Mixed Killing Vector Fields on the Cigar Ricci-Bourguignon Soliton

In this article, we study mixed Killing vector fields, defined by the condition $L_V L_V g = f\, L_V g$, on the Cigar Ricci--Bourguignon soliton. While conformal vector fields are always mixed Killing, the converse fails in flat and open cylinder with base as manifold geometries, where the mixed Killing class is infinite-dimensional. We establish a rigidity phenomenon of the Cigar Ricci--Bourguignon soliton: any complete steady almost gradient Ricci--Bourguignon soliton on a surface with positive curvature is, up to homothety, Hamilton's Cigar soliton. We then characterise complete mixed Killing fields, and affirm that locally any mixed Killing field is the sum of a rotationally Killing field and a mixed Killing radial field. Finally, we establish that the dimension of the vector space of complete mixed Killing fields of the Cigar Ricci--Bourguignon soliton is $5$. Moreover, we explicitly determine its basis. Thus, the Cigar Ricci--Bourguignon soliton exhibits completely different behaviour in contrast to Euclidean space. Finally, we also provide a complete description of the geodesic structure of the Cigar Ricci--Bourguignon soliton.