相关论文: The Ricci decomposition of the inertia tensor for …

The Berger-Ebin and York $L^2$-orthogonal decompositions of the vector space of symmetric bilinear differential two-forms are fundamental tools in global Riemannian geometry. In this paper, we investigate the structure of Ricci tensors on…

We prove that a four-dimensional gradient shrinking Ricci soliton with $\delta W^{\pm}=0$ is either Einstein, or a finite quotient of $S^3\times\mathbb{R}$, $S^2\times\mathbb{R}^2$ or $\mathbb{R}^4$. We also prove that a four-dimensional…

We provide a step towards classifying Riemannian four-manifolds in which the curvature tensor has zero divergence, or -- equivalently -- the Ricci tensor Ric satisfies the Codazzi equation. Every known compact manifold of this type belongs…

We study the field equations of extensions of General Relativity formulated within a metric-affine formalism setting torsion to zero (Palatini approach). We find that different (second-order) dynamical equations arise depending on whether…

In a previous paper, the first two authors classified complete Ricci-flat ALF Riemannian 4-manifolds that are toric and Hermitian, but non-Kaehler. In this article, we consider general Ricci-flat deformations of such spaces, assuming only…

It is fairly well known that rotation in three dimensions can be expressed as a quadratic in a skew symmetric matrix via the Euler-Rodrigues formula. A generalized Euler-Rodrigues polynomial of degree 2n in a skew symmetric generating…

For the Bach-flat closed manifold with positive scalar curvature, we prove a rigidity result under a given inequality involving the Weyl curvature and the traceless Ricci curvature. Moveover, under an inequality involving…

For complete Riemannian manifolds with vanishing Bach tensor and positive constant scalar curvature, we provide a rigidity theorem characterized by some pointwise inequalities. Furthermore, we prove some rigidity results under an inequality…

We show that the only complete shrinking gradient Ricci solitons with vanishing Weyl tensor are quotients of the standard ones. This gives a new proof of the Hamilton-Ivey-Perel'man classification of 3-dimensional shrinking gradient…

We classify Einstein metrics on $\mathbb{R}^4$ invariant under a four-dimensional group of isometries including a principal action of the Heisenberg group. The metrics are either Ricci-flat or of negative Ricci curvature. We show that all…

We investigate four-dimensional gradient shrinking Ricci solitons with positive modified sectional curvature. Our first main result shows that if the norm of the self-dual Weyl tensor and the scalar curvature satisfy a certain sharp…

We study the geometrical properties of null congruences generated by an aligned null direction of the Weyl tensor (WAND) in spacetimes of the Weyl and Ricci type N (possibly with a non-vanishing cosmological constant) in an arbitrary…

Starting with the curvature 2-form a recursive construction of totally antisymmetrised 2p-forms is introduced, to which we refer as p-Riemann tensors. Contraction of indices permits a corresponding generalisation of the Ricci tensor.…

In this paper, we prove rigidity results on gradient shrinking Ricci solitons with weakly harmonic Weyl curvature tensors. Let $(M^n, g)$ be a compact gradient shrinking Ricci soliton satisfying ${\rm Ric}_g + Ddf = \rho g$ with $\rho >0$…

We consider $n$-dimensional asymptotically anti-de Sitter spacetimes in higher curvature gravitational theories with $n \geq 4$, by employing the conformal completion technique. We first argue that a condition on the Ricci tensor should be…

We describe the structure of the Ricci tensor on a locally homogeneous Lorentzian gradient Ricci soliton. In the non-steady case, we show the soliton is rigid in dimensions three and four. In the steady case, we give a complete…

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…

It is known that on $\mathrm{RCD}$ spaces one can define a distributional Ricci tensor ${\bf Ric}$. Here we give a fine description of this object by showing that it admits the polar decomposition $${\bf Ric}=\omega\,|{\bf Ric}|$$ for a…

Using the harmonic superspace approach, we construct, at the linearized level, $\mathcal{N}=2$ supersymmetric curvatures generalizing scalar curvature, Ricci curvature and Weyl tensor. These supercurvatures are the building blocks of…

We determine the isomorphism classes of symmetric symplectic manifolds of dimension at least 4 which are connected, simply-connected and have a curvature tensor which has only one non-vanishing irreducible component -- the Ricci tensor.