We try to provide a visual introduction to some objects used in Riemannian geometry: parallel transport, sectional curvature, Ricci curvature, Bianchi identities... We then explain some of the strategies used to define analogues of curvature in non-smooth or discrete spaces, beginning with Alexandrov curvature and δ-hyperbolic spaces, and insisting on various notions of generalized Ricci curvat...

It is proved that a compact Kähler manifold whose Ricci tensor has two distinct constant non-negative eigenvalues is locally the product of two Kähler-Einstein manifolds. A stronger result is established for the case of Kähler surfaces. Irreducible Kähler manifolds with two distinct constant eigenvalues of the Ricci tensor are shown to exist in various situations: there are homogeneous examples...

Within a framework of noncommutative geometry, we develop an analogue of (pseudo) Riemannian geometry on nite and discrete sets. On a nite set, there is a counterpart of the continuum metric tensor with a simple geometric interpretation. The latter is based on a correspondence between rst order di erential calculi and digraphs (the vertices of the latter are given by the elements of the nite se...

We consider a closed manifold M with a Riemannian metric gij(t) evolving by ∂t gij = −2Sij where Sij(t) is a symmetric two-tensor on (M, g(t)). We prove that if Sij satisfies the tensor inequality D(Sij , X) ≥ 0 for all vector fields X on M , where D(Sij , X) is defined in (1.6), then one can construct a forwards and a backwards reduced volume quantity, the former being non-increasing, the latt...

The aim of this paper is to determine left-invariant strictly almost Kähler structures on 4-dimensional Lie groups (g, J,Ω) such that the Ricci tensor is J-invariant.

We use the recent discovered x−ψ duality to parameterize spacetime in terms of wave functions and quantum prepotentials. The components of metric, connection, curvature, Ricci tensor and scalar curvature as well as Einstein equations in this parameterization are

We establish a link between a connection symmetry, called conformal collineation, and almost Ricci soliton (in particular Ricci soliton) in reducible Ricci symmetric semi-Riemannian manifolds. As a physical application, by investigating the kinematic and dynamic properties of almost Ricci soliton manifolds, we present a physical model of imperfect fluid spacetimes. This model gives a general re...

We construct a 2-parameter family FZ of Riemannian metrics on the twistor space Z of a positive quaternion Kähler manifold M satisfying the following properties : (1) the family FZ contains an Einstein metric gZ and its scalings, (2) the family FZ is closed under the operation of making the convex sums, (3) the Ricci map g 7→ Ric(g) defines a dynamical system on the family FZ, (4) the Ricci flo...

We discuss recent calculations of the mass spectrum, electromagnetic and strong couplings of baryon resonances. The calculations are done in a collective constituent model for the nucleon, in which the resonances are interpreted as rotations and vibrations of a symmetric top with a prescribed distribution of the charge and magnetization. We analyze recent data on eta-photoand eta-electroproduct...