By resolving the Riemann curvature relative to a unit timelike vector into electric and magnetic parts, we define a duality transformation which interchanges active and passive electric parts. It implies interchange of roles of Ricci and Einstein curvatures. Further by modifying the vacuum/flat equation we construct spacetimes dual to the Schwarzschild solution and flat spacetime. The dual spac...

We study canonical paracontact connection on a para-Sasakian manifold. We prove that a Ricci-flat para-Sasakian manifold with respect to canonical paracontact connection is an η-Einstein manifold.We also investigate some properties of curvature tensor, conformal curvature tensor,W2curvature tensor, concircular curvature tensor, projective curvature tensor, and pseudo-projective curvature tensor...

This note is a study of nonnegativity conditions on curvature which are preserved by the Ricci flow. We focus on specific kinds of curvature conditions which we call noncoercive, these are the conditions for which nonnegative curvature and vanishing scalar curvature doesn’t imply flatness. We show that, in dimensions greater than 4, if a Ricci flow invariant condition is weaker than “Einstein w...

In this paper, we will study the asymptotic geometry of 4-dimensional steady gradient Ricci solitons under condition that they dimension reduce to 3-manifolds. We show such either strongly a spherical space form S3/Γ or weakly 3-dimensional Bryant soliton. also soliton singularity models with nonnegative curvature outside compact set are Ricci-flat ALE 4-manifolds manifolds. As further applicat...

We consider the sigma models where the base metric is proportional to the metric of the configuration space. We show that the corresponding sigma model equation admits a Lax pair. We also show that this type of sigma models in two dimensions are intimately related to the minimal surfaces in a flat pseudo Riemannian 3-space. We define two dimensional surfaces conformally related to the minimal s...

These notes provides some details on the lectures 2,3,4 on the Ricci flow with surgery. They are not complete and probably contains some inaccuracies. Interested readers can find most exhaustives explanations on the Perelman's papers in [KL]. The aim of these lecture is to give the classification and the description of 3-dimensional κ-solutions. Let κ > 0 and (M n , g(t)) a solution of the Ricc...

We show that the " Taub-bolt instantons " are the only Riemannian, Ricci-flat and asymptotically locally flat (ALF) C 2-metrics on a 4-dimensional, complete, simply connected man-ifold having (at least) a one-parameter group of isometries which are compatible with ALF, have trajectories of bounded length at infinity and no isolated fixed points. Attempts of estimating the path integral of Quant...

In the present study, we considered 3-dimensional generalized (κ, μ)-contact metric manifolds. We proved that a 3-dimensional generalized (κ, μ)-contact metric manifold is not locally φ-symmetric in the sense of Takahashi. However such a manifold is locally φ-symmetric provided that κ and μ are constants. Also it is shown that if a 3-dimensional generalized (κ, μ) -contact metric manifold is Ri...

Explicit solutions to the conifold equations with complex dimension n = 3, 4 in terms of complex coordinates (fields) are employed to construct the Ricci-flat Kähler metrics on these manifolds. The Kähler 2-forms are found to be closed. The complex realization of these conifold metrics are used in the construction of 2-dimensional non-linear sigma model with the conifolds as target spaces. The ...

The Ricci flow equation of a conformally flat Riemannian metric on a closed 2–dimensional configuration space is analysed. It turns out to be equivalent to the classical Hamilton–Jacobi equation for a point particle subject to a potential function that is proportional to the Ricci scalar curvature of configuration space. This allows one to obtain Schroedinger quantum mechanics from Perelman’s a...