Nielsen Theory on 3-manifolds Covered by S~2× R

Let f : M → M be a self-map of a closed manifold M of dimension dim M ≥ 3. The Nielsen number N(f) of f is equal to the minimal number of fixed points of f among all self-maps f in the homotopy class of f. In this paper, we determine N(f) for all self-maps f when M is a closed3-manifold with S2× R geometry. The calculation of N(f) relies on the induced homomorphisms of f on the fundamental group and on the second homotopy group of M.     

Nonnegatively Curved Metrics on S 2 × ℝ2

We classify the complete metrics of nonnegative sectional curvature on M ² × ², where M ² is any compact 2-manifold.     

Free involutions on S 2 × S 3

In this paper, we classify smooth 5-manifolds with fundamental group isomorphic to ${\mathbb{Z}/2}$ and universal cover diffeomorphic to S ² × S ³. This gives a classification of smooth free involutions on S ² × S ³ up to conjugation.     

K-loops and Bruck loops on ℝ×ℝ

Using the function (x)=cosh x, K-loops on × are constructed. Since every K-loop is a Bruck loop, we have also examples for Bruck loops. Furthermore we investigate the group of the automorphisms _a,b of the K-loop which satisfy the equation a(bc)=(ab)_a,b(c).Dedicated to Professor Dr. Oswald Giering on the occasion of his 60 th birthday     

On a new family of complete G 2-holonomy Riemannian metrics on S 3 × ℝ4

Studying a system of first-order nonlinear ordinary differential equations for the functions determining a deformation of the standard conic metric over S ³ × S ³, we prove the existence of a one-parameter family of complete G ₂-holonomy Riemannian metrics on S ³ × ?⁴.     

Group Actions on Hyperbolic 3-manifolds

Suppose M is a nonorientable closed hyperbolic 3-manifold or an orientable closed hyperbolic 3-manifold with odd first Betti number. Then any smooth action of finite cyclic group can be conjugated to preserve the hyperbolic structure. This result supports a main conjecture in 3-manifold theory.     

Isotrope Kurventheorie aufS 2 × ℝ

In this note we define an isotropic metric on the threedimensional manifoldS ² × . This metric will allow an symmetric riemannian connection , wich will be used to do differential geometry on S² × . We develope theory of curves onS ² × and show some relations to the theory of curves of threedimensional isotropic spaceI ₃.     

On the Yamabe constants of S2×R3 and S3×R2

We compare the isoperimetric profiles of $S^2\times {\mathbb{R}}^3$ and of $S^3\times {\mathbb{R}}^2$ with that of a round 5-sphere (of appropriate radius). Then we use this comparison to obtain lower bounds for the Yamabe constants of $S^2\times {\mathbb{R}}^3$ and $S^3\times {\mathbb{R}}^2$. Explicitly we show that $Y(S^3\times {\mathbb{R}}^2,[g_0^3+d{x}^2])>(3/4)Y(S^5)$ and $Y(S^2\times {\mathbb{R}}^3,[g_0^2+d{x}^2])>0.63Y(S^5)$. We also obtain explicit lower bounds in higher dimensions and for products of Euclidean space with a closed manifold of positive Ricci curvature. The techniques are a more general version of those used by the same authors in Petean and Ruiz (2011) [15] and the results are a complement to the work developed by B. Ammann, M. Dahl and E. Humbert to obtain explicit gap theorems for the Yamabe invariants in low dimensions.     

Volume-preserving Fields and Reeb Fields on 3-manifolds

Volume-preserving field X on a 3-manifold is the one that satisfies LxΩ = 0 for some volume Ω. The Reeb vector field of a contact form is of volume-preserving, but not conversely. On the basis of Geiges-Gonzalo＇s parallelization results, we obtain a volume-preserving sphere, which is a triple of everywhere linearly independent vector fields such that all their linear combinations with constant coefficients are volume-preserving fields. From many aspects, we discuss the distinction between volume-preserving fields and Reeb-like fields. We establish a duality between volume-preserving fields and h-closed 2-forms to understand such distinction. We also give two kinds of non-Reeb-like but volume-preserving vector fields to display such distinction.     

Invariant Sp(1)-instantons on S 2 × S 2

In this paper all (anti)self-dual invariant connections on homogeneous quaternionic line bundles over S ² × S ² are calculated and described in terms of the isotropy homomorphism of the bundle using Wang's theorem. These are the canonical connections on bundles with an (anti)symmetric twist and an S ¹-parametrized family of flat structures on bundles with a simple twist.     

Complete Riemannian metrics with holonomy group G 2 on deformations of cones over S 3 × S 3

Complete Riemannian metrics with holonomy group G ₂ on manifolds obtained by deformation of cones over S ³ × S ³ are constructed.     

Criterion of uniform approximability by harmonic functions on compact sets in ℝ3

For a function continuous on a compact set X ? ?³ and harmonic inside X, we obtain a criterion of uniform approximability by functions harmonic in a neighborhood of X in terms of the classical harmonic capacity. The proof is based on an improved localization scheme of A.G. Vitushkin, on a special geometric construction, and on the methods of the theory of singular integrals.     

Flächen im isotropen s2×ℝ

In [4] the author published the theory of curves in isotropic S² × . New results of Pottmann [1] show, that isotropic geometry has a meaning in CAGD, especially in questions on scattered data and visualisation. These are not only considered in euclidean space, but also on manifolds. So it may be interesting to look at the theory of surfaces in isotropic manifolds. This will be done in this paper for the manifold S² × by embedding it in I₄. Special surfaces on isotropic S² × will be geometrically interpreted.

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Herrn Prof. Dr. Oswald Giering zum 60. Geburtstag gewidmet     

An analog of bianchi transformations for two-dimensional surfaces in the space S 3 × ℝ1

Bianchi-type transformations are constructed for two-dimensional surfaces with constant negative intrinsic curvature in the space S ³ × ?¹.     

Quasispaces induced by vector fields measurable in ℝ3

We study some metric functions that are induced by a class of basis vector fields in ?³ with measurable coordinates. These functions are proved to be quasimetrics in the domain of definition of the vector fields. Under some natural constraints, the Rashevsky-Chow Theorem and the Ball-Box Theorem are established for the classes of vector fields we consider.     

Linear structures on the collections of minimal surfaces inℝ 3 andℝ 4

The collection of minimal herissons in ³ is endowed with a vector space structure. The existence of this structure is related to the fact that null curves inC ³ are described by a single map from the étalé space of the sheaf of germs of holomorphic sections of the line bundle of degree 2 over ₁ to C³, which islinear on stalks. There is an analogous construction for null curves inC ⁴. This gives a similar class of minimal surfaces in ⁴.     

Gradient-like and integrable vector fields on ℝ2

Recently, the authors have obtained criteria for the integral curves of a nonsingular smooth vector field X on a smooth manifold M to be timelike, null or spacelike geodesics for some Lorentzian metric g for M. In this paper, we show that for smoothly contractible subsets S of ² null geodesibility of a vector field X is equivalent to X being preHamiltonian on S and timelike, spacelike or Riemannian pregeodesibility of X are all equivalent to X being gradient-like. It turns out that null geodesibility is quite rare as we prove that even among real analytic vector fields on S there are many open sets of vector fields which fail to be preHamiltonian.Partially supported by a grant from the Weldon Springs endowment of the University of Missouri-Columbia