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Integral formulas in riemannian geometry

From infinitesimal harmonic transformations to Ricci solitons

In this paper 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 of 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. The similar flow to finding holomorphic vector fields on K\"ahler manifolds will be studied in \cite{LL2}.

A geometric heat flow for vector fields

On quasiconformal deformations on manifolds

The object of the present paper is to characterize two classes of almost Kenmotsu manifolds admitting Ricci-Yamabe soliton. It is shown that a $(k,\mu)'$-almost Kenmotsu manifold admitting a Ricci-Yamabe soliton or gradient Ricci-Yamabe soliton is locally isometric to the Riemannian product $\mathbb{H}^{n+1}(-4) \times \mathbb{R}^n$. For the later case, the potential vector field is pointwise collinear with the Reeb vector field. Also, a $(k,\mu)$-almost Kenmotsu manifold admitting certain Ricci-Yamabe soliton with the curvature property $Q \cdot P = 0$ is locally isometric to the hyperbolic space $\mathbb{H}^{2n+1}(-1)$ and the non-existense of the curvature property $Q \cdot R = 0$ is proved.

Integral formulas in riemannian geometry

Citations

From infinitesimal harmonic transformations to Ricci solitons

A geometric heat flow for vector fields

On quasiconformal deformations on manifolds

Almost Kenmotsu metric as quasi Yamabe soliton

Ricci Solitons and Gradient Ricci Solitons on N(k)-Paracontact Manifolds

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