Codimension three nonnegatively curved submanifolds with infinite fundamental group

We show that a complete, codimension three submanifold M of nonnegative sectional curvature that isometrically splits as ${\bar{M} \times \mathbb{R}}$ has nonnegative curvature operator. We apply this result to obtain a classification of codimension three nonflat manifolds of nonnegative sectional curvature and infinite fundamental group.     

Harmonic 2-forms and positively curved 4-manifolds

Science China Mathematics - We prove that if a compact Riemannian 4-manifold with positive sectional curvature satisfies a strengthened Kato type inequality, then it is definite. We also discuss...     

On 4-manifolds with zero-entropy metrics

We sharpen a recent result of Paternain and Petean by showingthat a closed 4-manifold which admits a Riemannian metric withzero topological entropy and infinite fundamental group eitheris aspherical or has a finite covering space homeomorphic toS³ x S¹ or S² x S¹ x S¹.     

A splitting theorem for open nonnegatively curved manifolds

We prove that an open nonnegatively curved manifold Mⁿ with soul S^k splits isometrically as S × ^2–k if (and only if) the holonomy group of the normal bundle NS is trivial.Supported by the Heinrich Hertz foundation     

On 4-manifolds with singular torus actions

Mathematische Annalen -     

On symmetry of discrete polynomial hypergroups

Let be a discrete polynomial hypergoup on with Plancherel measure If the hypergroup is symmetric, the set of characters can be identified with a compact subset of the real line which contains the support of We show that the lower and upper bounds of and coincide. In particular, the trivial character belongs to the support of the Plancherel measure.

     

Totally geodesic embeddings of 7-manifolds in positively curved 13-manifolds

We demonstrate a correspondence between the set of Bazaikin spaces with a subset of Aloff-Wallach and Eschenburg spaces. We show that each Bazaikin space contains at least one and generically 10 totally geodesically embedded Aloff-Wallach or Eschenburg spaces. We use these embeddings to get a sharp upper bound for the pinching of the standard biquotient metrics over the whole family of Bazaikin spaces, and to characterise the curvature properties and regularity of the Bazaikin space in terms of the curvature and regularity of the embedded spaces.Mathematics Subject Classification (2000): 53C20, 53C30, 53C40Acknowledgements. It is a pleasure for us to thank I. Taimanov and W. Ziller for valuable hints and discussion.Dedicated to Ernst Heintze on occasion of his 60th birthday     

On 4-manifolds homotopy equivalent to circle bundles over 3-manifolds

We give criteria for a closed 4-manifold to be homotopy equivalent to the total space of an S¹-bundle over a closed 3-manifold. In the aspherical case the conditions are that the Euler characteristic be 0 and that the fundamental group have an infinite cyclic normal subgroup such that the quotient group has one end and finite cohomological dimension. Under further assumptions on this quotient group we characterize the total spaces of such bundles over -or H² × E¹-manifolds and over E³-, Nil³- or Sol³-manifolds up to s-cobordism and homeomorphism respectively.     

On Almost Hermitian 4-manifolds with J-invariant Ricci Tensor

We prove that every almost Hermitian 4-manifold with J-invariant Ricci tensor which is conformally flat or has harmonic curvature is either a space of constant curvature or a Kähler manifold. We also obtain analogous results on almost Kähler 4-manifolds.     

Nonnegatively curved fixed point homogeneous 5-manifolds

Let G be a compact Lie group acting effectively by isometries on a compact Riemannian manifold M with nonempty fixed point set Fix(M, G). We say that the action is fixed point homogeneous if G acts transitively on a normal sphere to some component of Fix(M, G), equivalently, if Fix(M, G) has codimension one in the orbit space of the action. We classify up to diffeomorphism closed, simply connected 5-manifolds with nonnegative sectional curvature and an effective fixed point homogeneous isometric action of a compact Lie group.     

Hamiltonian systems with a discrete symmetry

If $\text{?: M → M}$ is an antisymplectic involution of a symplectic manifold M then the fixed set of ? is a Lagrangian submanifold L ? M. Moreover there exist cotangent bundle coordinates in a neighborhood of L in M such that ? in these coordinates maps a covector into its negative. Thus classical examples which have a discrete symmetry such as the restricted three-body problems are locally like a reversible system.