Mixed Killing Vector Fields on Cigar Ricci-Bourguignon Solitons
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| Format: | Preprint |
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2026
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| _version_ | 1866918533460393984 |
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| author | Aqib, Mohammad Shah, Hemangi Madhusudan |
| author_facet | Aqib, Mohammad Shah, Hemangi Madhusudan |
| contents | In this article, we study mixed Killing vector fields, defined by the condition $L_V L_V g = f\,L_V g$, on Cigar Ricci--Bourguignon solitons. While conformal vector fields are always mixed Killing, the converse fails in flat and open cylinders with base manifold geometries, where the mixed Killing class is infinite-dimensional. We establish a rigidity phenomenon for Cigar Ricci--Bourguignon solitons: 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 show that locally any mixed Killing field is the sum of a rotational Killing field and a mixed Killing radial field. Finally, we establish that the dimension of the vector space of complete mixed Killing fields of Cigar Ricci--Bourguignon solitons is $5$. Moreover, we explicitly determine a basis. Our results show that Cigar Ricci--Bourguignon solitons exhibit behaviour completely different from that of Euclidean space. Finally, we provide a complete description of the geodesic structure of Cigar Ricci--Bourguignon solitons. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_28970 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Mixed Killing Vector Fields on Cigar Ricci-Bourguignon Solitons Aqib, Mohammad Shah, Hemangi Madhusudan Differential Geometry 53B20, 53B25, 53C18, 53C35 In this article, we study mixed Killing vector fields, defined by the condition $L_V L_V g = f\,L_V g$, on Cigar Ricci--Bourguignon solitons. While conformal vector fields are always mixed Killing, the converse fails in flat and open cylinders with base manifold geometries, where the mixed Killing class is infinite-dimensional. We establish a rigidity phenomenon for Cigar Ricci--Bourguignon solitons: 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 show that locally any mixed Killing field is the sum of a rotational Killing field and a mixed Killing radial field. Finally, we establish that the dimension of the vector space of complete mixed Killing fields of Cigar Ricci--Bourguignon solitons is $5$. Moreover, we explicitly determine a basis. Our results show that Cigar Ricci--Bourguignon solitons exhibit behaviour completely different from that of Euclidean space. Finally, we provide a complete description of the geodesic structure of Cigar Ricci--Bourguignon solitons. |
| title | Mixed Killing Vector Fields on Cigar Ricci-Bourguignon Solitons |
| topic | Differential Geometry 53B20, 53B25, 53C18, 53C35 |
| url | https://arxiv.org/abs/2605.28970 |