Different frameworks for defining energy at spatial infinity are reviewed. Some new results concerning the definitional uniqueness of energy-momentum are presented.

We study general properties of the partial differential equations for generalized gravity arising from the Lovelock Lagrangian.

The level sets of the momenta for GR are analyzed.

In his well-known problem section appearing in [Yl], s.-T. Yau posed the problem of establishing a Lorentzian analogue of the Cheeger-Gromoll [CG] splitting theorem of Riemannian geometry. Although stated in [Yl] as a problem in pure Lorentzian geometry, there is a physical, as well as mathematical, motivation for considering this problem [Y2). The purpose of this paper is to discuss the physical motivation, and to survey the progress that has been made on this and related problems. We will also briefly describe some of the methods which have been employed to attack such problems.

It is proved that all stationary and axially symmetric metrics that are rational in prolate-spheroidal coordinates have a simple canonical form. This follows directly from a theorem for 2 x 2 matrices with multinomial elements and has nothing to do with the field equations.

The main thrust of this paper is on real vacuum (single) Kerr-Schild (VKS) metrics. The conditions are given for real VKS metrics to have twistfree expanding principal null directions and to have Weyl tensors of Petrov type $D$. It follows that there are no twistfree type $II$ metrics i.e. no Robinson-Trautman ones. VKS metrics with twist either belong to the Kerr family or else are of type $II$. Non-expanding VKS metrics are of type $N$. The condition for a VKS metric to be of type $D$ leads naturally to a complex translation which has been used to obtain the Kerr metric from the Schwarzschild metric.

A survey is given of the structure and applications of spinor fields in three-dimensional (pseudo-) Riemannian manifolds. A systematic treatment, independent of the metric signature, is possible since there exists a fairly general structure, to be associated with unitary spinors, which encompasses all but the reality properties. The discussion begins with the algebraic and analytic properties of unitary spinors, the Ricci identities and curvature spinor, followed by the spinor adjungation as space reflection, and the SU(2) and SU(l,l) spin coefficients with some applications. The rapidly increasing range of applications includes space-times with Killing symmetries, the initial-value formulation, positivity theorems on gravitational energy and topologically massive gauge theories.

One of the outstanding problems in cosmology is the presence of inhomogeneities, which are characterized by an anisotropic pressure and density distribution. Following a method developed by McVittie (1967, 1968) it has been possible to find time-dependent spherically symmetric solutions of Einstein's field equations containing an arbitrary pressure and density distribution which connect smoothly to a Friedman universe for any desired equation of state.

We investigate supersymmetric Kaluza-Klein theory for a realistic unification theory. To account for chiral phenomenology of low energy physics, the Kaluza-Klein world must be of even dimensions, and for a theory beyond ten dimensions, we must introduce extra time-like dimension. However, we can avoid massless ghosts in 4-dimensional spacetime after spontaneous compactification of the ground state, and the cosmological constant is zero.

The equation of a hypersurface interacting with external fields in space-time is established. A class of models is based on the properties of a symmetric divergence-free distribution-valued stress tensor for the interacting system. The elastodynamic properties are encoded into the induced metric and shape tensor of the immersion together with invariants constructed from the Weingarten map. Two particular examples are discussed that are relevant to recent developments in the theories of relativistic extended particles.