Many triangulated 3-spheres

We construct combinatorial types of triangulated 3-spheres on n vertices. Since by a result of Goodman and Pollack (1986) there are no more than 2^{O(n log n)} combinatorial types of simplicial 4-polytopes, this proves that asymptotically there are far more combinatorial types of triangulated 3-spheres than of simplicial 4-polytopes on n vertices. This complements results of Kalai (1988), who had proved a similar statement about d-spheres and (d+1)-polytopes for fixed d4.Mathematics Subject Classification (1991): 52B11, 52B70, 57Q15Supported by the Deutsche Forschungsgemeinschaft within the European graduate program Combinatorics, Geometry, and Computation (GRK 588/1) and an MSRI post-doctoral fellowship.Partially supported by Deutsche Forschungs-Gemeinschaft (DFG), via the DFG Research Center Mathematics in the Key Technologies (FZT86), the Research Group Algorithms, Structure, Randomness (Project ZI 475/3), and a Leibniz grant (ZI 475/4).     

Covering space for monotonic homotopy of trajectories of control systems

This paper considers monotonic (or causal) homotopy between trajectories of control systems. The main result is the construction of an analogue of the simply connected covering space. The constructed covering Γ(Σ,x) has the structure of a manifold and satisfies the property that two trajectories are monotonic homotopic if and only if the end points of their liftings coincide.     

Multiple closed geodesics on 3-spheres

This paper is devoted to a study on closed geodesics on Finsler and Riemannian spheres. We call a prime closed geodesic on a Finsler manifold rational, if the basic normal form decomposition (cf. [Y. Long, Bott formula of the Maslov-type index theory, Pacific J. Math. 187 (1999) 113-149]) of its linearized Poincaré map contains no 2×2 rotation matrix with rotation angle which is an irrational multiple of π, or irrational otherwise. We prove that if there exists only one prime closed geodesic on a d-dimensional irreversible Finsler sphere with d?2, it cannot be rational. Then we further prove that there exist always at least two distinct prime closed geodesics on every irreversible Finsler 3-dimensional sphere. Our method yields also at least two geometrically distinct closed geodesics on every reversible Finsler as well as Riemannian 3-dimensional sphere. We prove also such results hold for all compact simply connected 3-dimensional manifolds with irreversible or reversible Finsler as well as Riemannian metrics.     

Reducing a Representation of Homotopy 3-spheres

In the present note we show that a representation of homotopy 3-spheres can be somewhat simplified under some circumstances.     

Casson's invariant of Seifert homology 3-spheres

Dedicated to Professor Akio Hattori on his sixtieth birthday     

三维同伦球表示的约化

In the present note we show that a representation of homotopy 3-spheres can be somewhat simplified under some circumstances.     

Multiple closed geodesics on Riemannian 3-spheres

In this paper, we prove that for every Riemannian Q-homological 3-sphere (M, g) with injectivity radius and the sectional curvature K satisfying there exist at least two geometrically distinct closed geodesics. Yiming Long was partially supported by the 973 Program of MOST, Yangzi River Professorship, NNSF, MCME, RFDP, LPMC of MOE of China, S. S. Chern Foundation, and Nankai University. Wei Wang was partially supported by NNSF, RFDP of MOE of China.     

Trivial cocycles and invariants of homology 3-spheres

We study the relationship between trivial cocycles on the Torelli group and invariants of oriented integral homology 3-spheres. We apply this study to give a new purely algebraic construction of the Casson invariant. As a by-product we get a new 2-torsion cohomology class in the second integral cohomology of the Torelli group.     

Mayberry-Murasugi's formula for links in homology 3-spheres

We prove the Mayberry-Murasugi formula for links in homology 3-spheres, which was proved before only for links in the 3-sphere. Our proof uses Franz-Reidemeister torsions.

     

Explicit formulas for biharmonic submanifolds in non-Euclidean 3-spheres

We obtain the parametric equations of all biharmonic Legendre curves and Hopf cylinders in the 3-dimensional unit sphere endowed with the modified Sasakian structure defined byTanno.     

Equifacetted 3-spheres as topes of nonpolytopal matroid polytopes

We investigate simplicial 3-manifolds, in particular 3-spheres, with few vertices such that the links of all vertices are combinatorially equivalent (equilinked 3-spheres), and, simple 3-manifolds, in particular 3-spheres, with few facets such that all facets are combinatorially equivalent (equifacetted 3-spheres).     

Notes on homology cobordisms of plumbed homology 3-spheres

The gauge-theoretical invariants of Donaldson and Seiberg-Witten are used to detect some infinite order elements in the homology cobordism group of integral homology 3-spheres.

     

On minimal surfaces in a class of Finsler 3-spheres

Perturbing the classical metric on the round 3-sphere $S^3$ by the Killing vector fields tangent to Hopf fibers, one gets a class of Finsler metrics of Randers type with constant flag curvature $\mathbf{K}=1$ , depending on one parameter $k>1$ , called Bao–Shen’s (J Lond Math Soc 66:453–467, 2002) metrics. The corresponding spheres will be called Bao–Shen’s spheres, which are proper candidates of positively curved Finsler space forms. In this paper, we study the minimal surfaces in Bao–Shen’s spheres. We first study submanifolds isometrically immersed in a Randers manifold by the method of Zermelo’s navigation. Then we give a clear formula of the mean curvature of the surface in a Bao–Shen’s sphere by introducing the volume ratio function to show its relation with the mean curvature of the surface in round 3-sphere. As an application, we find an interesting family of minimal surfaces with respect to Busemann–Hausdorff volume form in Bao–Shen’s sphere called helicoids. This family contains the compact minimal surfaces $\tau _{m,n}$ in round 3-sphere constructed by Lawson (Ann Math 92(3):335–374, 1970), including great 2-spheres, Clifford torus, Klein bottles, etc. Moreover, two rigidity results are given.     

On SL(3, ℝ)-actions on 4-spheres

We construct a natural, continuous SL(3, ℝ)-action on S ⁴ which is an extension of the SO(3)-action ψ of Uchida. The construction is based on the Kuiper theorem asserting that the quotient space of ℂP(2) by complex conjugation is S ⁴. We also give a new proof of the Kuiper theorem. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 5, pp. 99–105, 2005.