2 Ted Chinburg ,
نویسنده
چکیده
Riemannian manifolds with positive sectional curvature have been a frequent topic of global Riemannian geometry for over 40 years. Nevertheless, there are relatively few known examples of such manifolds. The purpose of this article is to study the topological properties of some of these examples, the so-called Eschenburg spaces, in detail. In addition to positively curved metrics, some Eschenburg spaces also carry another special geometric structure, namely a 3-Sasakian metric, i.e. a metric whose Euclidean cone is Hyperkähler [BG]. 3-Sasakian spaces are interesting since they are Einstein manifolds and are connected to several other geometries: They admit an almost free, isometric action by SU(2) whose quotient is a quaternionic Kähler orbifold. The twistor space of this orbifold, which can be viewed as an S-quotient of the 3-Sasakian manifold, carries a natural Kähler-Einstein orbifold metric with positive scalar curvature. 3-Sasakian structures are rare and rigid, in fact the moduli space of such metrics on a fixed manifold consists of at most isolated points. This motivated C. Boyer and K. Galicki to pose the question in [BG][Question 9.9, p. 52] whether a manifold can admit more than one 3-Sasakian structure. Natural candidates for such examples are the 3Sasakian metrics discovered in [BGM]. They are defined on the Eschenburg biquotients Ea,b,c = diag(z , z, zc)\SU(3)/ diag(z, 1, 1), where a, b, c are positive, pairwise relatively prime integers. The simplest topological invariant of these spaces is the order of the fourth cohomology group, which is a finite cyclic group of order r = ab+ ac+ bc. By studying further topological invariants of these manifolds we show:
منابع مشابه
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 carri...
متن کاملThe Arithmetic Hyperbolic 3-manifold of Smallest Volume
We show that the arithmetic hyperbolic 3-manifold of smallest volume is the Weeks manifold. The next smallest one is the Meyerhoff manifold.
متن کاملMixed Zeta Functions
MIXED ZETA FUNCTIONS Pieter Mostert Ted Chinburg We examine Dirichlet series which combine the data of a distance function, u, a homogeneous degree zero function, φ, and a multivariable Dirichlet series, K. By using an integral representation and Cauchy’s residue formula, we show that under certain conditions on K, such functions extend to meromorphic functions on C, or to some region strictly ...
متن کاملPfaffians, the G-Signature Theorem and Galois Hodge discriminants
Let G be a finite group acting freely on a smooth projective scheme X over a locally compact field of characteristic 0. We show that the ε0-constants associated to symplectic representations V of G and the action of G on X may be determined from Pfaffian invariants associated to duality pairings on Hodge cohomology. We also use such Pfaffian invariants, along with equivariant Arakelov Euler cha...
متن کامل