Immersions of Sasaki-Ricci solitons into homogeneous Sasakian manifolds

TL;DR

This paper investigates the conditions under which Sasaki-Ricci solitons can be immersed into homogeneous Sasakian manifolds, revealing that many are actually $ ext{η}$-Einstein and providing examples of non-embeddable solitons.

Contribution

It establishes new results on local immersions of Sasaki-Ricci solitons into fiber products of homogeneous Sasakian manifolds and Sasakian space forms, including non-embeddability examples.

Findings

SRS immersed in fiber products are often $ ext{η}$-Einstein.

Stronger results for immersions into Sasakian space forms.

Existence of Kähler-Ricci solitons with no local holomorphic isometry into certain product manifolds.

Abstract

We discuss local Sasakian immersion of Sasaki-Ricci solitons (SRS) into fiber products of homogeneous Sasakian manifolds. In particular, we prove that SRS locally induced by a large class of fiber products of homogeneous Sasakian manifolds are, in fact, $\eta$-Einstein. The results are stronger for immersions into Sasakian space forms. Moreover, we show an example of a K\"ahler-Ricci soliton on $\mathbb C^n$ which admits no local holomorphic isometry into products of homogeneous bounded domains with flat K\"ahler manifolds and generalized flag manifolds.