A note on smooth toral reductions of spheres

In this paper we show that there exist mod 2 obstructions to the smoothness of 3-Sasakian reductions of spheres. Specifically, if is a smooth 3-Sasakian manifold obtained by reduction of the 3-Sasakian sphere S _{4 n -1} by a torus, and if the second Betti number then 7, 11, 15, whereas, if then . We also show that the above bounds are sharp, in that we construct explicit examples of 3-Sasakian manifolds in the cases not excluded by these bounds. Received: 6 January 1997 / Revised version: 11 June 1997     

Minimal sphere bundles in Euclidean spheres

The purpose of this paper is to pursue to work initiated by Hsiang-Lawson and study cohomogeneity 1 minimal hypersurfaces in Euclidean spheres which are equivariant under the linear isotropy representation of a rank 3 compact symmetric space.Supported by the grant NSF DMS 90-01089 and by CNPq (Brazil)     

Fixity and free group actions on products of spheres

A representation G U(n) of degree n has fixity equal to the smallest integer f such that the induced action of G on U(n) /U(n-f-1) is free. Using bundle theory we show that if G admits a representation of fixity one, then it acts freely and smoothly on We use this to prove that a finite p-group (for p > 3) acts freely and smoothly on a product of two spheres if and only if it does not contain ( /p)³ as a subgroup. We use propagation methods from surgery theory to show that a representation of fixity f < n - 1 gives rise to a free action of G on a product of f + 1 spheres provided the order of G is relatively prime to (n - 1)!. We give an infinite collection of new examples of finite p-groups of rank r which act freely on a product of r spheres, hence verifying a strong form of a well-known conjecture for these groups. In addition we show that groups of fixity two act freely on a finite complex with the homotopy type of a product of three spheres. A number of examples are explicitly described.     

Three-dimensional Lie group actions on compact (4n+3)-dimensional geometric manifolds

The (4n+3)-dimensional sphere S⁴ⁿ⁺³ can be viewed as the boundary of the quaternionic hyperbolic space and the group PSp(n+1,1) of quaternionic hyperbolic isometries extends to a real analytic transitive action on S⁴ⁿ⁺³. We call the pair (PSp(n+1,1),S⁴ⁿ⁺³) a spherical Q C-C geometry. A manifold M locally modelled on this geometry is said to be a spherical Q C-C manifold. We shall classify all pairs (G,M) where G is a three-dimensional connected Lie group which acts smoothly and almost freely on a compact spherical Q C-C manifold M, preserving the geometric structure. As an application, we shall determine all compact 3-pseudo-Sasakian manifolds admitting spherical Q C-C structures.     

Cohomogeneity one Riemannian manifolds of non-positive curvature

We study a G-manifold M which admits a G-invariant Riemannian metric g of non-positive curvature. We describe all such Riemannian G-manifolds (M,g) of non-positive curvature with a semisimple Lie group G which acts on M regularly and classify cohomogeneity one G-manifolds M of a semisimple Lie group G which admit an invariant metric of non-positive curvature. Some results on non-existence of invariant metric of negative curvature on cohomogeneity one G-manifolds of a semisimple Lie group G are given.     

Isoparametric functions on exotic spheres

This paper extends widely the work in [11]. Existence and non-existence results of isoparametric functions on exotic spheres and Eells–Kuiper projective planes are established. In particular, every homotopy n  -sphere (n>4$n>4$) carries an isoparametric function (with certain metric) with 2 points as the focal set, in strong contrast to the classification of cohomogeneity one actions on homotopy spheres [26] (only exotic Kervaire spheres admit cohomogeneity one actions besides the standard spheres). As an application, we improve a beautiful result of Bérard-Bergery [2] (see also pp. 234–235 of [3]).     

A topological minorization for the volume of vector fields on 5-manifolds

A vector field X on a Riemannian manifold determines a submanifold in the tangent bundle. The volume of X is the volume of this submanifold for the induced Sasaki metric. When M is compact, the volume is well defined and, usually, this functional is studied for unit fields. Parallel vector fields are trivial minima of this functional.For manifolds of dimension 5, we obtain an explicit result showing how the topology of a vector field with constant length influences its volume. We apply this result to the case of vector fields that define Riemannian foliations with all leaves compact.Received: 29 April 2004     

Einstein metrics and the number of smooth structures on a four-manifold

We prove that for every natural number k there are simply connected topological four-manifolds which have at least k distinct smooth structures supporting Einstein metrics, and also have infinitely many distinct smooth structures not supporting Einstein metrics. Moreover, all these smooth structures become diffeomorphic to each other after connected sum with only one copy of the complex projective plane. We prove that manifolds with these properties cover a large geographical area.     

Circles and spheres in pseudo-Riemannian geometry

Summary A totally umbilical pseudo-Riemannian submanifold with the parallel mean curvature vector field is said to be an extrinsic sphere. A regular curve in a pseudo-Riemannian manifold is called a circle if it is an extrinsic sphere. LetM be ann-dimensional pseudo-Riemannian submanifold of index (0n) in a pseudo-Riemannian manifold with the metricg and the second fundamental formB. The following theorems are proved. For ₀ = +1 or –1, ₁ = +1, –1 or 0 (2–2 ₀+ ₁2n–2–2) and a positive constantk, every circlec inM withg(c, c) = ₀ andg( _c c, _c c) = ₁ k ² is a circle in iffM is an extrinsic sphere. For ₀ = +1 or –1 (–₀n–), every geodesicc inM withg(c, c) = ₀ is a circle in iffM is constant isotropic and B(x,x,x) = 0 for anyx T(M). In this theorem, assume, moreover, that 1n–1 and the first normal space is definite or zero at every point. Then we can prove thatM is an extrinsic sphere. When = 0 orn, this fact does not hold in general.     

How tightly can you fold a sphere?

Consider a compact, connected Lie group G acting isometrically on a sphere Sn of radius 1. The quotient of Sn by this group action, Sn/G, has a natural metric on it, and so we may ask what are its diameter and q-extents. These values have been computed for cohomogeneity one actions on spheres. In this paper, we compute the diameters, extents, and several q-extents of cohomogeneity two orbit spaces resulting from such actions, and we also obtain results about the q-extents of Euclidean disks. Additionally, via a simple geometric criterion, we can identify which of these actions give rise to a decomposition of the sphere as a union of disk bundles. In addition, as a service to the reader, we give a complete breakdown of all the isotropy subgroups resulting from cohomogeneity one and two actions.     

Real, complex and quaternionic equivariant vector fields on spheres

The equivariant real, complex and quaternionic vector fields on spheres problem is reduced to a question about the equivariant J-groups of the projective spaces. As an application of this reduction, we give a generalization of the results of Namboodiri [U. Namboodiri, Equivariant vector fields on spheres, Trans. Amer. Math. Soc. 278 (2) (1983) 431-460], on equivariant real vector fields, and Önder [T. Önder, Equivariant cross sections of complex Stiefel manifolds, Topology Appl. 109 (2001) 107-125], on equivariant complex vector fields, which avoids the restriction that the representation containing the sphere has enough orbit types.     

Smooth actions of finite Oliver groups on spheres

In this article, we deal with the following two questions. For smooth actions of a given finite group G on spheres S, which smooth manifolds F occur as the fixed point sets in S, and which real G-vector bundles ν over F occur as the equivariant normal bundles of F in S? We focus on the case G is an Oliver group and answer both questions under some conditions imposed on G, F, and ν. We construct smooth actions of G on spheres by making use of equivariant surgery, equivariant thickening, and Oliver's equivariant bundle extension method modified by an equivariant wegde sum construction and an equivariant bundle subtraction procedure.     

Minimal hyperspheres in rank two compact symmetric spaces

We describe a method to construct embedded, minimal hyperspheres in rank two compact symmetric spaces which are equivariant under the isotropy action of the symmetric space, and we supply the details of the construction for the exceptional Lie groupG ₂.Partially supported by CNPq (brazil)     

Diameters of 3-sphere quotients

Let G⊂O(4) act isometrically on S³. In this article we calculate a lower bound for the diameter of the quotient spaces S³/G. We find it to be , which is exactly the value of the lower bound for diameters of the spherical space forms. In the process, we are also able to find a lower bound for diameters for the spherical Aleksandrov spaces, Sn/G, of cohomogeneities 1 and 2, as well as for cohomogeneity 3 (with some restrictions on the group type). This leads us to conjecture that the diameter of Sn/G is increasing as the cohomogeneity of the group G increases.     

Topological properties of Eschenburg spaces and 3-Sasakian manifolds

We examine topological properties of the seven-dimensional positively curved Eschenburg biquotients and find many examples which are homeomorphic but not diffeomorphic. A special subfamily of these manifolds also carries a 3-Sasakian metric. Among these we construct a pair of 3-Sasakian spaces which are diffeomorphic to each other, thus giving rise to the first example of a manifold which carries two non-isometric 3-Sasakian metrics. Christine Escher was supported by a grant from the Association for Women in Mathematics. Wolfgang Ziller was supported by the Francis J. Carey Term Chair, and Ted Chinburg and Wolfgang Ziller were supported by a grant from the National Science Foundation.     

A Kummer-type construction of self-dual 4-manifolds

Supported in part by NSF grant DMS-9204093     

Homotopy types of locally linear representation forms

Summary The authors of [6] investigated certain locally linear actions of a cyclic groupG of odd order on homotopy spheres, the so-calledG-representation forms [16]. In particular, several conditions on a dimension function were described that made sure that it can be realized as the dimension function of aG-representation form. It remained unclear, whether all homotopy types with those dimension functions would support a locally linear structure. It is the aim of this note to show that this is not the case, i.e., to give examples of homotopy representations [17] with the same dimension functions some of which support a locally linear structure with stably trivial tangent bundle and others do not. The main tools are formulated as general splitting principles for fixed point and restriction functors that may have some interest in their own right, too. Part of the work with this paper was assembled while the authors were visiting Institut Mittag-Leffler at Djursholm, Sweden, whose support is gratefully acknowledged. This article was processed by the author using the LATEX style filecljour1 from Springer-Verlag.     

Minimal two spheres in Kähler-Einstein Fano manifolds

In this paper we show the existence of stable symplectic non-holomorphic two-spheres in Kähler manifolds of positive constant scalar curvature of real dimension four and in Kähler-Einstein Fano manifolds of real dimension six. Some of the techniques used involve deformation theory of algebraic cycles.