The aim of this paper is to find some equations of structure for almost Ricci solitons which generalize the equivalent for Ricci solitons. As a consequence of these equations we derive an integral formula for the compact case which enables us to show that a compact nontrivial almost Ricci soliton is isometric to a sphere provided either it has constant scalar curvature or its associated vector field is conformal. Moreover, we also use the Hodge-de Rham decomposition theorem to make a link with the associated vector field of an almost Ricci soliton.

The purpose of this article is to investigate Bach-flat critical metrics of the volume functional on a compact manifold M with boundary ∂ M. Here, we prove that a Bach-flat critical metric of the volume functional on a simply connected 4-dimensional manifold with boundary isometric to a standard sphere must be isometric to a geodesic ball in a simply connected space form R 4 , H 4 or S 4 . Moreover, we show that in dimension three the result even is true replacing the Bach-flat condition by the weaker assumption that M has divergence-free Bach tensor.

The aim of this note is to prove that any compact non-trivial almost Ricci soliton M n , g, X, λ with constant scalar curvature is isometric to a Euclidean sphere S n . As a consequence we obtain that every compact non-trivial almost Ricci soliton with constant scalar curvature is gradient. Moreover, the vector field X decomposes as the sum of a Killing vector field Y and the gradient of a suitable function.

A b s tra c t Complex common names such as Indian elephant or green tea denote a certain type o f entity, viz. kinds. Moreover, those kinds are always subkinds o f the kind denoted by their head noun. Establishing such subkinds is essentially the task o f classifying modifiers that are a defining trait of endocentrically structured complex common names. Examining complex common names of different lexico-syntactic types (N N compounds, N + N syntagmas, N P /P P syntagmas, A + N syntagmas) and from different languages (particularly English, G erm an and French) it can be shown that complex common names are subject to language-independent formal and semantic constraints. In particular, complex common names qualify as name-like expressions in that they tend to be deficient in term s of formal complexity and semantic compositionality.

The aim of this article is to study the space of metrics with constant scalar curvature of volume 1 that satisfies the critical point equation frakturLg*(f)=Ric˚, for simplicity CPE metrics. It has been conjectured that every CPE metric must be Einstein. Here, we shall focus our attention for 4‐dimensional half conformally flat manifolds M4. In fact, we shall show that for a nontrivial f,M4 must be isometric to a sphere double-struckS4 and f is some height function on S4.

Given a generalized M n+1 = I × φ F n Robertson-Walker spacetime we will classify strongly stable spacelike hypersurfaces with constant mean curvature whose warping function verifies a certain convexity condition. More precisely, we will show that given x : M n → M n+1 a closed spacelike hypersurfaces of M n+1 with constant mean curvature H and the warping function φ satisfying φ ′′ ≥ max{Hφ ′ , 0}, then M n is either minimal or a spacelike slice Mt 0 = {t 0 } × F , for some t 0 ∈ I.

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The aim of this paper is to present a link between the Perelman potential for a compact Ricci soliton M n and the Hodge-de Rham decomposition theorem, we shall use this result to present an integral formula which enables us to establish conditions under which the Ricci soliton is trivial. Moreover, given a Ricci soliton such that its associated vector field X is a conformal vector field we show that in the compact case X is a Killing vector field, while for the non-compact case, either the soliton is Gaussian or X is a Killing vector field.Mathematics Subject Classification (2010). Primary 53C25, 53C20, 53C21; Secondary 53C65.