Einstein-Thorpe manifolds

One of the most fascinating facts in mathematics is that the local geometry of a manifold provides us with information about its global topology. For instance, the Euler-Poincaré characteristic χ of a compact oriented Riemannian manifold $M4k$ of even dimension can be written as an integral c=2V 2k! ²4k Mtrace*R2k *R2kdV where $R2k$ is the 2kth-curvature operator and $*$ is the Hodge star operator and V is the volume of the Euclidean unit 4k sphere and dV is the volume element of M.; If $R2k$commutes with $*$, we say that the metric is a Thorpe metric. In the 4-dimensional case, a metric is Thorpe metric if and only if it is Einstein. On the other hand, as we shall see in Section 2.4, Thorpe metrics need not be Einstein in higher dimensions.; We shall say that a Riemannian 4k manifold is Einstein-Thorpe if it is both Einstein and Thorpe.; In this dissertation, we shall see that (1) There is an infinite dimensional moduli space of Thorpe metrics on $T4k$ (k > 1). Most of these are not Einstein metrics. The same construction also yields Thorpe metrics on where $M4k-p$ is any compact oriented manifold. (2) However, every Einstein-Thorpe metric on T₈ must be flat. On compact oriented hyperbolic manifolds of dimension 8, every Einstein-Thorpe metric is a hyperbole metric up to rescalings and diffeomorphisms. (3) There are some manifolds of dimension 8 which have χ = 0 and P₂ = 0 but which never carry an Einstein-Thorpe metric. In particular, a compact orientable Einstein-Thorpe manifold ( M8, g) that satisfies c=2!2!4!∣ P2 ∣ must be ($T8/G$, flat) where Γ is of finite order.