Ricci–Yamabe maps for Riemannian flows and their volume variation and volume entropy

Abstract

The aim of this short note is to produce new examples of geometrical flows associated to a given Riemannian flow g(t). The considered flow in covariant symmetric 2-tensor fields will be called Ricci-Yamabe map since it involves a scalar combination of Ricci tensor and scalar curvature of g(t). Due to the signs of considered scalars the Ricci-Yamabe flow can be also a Riemannian or semi-Riemannian or singular Riemannian flow. We study the associated function of volume variation as well as the volume entropy. Finally, since the two-dimensional case was the most commonly addressed situation we express the Ricci flow equation in all four orthogonal separable coordinate systems of the plane.

Geometry of α-Cosymplectic Metric as ∗-Conformal η-Ricci–Yamabe Solitons Admitting Quarter-Symmetric Metric Connection

“…In 2019, S. Güler et al [21] introduced a new geometry flow, which is a scalar combination of the Ricci and Yamabe flow under the Ricci-Yamabe map. The new flow is also known as the Ricci-Yamabe flow for type (ρ, q).…”

“…Suppose that M is a Riemannian manifold of dimension n and T s 2 (M) is a linear space of its symmetric tensor fields for (0, 2)-type and Riem(M) T s 2 (M) is infinite space of its Riemannian metrics. In the paper [21], the authors have stated the below definition: Definition 1. [21] Suppose that a Riemannian flow on M is a smooth map:…”

“…In the paper [21], the authors have stated the below definition: Definition 1. [21] Suppose that a Riemannian flow on M is a smooth map:…”

“…Definition 2. [21] We call the map RY (ρ,q,g) : I → T s 2 (M) which is defined by: RY (ρ,q,g) := 2ρS(t) + qr(t)g(t) + ∂ ∂t g(t), is (ρ, q)-Ricci-Yamabe map of (M n , g), where ρ, q are some scalars. If RY (ρ,q,g) ≡ 0, after that, call g(•) an (ρ, q)-Ricci-Yamabe flow.…”

On Ricci–Yamabe soliton and geometrical structure in a perfect fluid spacetime

“…Recently, Guler and Crasmareanu [10] introduced a new geometric flow which is a scalar combination of Ricci flow and Yamabe flow, and called it as Ricci-Yamabe map. The Ricci-Yamabe flow of type (α, β) is defined as follows:…”