Showing papers in "Surveys in differential geometry in 2001"

3-Sasakian manifolds

Abstract: This is an expository paper describing the geometry of certain Sasakian-Einstein manifolds. Such manifolds have recently become of interest due to Maldacena's AdS/CFT conjecture. They describe near-horizon geometries of branes at conical singularities.

Einstein-Weyl Geometry

Abstract: A Weyl manifold is a conformal manifold equipped with a torsion free connection preserving the conformal structure, called a Weyl connection. It is said to be Einstein-Weyl if the symmetric tracefree part of the Ricci tensor of this connection vanishes. In particular, if the connection is the Levi-Civita connection of a compatible Riemannian metric, then this metric is Einstein. Such an approach has two immediate advantages: firstly, the homothety invariance of the Einstein condition is made explicit by focusing on the connection rather than the metric; and secondly, not every Weyl connection is a Levi-Civita connection, and so Einstein-Weyl manifolds provide a natural generalisation of Einstein geometry. The simplest examples of this generalisation are the locally conformally Einstein manifolds. A Weyl connection on a conformal manifold is said to be closed if it is locally the Levi-Civita connection of a compatible metric; but it need not be a global metric connection unless the manifold is simply connected. Closed EinsteinWeyl structures are then locally (but not necessarily globally) Einstein, and provide an interpretation of the Einstein condition which is perhaps more appropriate for multiply connected manifolds. For example, S1×Sn−1 admits flat Weyl structures, which are therefore closed Einstein-Weyl. These closed structures arise naturally in complex and quaternionic geometry. Einstein-Weyl geometry not only provides a different way of viewing Einstein manifolds, but also a broader setting in which to look for and study them. For instance, few compact Einstein manifolds with positive scalar curvature and continuous isometries are known to have Einstein deformations, yet we shall see that it is precisely under these two conditions that nontrivial Einstein-Weyl deformations can be shown to exist, at least infinitesimally. The Einstein-Weyl condition is particularly interesting in three dimensions, where the only Einstein manifolds are the spaces of constant curvature. In contrast, three dimensional Einstein-Weyl geometry is extremely rich [16, 68, 72], and has an equivalent formulation in twistor theory [34] which provides a tool for constructing selfdual four dimensional geometries. In section 10, we shall discuss a construction relating Einstein-Weyl 3-manifolds and hyperKahler 4-manifolds [40, 29, 50, 79]. Twistor methods also yield complete selfdual Einstein metrics of negative scalar curvature with prescribed conformal infinity [48, 35]. An important special case of this construction is the case of an Einstein-Weyl conformal infinity [34, 61]. Although Einstein-Weyl manifolds can be studied, along with Einstein manifolds, in a Riemannian framework, the natural context is Weyl geometry [23]. We

Einstein deformations of hyperbolic metrics

Abstract: The simplest nontrivial examples of Einstein metrics are certainly rank one symmetric spaces. We are interested in those of negative curvature, that is the hyperbolic spaces KH (m > 2), where K is the field of real numbers (R), complex numbers (C), quaternions (H) or the algebra of octonions (O); in the last case we have only the Cayley hyperbolic plane OH. They are the noncompact duals of the projective spaces KP. We normalize the metric so that the maximum of the sectional curvature is −1. We denote by d the real dimension of K (so d = 1, 2, 4 or 8) and by n = md the real dimension of KH. The boundary sphere Sn−1 of a hyperbolic space carries a rich geometric structure, namely a conformal Carnot-Caratheodory metric. Let see this first in the real and complex examples. The real hyperbolic space (with constant sectional curvature −1) is the unit ball B in R, with the metric

Compact Riemannian manifolds with exceptional holonomy

Abstract: Suppose that M is an orientable n-dimensional manifold, and g a Riemannian metric on M . Then the holonomy group Hol(g) of g is an important invariant of g. It is a subgroup of SO(n). For generic metrics g on M the holonomy group Hol(g) is SO(n), but for some special g the holonomy group may be a proper Lie subgroup of SO(n). When this happens the metric g is compatible with some extra geometric structure on M , such as a complex structure. The possibilities for Hol(g) were classified in 1955 by Berger. Under conditions on M and g given in §1, Berger found that Hol(g) must be one of SO(n), U(m), SU(m), Sp(m), Sp(m)Sp(1), G2 or Spin(7). His methods showed that Hol(g) is intimately related to the Riemann curvature R of g. One consequence of this is that metrics with holonomy Sp(m)Sp(1) for m > 1 are automatically Einstein, and metrics with holonomy SU(m), Sp(m), G2 or Spin(7) are Ricci-flat. Now, people have found many different ways of producing examples of metrics with these holonomy groups, by exploiting the extra geometric structure – for example, quotient constructions, twistor geometry, homogeneous and cohomogeneity one examples, and analytic approaches such as Yau’s solution of the Calabi conjecture. Naturally, these methods yield examples of Einstein and Ricci-flat manifolds. In fact, metrics with special holonomy groups provide the only examples of compact, Ricci-flat Riemannian manifolds that are known (or known to the author). The holonomy groups G2 and Spin(7) are known as the exceptional holonomy groups, since they are the exceptional cases in Berger’s classification. Here G2 is a holonomy group in dimension 7, and Spin(7) is a holonomy group in dimension 8. Thus, metrics with holonomy G2 and Spin(7) are examples of Ricci-flat metrics on 7and 8-manifolds. The exceptional holonomy groups are the most mysterious of the groups on Berger’s list, and have taken longest to reveal their secrets – it was not even known until 1985 that metrics with these holonomy groups existed. The purpose of this chapter is to describe the construction of compact Riemannian manifolds with holonomy G2 and Spin(7). These constructions were found in 1994-5 by the present author, and appear in [16], [17] for the case of G2, and in

The stability of Minkowski Space-Time

Abstract: These equations are at first sight a degenerate differential system. That is, the null space of the symbol o^ at a given covector £ is nonzero for all covectors £. This is due to the fact that the equations are generally covariant; proper account must be taken of the geometric equivalence of metrics related by diffeomorphism. This is done by considering o not on the space of 2-covariant symmetric tensors at a point but rather on the quotient of this space by the equivalence relation induced by the symbol of the diffeomorphisms; two such tensors are equivalent if they differ by f X + X C for some covector X. The null space of a^ is then found to be nonzero if and only if £ belongs to the null cone defined by the metric g. The Einstein equations are therefore of hyperbolic character. The central mathematical problem of the theory is the initial value problem. An initial data set is a 3-dimensional manifold E with a positive definite metric gtj and a 2-covariant symmetric tensorfield fcy. The problem is to find a 4-dimensional manifold M with a metric gsV of signature (3, 1) satisfying the Einstein vacuum equations and an imbedding of 27 into M such that gtj and fcy are respectively the first and second fundamental forms induced on E by the imbedding. The Einstein vacuum equations impose on the initial data set the constraint equations: