We introduce a geometrical framework for double field theory in which generalized Riemann and torsion tensors are defined without reference to a particular basis. This invariant geometry provides a unifying framework for the frame-like and metric-like formulations developed before. We discuss the relation to generalized geometry and give an ‘index-free’ proof of the algebraic Bianchi identity. ...

For a positive definite fundamental tensor all known examples of Osserman algebraic curvature tensors have a typical structure. They can be produced from a metric tensor and a finite set of skew-symmetric matrices which fulfil Clifford commutation relations. We show by means of Young symmetrizers and a theorem of S. A. Fulling, R. C. King, B. G. Wybourne and C. J. Cummins that every algebraic c...

A theory of gravitation is proposed, modeled after the notion of a Ricci flow. In addition to the metric an independent volume enters as a fundamental geometric structure. Einstein gravity is included as a limiting case. Despite being a scalar-tensor theory the coupling to matter is different from Jordan-Brans-Dicke gravity. In particular there is no adjustable coupling constant. For the solar ...

Which smooth compact 4-manifolds admit an Einstein metric with non-negative Einstein constant? A complete answer is provided in the special case of 4-manifolds that also happen to admit either a complex structure or a symplectic structure. A Riemannian manifold (M, g) is said to be Einstein if it has constant Ricci curvature, in the sense that the function v −→ r(v, v) on the unit tangent bundl...

The main result of this paper is that a Lorentzian manifold is locally conformally equivalent to a manifold with recurrent lightlike vector field and totally isotropic Ricci tensor if and only if its conformal tractor holonomy admits a 2-dimensional totally isotropic invariant subspace. Furthermore, for semi-Riemannian manifolds of arbitrary signature we prove that the conformal holonomy algebr...

We study the decomposition of the Riemannian curvature R tensor of an almost quaternion-Hermitian manifold under the action of its structure group Sp(n)Sp(1). Using the minimal connection, we show that most components are determined by the intrinsic torsion ξ and its covariant derivative ∇̃ξ and determine relations between the decompositions of ξ ⊗ ξ, ∇̃ξ and R. We pay particular attention to the...

We investigate brane world models in higher-derivative gravity theories where the gravitational Lagrangian is an arbitrary function of the Ricci scalar. Making use of the conformal equivalence of such gravity models and Einstein-Hilbert gravity with a scalar field, we deduce the main features of higher-derivative gravity brane worlds. We solve for a gravity model that has corrections quadratic ...

We prove the following results: (i) A Sasakian metric as a nontrivial Ricci soliton is null η-Einstein, and expanding. Such a characterization permits to identify the Sasakian metric on the Heisenberg group H as an explicit example of (non-trivial) Ricci soliton of such type. (ii) If an ηEinstein contact metric manifold M has a vector field V leaving the structure tensor and the scalar curvatur...

The Palatini gravitational action is enlarged by an arbitrary function $f(X)$ of the determinants Ricci tensor and metric, $X=|\textbf{det}.R|/|\textbf{det}.g|$. resulting Ricci-determinant theory exhibits novel deviations from general relativity. We study a particular realization where extension characterized square-root Ricci-determinant, $f(X)=\lambda_\text{Edd}\sqrt{X}$, which corresponds t...

Applying a well known result for attracting fixed points of biholomorphisms [4, 6], we observe that one immediately obtains the following result: if (Mn, g) is a complete non-compact gradient Kähler-Ricci soliton which is either steady with positive Ricci curvature so that the scalar curvature attains its maximum at some point, or expanding with non-negative Ricci curvature, then M is biholomor...