irreducible. It is possible to exploit our existence theorem to study

A topological version of Obata's sphere theorem

A topological version of Obata's sphere theorem

A Generalized Sphere Theorem

A basic problem in Riemannian geometry is the study of relations between the topological structure and the Riemannian structure of a complete, connected Riemannian manifold M of dimension n > 2. By a classical theorem of Myers [10] such a manifold is compact if the sectional curvature K of M satisfies K > a > O. More precisely, the diameter d(M) of M satisfies d(M) < zc/< 8 . After the pioneering work of Rauch [11] the following result, known as the sphere theorem, was proved first by Berger [1] in even dimensions and finally by Klingenberg [8] as stated.

An annulus theorem for suspension spheres

An annulus theorem for suspension spheres

Dynamics of Fluid Spheres of Uniform Density

The adiabatic collapse of spheres of uniform density made up of perfect fluids either electrically neutral or charged has been studied. A set of field equations equivalent to Einstein's field equations are derived in terms of the energy-momentum tensor. With the help of these equations, the following theorems for perfect fluid spheres have been established: (1) For the gravitational collapse or bounce of a homogeneous sphere of perfect fluid the only motions possible are shear-free, uniform contraction or expansion. (2) For a charged fluid sphere embedded in empty space the conditions $\ensuremath{\rho}=\ensuremath{\rho}(t)$ and $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\lambda}}=\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\mu}}$ are inconsistent in the sense that either the solutions are static or matter is uncharged.

ULC Properties in Neighbourhoods of Embedded Surfaces and Curves in E3

In this paper we derive those properties of topologically embedded curves and surfaces in E3 which can be obtained without use of Bing's Side Approximation Theorem [3] for surfaces. The local homology and homotopy properties studied classically play the largest role in the paper, but the final geometrization of some of the results requires theorems such as the PL Schoenflies Theorem, Dehn's Lemma, the Loop Theorem, the Sphere Theorem, and Waldhausen's generalization of the Loop Theorem (n.b., one application of Waldhausen's theorem (in (3.4)) requires use of the nontrivial normal subgroup in the statement of that theorem).

On Embedding Annuli in M 3

Let M be a 3-manifold. In this note we give a condition when a proper map of an into M3 can be replaced by a proper embedding of an in M3. Let 2L1 and A2 be disjoint simple loops embedded in the boundary of an orientable 3-manifold M. We suppose that 2, and 22 are freely homotopic in M and that )1 is not nullhomotopic in M. Then it is a consequence of a well-known theorem in [3] that L1u)2 is the boundary of an embedded in M. Unfortunately we can obtain no new information from this theorem when )1 and AS bound an in the boundary of M. The following theorem allows us to construct a annulus in a few special cases when the free homotopy of 2, and ii2 cannot be deformed to bd(M). This theorem seems to be closely related to a problem of R. H. Fox discussed in [1]. The proof of Theorem I is basically an application of the proof of the sphere theorem in [2] and [4]. One might hope that one could use a tower of two sheeted coverings and obtain a more general result. The difficulty here is analogous to the one encountered when trying to prove the sphere theorem via a tower of two sheeted coverings, i.e. one obtains an embedded surface at the top of the tower, but the projection of the surface may not be nontrivial in the original sense. Throughout the remainder of this paper all spaces will be simplicial complexes and all maps will be piecewise linear. Let I be a simple loop embedded in a 3-manifold M and based at a point x. Let ([l]) be the subgroup of 7r,(M, x) generated by [1]. Let N(l) be the normal closure of ([l]) in 7T1(M, x) and C(l) the centralizer of K[I]) in 71(M, X). THEOREM 1. Let M be a compact, orientable 3-mnanifoldsuch that 7T2(M) = 0. Let F be a surface, other than a torus, which is a boundary component of M. Let I be a simple loop embedded in F and not nullhomotopic in M. Then Received by the editors January 14, 1972. AMS 1970 subject classifications. Primary 55A35; Secondary 57C35. ' The author is partially supported by NSF Grant #GP-15357. Q American Mathematical Society 1972

Singularité générique des applications différentiables de la 2-sphère dans une 3-variété différentiable

In this article we have studied the singularities of smooth applications of the two sphere in a three manifold with the transversality methods and then we use the results obtained to reformulate in the differentiable case, the proof of Papakyriakopoulos and Whitehead of the sphere theorem.

The Iterated Equation of Generalized Axially Symmetric Potential Theory

Milne-Thomson's well-known circle theorem [1] gives the stream function for steady two-dimensional irrotational flow of a perfect fluid past a circular cylinder when the flow in the absense of the cylinder is known. Butler's sphere theorem [2] gives the corresponding result for axially symmetric irrotational flow of a perfect fluid past a sphere. Collins [3] has obtained a sphere theorem for axially symmetric Stokes flow of a viscous liquid which gives a stream function satisfying the appropriate viscous boundary conditions on the surface of a sphere when the stream function for irrotational flow in the absence of the sphere is known.

Correction to the paper:Circle and Sphere Theorems for the Biharmonic Equation (Interior and Exterior Problems)

Correction to the paper:Circle and Sphere Theorems for the Biharmonic Equation (Interior and Exterior Problems)

A note concerning the Sphere Theorem for pinched manifolds

A note concerning the Sphere Theorem for pinched manifolds

Circle and sphere theorems for the biharmonic equation (interior and exterior problems)

In dieser Arbeit werden Kreis- und Kugelsatze hergeleitet, von denen der erste im Zusammenhang mit den Verschiebungsfeldern einer eingespannten Platte unter reiner Biegung interessiert. Mit Hilfe des biharmonischen Feldes im unbeschrankten Bereich werden Losungen angegeben, wobei das Endergebnis fur den beschrankten Bereich innerhalb des Kreises bzw. der Kugel die gleichen Singularitaten aufweist wie dasjenige fur den unbeschrankten Bereich.

Weakly compact sets

It has been conjectured that a closed convex subset C of a Banach space B is weakly compact if and only if each continuous linear functional on B attains a maximum on C [5]. This reduces easily to the case in which C is bounded, and will be answered in the affirmative [Theorem 4] after some preliminary results are established. Following suggestions by Namioka and Peck, the result is then generalized, first to weakly closed subsets of Banach spaces and then to weakly closed subsets of complete locally convex linear spaces. The original motivation for this conjecture was the knowledge that it is true when C is the unit sphere of a separable Banach space [2], which was later extended to arbitrary Banach spaces [3, Theorem 5]. Additional support was given the conjecture when V. L. Klee [5] proved a seemingly related theorem-namely, that if C is a bounded closed non-weakly-compact convex subset of a Banach space, then there is a decreasing sequence {K1} of nonempty closed convex subsets of C such that, for each x E C and each m E [0, 1), the set x + m(C x) meets only finitely many of the sets {Ki}. Iff is a continuous linear functional whose sup on C is M and if there is no x in C for which f(x) = M, then a suitable choice for Klee's sequence {Ki} is to let Kn = C r1 {x:f (x) > M l/n} for each n. It is interesting to note that the conjecture can be verified easily for a bounded closed convex set that is symmetric about an interior point x, since if x is translated to 0, then the convex set as a unit sphere induces a norm for which the new Banach space is isomorphic to the original space. More generally, if 0 is an interior point of a bounded convex set C and K is the closed convex span of Cu ( C), then K is symmetric about 0 and the sup on K of a continuous linear functional is the larger of its sups on C and C. Therefore for convex bodies the conjecture can be established by using the known theorem for unit spheres [3, Theorem 5, p. 215]. The following theorem is a generalization of a characterization of weak compactness of the unit sphere that was useful in [2]. In the proof of this theorem and thereafter, we shall use the convention that a sequence of nonoverlapping

Eccentric multipole representations of current generators in a spherical volume conductor

In Yeh, Martinek and de Beaumont (Bull. Math. Biophysics,20, 203–16, 1958), a method is presented for determining successively better central multipole representations of the current generators in a homogeneous conducting sphere by measuring surface potentials at a successively increasing number of points. This paper generalizes the method such that the multipoles may be located at any chosen point in the conductor. The spherical harmonic expansion is advantageously used and the “interior sphere theorem” of Ludford, Martinek and Yeh (Proc. Cambridge Philos. Soc.,51, 389–93, 1955) makes possible disturbance potential expressions in closed forms. A method for approximate determination of the eccentricity is also presented. In the theory of electrocardiography, the eccentric multipoles can more accurately represent the heart as a current generator with fewer surface potential measurements than the central multipoles.