Recently, the study of three-dimensional spaces is becoming of great interest. In these dimensions the Cotton tensor is prominent as the substitute for the Weyl tensor. It is conformally invariant and its vanishing is equivalent to conformal flatness. However, the Cotton tensor arises in the context of the Bianchi identities and is present in any dimension n. We present a systematic derivation of the Cotton tensor. We perform its irreducible decomposition and determine its number of independent components as n(n 2 − 4)/3 for the first time. Subsequently, we exhibit its characteristic properties and perform a classification of the Cotton tensor in three dimensions. We investigate some solutions of Einstein's field equations in three dimensions and of the topologically massive gravity model of Deser, Jackiw, and Templeton. For each class examples are given. Finally we investigate the relation between the Cotton tensor and the energy-momentum in Einstein's theory and derive a conformally flat perfect fluid solution of Einstein's field equations in three dimensions. file cott16.tex,Recently, the study of three-dimensional spaces is becoming of great interest; for these spaces the Weyl tensor is always zero and the vanishing of the Cotton tensor depends on the type of relation between the Ricci tensor and the energy-momentum tensor of matter.Any three-dimensional space is conformally flat if the Cotton tensor vanishes. If matter is present, the Ricci tensor is related to the energy-momentum tensor of matter by means of the Einstein equations. Then the vanishing of the Cotton tensor imposes severe restrictions on the energy-momentum tensor. The Cotton tensor also plays a role in the context of the Hamiltonian formulation of general relativity, see [10].The outline of the article is as follows. First we derive the Cotton 2-form in the context of the Bianchi identities. Subsequently we describe its characteristic properties and perform

In the framework of (1+2)-dimensional Poincaré gauge gravity, we start from the Lagrangian of the Mielke-Baekler (MB) model that depends on torsion and curvature and includes translational and Lorentzian Chern-Simons terms. We find a general stationary circularly symmetric vacuum solution of the field equations.We determine the properties of this solution, in particular its mass and its angular momentum. For vanishing torsion, we recover the BTZ-solution. We also derive the general conformally flat vacuum solution with torsion. In this framework, we discuss Cartan's (3-dimensional) spiral staircase and find that it is not only a special case of our new vacuum solution, but can alternatively be understood as a solution of the 3-dimensional Einstein-Cartan theory with matter of constant pressure and constant torque. file 3dexact19.tex, 2003-06-21

We show that the Einstein-aether theory of Jacobson and Mattingly (J&M) can be understood in the framework of the metric-affine (gauge theory of) gravity (MAG).We achieve this by relating the aether vector field of J&M to certain post-Riemannian nonmetricity pieces contained in an independent linear connection of spacetime.Then, for the aether, a corresponding geometrical curvature-square Lagrangian with a massive piece can be formulated straightforwardly. We find an exact spherically symmetric solution of our model.

Starting from Newton's gravitational theory, we give a general introduction into the spherically symmetric solution of Einstein's vacuum field equation, the Schwarzschild(-Droste) solution, and into one specific stationary axially symmetric solution, the Kerr solution. The Schwarzschild solution is unique and its metric can be interpreted as the exterior gravitational field of a spherically symmetric mass. The Kerr solution is only unique if the multipole moments of its mass and its angular momentum take on prescribed values. Its metric can be interpreted as the exterior gravitational field of a suitably rotating mass distribution. Both solutions describe objects exhibiting an event horizon, a frontier of no return. The corresponding notion of a black hole is explained to some extent. Eventually, we present some generalizations of the Kerr solution.

Starting from Newton's gravitational theory, we give a general introduction into the spherically symmetric solution of Einstein's vacuum field equation, the Schwarzschild(-Droste) solution, and into one specific stationary axially symmetric solution, the Kerr solution. The Schwarzschild solution is unique and its metric can be interpreted as the exterior gravitational field of a spherically symmetric mass. The Kerr solution is only unique if the multipole moments of its mass and its angular momentum take on prescribed values. Its metric can be interpreted as the exterior gravitational field of a suitably rotating mass distribution. Both solutions describe objects exhibiting an event horizon, a frontier of no return. The corresponding notion of a black hole is explained to some extent. Eventually, we present some generalizations of the Kerr solution.

The definition of the Einstein 3-form Gα is motivated by means of the contracted 2nd Bianchi identity. This definition involves at first the complete curvature 2-form. The 1-form Lα is defined via Gα = L β ∧ ⋆ (ϑ β ∧ ϑα) (here ⋆ is the Hodge-star, ϑα the coframe). It is equivalent to the Einstein 3-form and represents a certain contraction of the curvature 2-form. A variational formula of Salgado on quadratic invariants of the Lα 1-form is discussed, generalized, and put into proper perspective.

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