Ricci--Bourguignon Almost Solitons with Special Potential on Sasaki-like Almost Contact Complex Riemannian Manifolds

TL;DR

This paper studies Ricci--Bourguignon almost solitons on Sasaki-like almost contact complex Riemannian manifolds, introducing a new class of geometric structures with self-similar metrics and providing explicit examples and characterizations.

Contribution

It introduces a new class of Ricci--Bourguignon almost solitons on Sasaki-like manifolds, generalizing known solitons using pairs of metrics and special potential functions.

Findings

Characterization of Ricci--Bourguignon almost solitons on Sasaki-like manifolds

Construction of explicit examples of such solitons

Confirmation of theoretical properties through examples

Abstract

Almost contact complex Riemannian manifolds, also known as almost contact B-metric manifolds, are equipped with a pair of pseudo-Riemannian metrics that are mutually associated with each other using the tensor structure. Here we consider a special class of these manifolds, those of the Sasaki-like type. They have an interesting geometric interpretation: the complex cone of such a manifold is a holomorphic complex Riemannian manifold (also called a K\"ahler-Norden manifold). The basic metric on the considered manifold is specialized here as a soliton, i.e. has an additional curvature property such that the metric is a self-similar solution of an intrinsic geometric flow. Almost solitons are more general objects than solitons because they use functions rather than constants as coefficients in the defining condition. A $\beta$-Ricci-Bourguignon-like almost soliton ($\beta$ is a real constant) is defined using the pair of metrics. The introduced soliton is a generalization of some well-known (almost) solitons (such as those of Ricci, Schouten, and Einstein), which in principle arise from a single metric rather than a pair of metrics. The soliton potential is chosen to be pointwise collinear to the Reeb vector field, or the Lie derivative of any B-metric along the potential to be the same metric multiplied by a function. The resulting manifolds equipped with the introduced almost solitons are characterized geometrically. Appropriate examples for two types of almost solitons are constructed and the properties obtained in the theoretical part are confirmed.