In this paper we introduce the concept of (ε)-almost paracontact manifolds, and in particular, of (ε)-para Sasakian manifolds. Several examples are presented. Some typical identities for curvature tensor and Ricci tensor of (ε)-para Sasakian manifolds are obtained. We prove that if a semi-Riemannian manifold is one of flat, proper recurrent or proper Ricci-recurrent, then it can not admit an (ε...

We consider analytic curves ∇ t of symplectic connections of Ricci type on the torus T 2n with ∇ 0 the standard connection. We show, by a recursion argument, that if ∇ t is a formal curve of such connections then there exists a formal curve of symplectomorphisms ψ t such that ψ t · ∇ t is a formal curve of flat T 2n-invariant symplectic connections and so ∇ t is flat for all t. Applying this re...

We prove uniqueness and existence theorems for four-dimensional asymptotically flat, Ricci-flat, gravitational instantons with a torus symmetry. In particular, we that such are uniquely characterised by their rod structure, which is data encode the fixed point sets of action. Furthermore, establish every admissible structure there exists an instanton smooth up to possible conical singularities ...

We propose a class of N = 2 supersymmetric nonlinear sigma models on the Ricci-flat Kähler manifolds with O(N) symmetry. ∗ E-mail: [email protected] † E-mail: [email protected] ‡ E-mail: [email protected] String theory propagating in a curved spacetime is described by a conformal invariant nonlinear sigma model in two dimensions. Spacetime supersym...

We obtain Schroedinger quantum mechanics from Perelman’s functional and from the Ricci flow equations of a conformally flat Riemannian metric on a closed 2– dimensional configuration space. We explore links with the recently discussed emergent quantum mechanics.

We study G-invariant Kähler metrics on G from the Hamiltonian point of view. As an application we show that there exist G × G-invariant Ricci-flat Kähler metrics on G for any compact semisimple Lie group G.

We solve the Ricci-flat equations of extended general relativity to obtain an interesting class of cosmological models. The solutions are analogous to the 4D ones of Bianchi type-I of Kasner type and have significant implications for astrophysics.

In this article we study the Kähler Ricci flow, the corresponding parabolic Monge Ampère equation and complete noncompact Kähler Ricci flat manifolds. In our main result Theorem 1 we prove that if (M, g) is sufficiently close to being Kähler Ricci flat in a suitable sense, then the Kähler-Ricci flow (1) has a long time smooth solution g(t) converging smoothly uniformly on compact sets to a comp...

We give a new construction of Ricci-flat self-dual metrics which is a natural extension of the Gibbons–Hawking ansatz. We also give characterisations of both these constructions, and explain how they come from harmonic morphisms.

Using geometric quantization procedure, the quantization of algebra of observables for physical system with Ricci-flat phase space is obtained. In the classical case the appointed physical system is reduced to harmonic oscillator when the one real parameter is vanished.