We introduce and study a geometric heat flow to find Killing vector fields on closed Riemannian manifolds with positive sectional curvature. We study its various properties, prove the global existence of the solution to this flow, discuss its convergence and possible applications, and its relation to the Navier-Stokes equations on manifolds and Kazdan-Warner-Bourguignon-Ezin identity for conformal Killing vector fields. We also provide two new criterions on the existence of Killing vector fields. A similar flow to finding holomorphic vector fields on Kahler manifolds will be studied by Li and Liu (2014).

/pdf/a-geometric-heat-flow-for-vector-fields-hxjtmqn2a0.pdf

Integral formulas in riemannian geometry

A geometric heat flow for vector fields

We study conformal properties of the quantum kinetic equations in curved spacetime. A transformation law for the covariant Wigner function under conformal transformations of a spacetime is derived by using the formalism of tangent bundles. The conformal invariance of the quantum corrected Vlasov equation is proven. This provides a basis for generating new solutions of the quantum kinetic equations in the presence of gravitational and other external fields. We use our method to find explicit quantum corrections to the class of locally isotropic distributions, to which equilibrium distributions belong. We show that the quantum corrected stress--energy tensor for such distributions has, in general, a non--equilibrium structure. Local thermal equilibrium is possible in quantum systems only if an underlying spacetime is conformally static (not stationary). Possible applications of our results are discussed.

Conformal Transformations of the Wigner Function and Solutions of the Quantum Corrected Vlasov Equation

New Methods of the Bochner Technique and Their Applications

This is a review paper of the essential research on metric (Killing, homothetic, and conformal) symmetries of Riemannian, semi-Riemannian, and lightlike manifolds. We focus on the main characterization theorems and exhibit the state of art as it now stands. A sketch of the proofs of the most important results is presented together with sufficient references for related results.

/pdf/a-review-on-metric-symmetries-used-in-geometry-and-physics-3bwnu08nue.pdf

A Review on Metric Symmetries Used in Geometry and Physics

In the present paper we study three dimensional cosymplectic manifolds admitting almost Ricci solitons. Among others we prove that in a three dimensional compact orientable cosymplectic manifold M^3 withoutboundary an almost Ricci soliton reduces to Ricci soliton under certain restriction on the potential function lambda. As a consequence we obtain several corollaries. Moreover we study gradient almost Ricci solitons.

/pdf/on-three-dimensional-cosymplectic-manifolds-admitting-almost-3uiv53oki4.pdf

Integral formulas in riemannian geometry

Citations

A geometric heat flow for vector fields

Conformal Transformations of the Wigner Function and Solutions of the Quantum Corrected Vlasov Equation

New Methods of the Bochner Technique and Their Applications

A Review on Metric Symmetries Used in Geometry and Physics

On Three Dimensional Cosymplectic Manifolds Admitting Almost Ricci Solitons

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