Mixed Killing Vector Fields on the Cigar Ricci-Bourguignon Soliton

Abstract: 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.

• The paper classifies complete mixed Killing fields on the Cigar Ricci–Bourguignon soliton and proves that they form a five-dimensional vector space with an explicit basis.
• The authors establish that positively curved, steady almost gradient Ricci–Bourguignon solitons on surfaces are rigid under the stated hypotheses and must be isometric to Hamilton’s Cigar soliton up to homothety.
• The paper derives the complete geodesic structure, showing that non-radial geodesics have one turning point and that all geodesics except those through the tip escape to infinity.

Overview

This paper studies mixed Killing vector fields—vector fields $V$ satisfying $\mathcal{L}_V\mathcal{L}_V g = f\,\mathcal{L}_V g$ for some smooth function $f$—on the Cigar Ricci–Bourguignon soliton. The notion, introduced by Ghosh, interpolates between Killing fields ($\mathcal{L}_V g = 0$), 2-Killing fields ($f \equiv 0$), and conformal fields. The paper's central finding is a rigidity phenomenon: whereas on flat and product geometries the class of mixed Killing fields is infinite-dimensional, on the complete Cigar Ricci–Bourguignon soliton it collapses to a five-dimensional vector space, for which an explicit basis is given. Alongside this classification, the authors prove a two-dimensional rigidity theorem for steady almost gradient Ricci–Bourguignon solitons and give a complete description of the geodesic structure of the soliton.

Mixed Killing fields: generalities and flat/product examples

A vector field $V$ is mixed Killing if its second Lie derivative of the metric is pointwise proportional to the first. Every conformal field is mixed Killing: if $\mathcal{L}_V g = 2\lambda g$, then on $\{\lambda \neq 0\}$ one has the mixed Killing factor $f = V(\lambda)/\lambda + 2\lambda$, which reduces to the constant $2c$ for homotheties. The converse fails dramatically in low-curvature settings.

For a Riemannian product $\mathcal{L}_V\mathcal{L}_V g = f\,\mathcal{L}_V g$0, any field of the form $\mathcal{L}_V\mathcal{L}_V g = f\,\mathcal{L}_V g$1 satisfies

$\mathcal{L}_V\mathcal{L}_V g = f\,\mathcal{L}_V g$2

so $\mathcal{L}_V\mathcal{L}_V g = f\,\mathcal{L}_V g$3 is mixed Killing on $\mathcal{L}_V\mathcal{L}_V g = f\,\mathcal{L}_V g$4 with factor $\mathcal{L}_V\mathcal{L}_V g = f\,\mathcal{L}_V g$5, but is conformal only when $\mathcal{L}_V\mathcal{L}_V g = f\,\mathcal{L}_V g$6. Consequently, on Euclidean space the strictly non-conformal mixed Killing class is infinite-dimensional. This contrast sets up the rigidity results that follow: curvature, as encoded in the Cigar geometry, severely constrains the mixed Killing condition.

Rigidity of the Cigar Ricci–Bourguignon soliton

The Cigar Ricci–Bourguignon soliton is the family of metrics on $\mathcal{L}_V\mathcal{L}_V g = f\,\mathcal{L}_V g$7,

$\mathcal{L}_V\mathcal{L}_V g = f\,\mathcal{L}_V g$8

with potential vector field $\mathcal{L}_V\mathcal{L}_V g = f\,\mathcal{L}_V g$9 and vanishing soliton function; it solves $f$0 for all $f$1. In geodesic polar coordinates $f$2 via $f$3, the metric takes the warped form $f$4: Euclidean near the tip and asymptotically cylindrical at infinity. Its Gaussian curvature is $f$5, decaying exponentially.

The main structural result is a classification theorem: any complete surface admitting a steady almost gradient Ricci–Bourguignon soliton with $f$6, positive Gaussian curvature, and a critical point of the potential, is isometric up to homothety to Hamilton's Cigar soliton. The proof exploits the fact that on a surface the soliton equation forces $f$7 to be gradient-conformal, so Tashiro's classical result yields rotational symmetry; the resulting ODE system for the warping function integrates to $f$8, giving $f$9 exactly. The assumptions are essential: the hypothesis $\mathcal{L}_V g = 0$0 has a zero and $\mathcal{L}_V g = 0$1 are used at specific steps, and the result is stated only for surfaces.

Classification of mixed Killing fields on the Cigar

The potential vector field $\mathcal{L}_V g = 0$2 itself is shown to be mixed Killing, with factor $\mathcal{L}_V g = 0$3, nonzero off the circle $\mathcal{L}_V g = 0$4—the observation motivating the entire analysis.

The classification proceeds by splitting fields into angular and radial components:

• Angular rigidity: any smooth angular field $\mathcal{L}_V g = 0$5 that is mixed Killing must have $\mathcal{L}_V g = 0$6 constant, hence is a genuine rotational Killing field. Unlike the product case, no strict (non-Killing) angular mixed Killing fields exist.
• Radial fields: a complete radial field $\mathcal{L}_V g = 0$7 is mixed Killing if and only if $\mathcal{L}_V g = 0$8 for constants $\mathcal{L}_V g = 0$9. It is conformal precisely when $f \equiv 0$0; for $f \equiv 0$1 it is strictly mixed Killing. Smooth extension across the tip and across zeros of $f \equiv 0$2 is verified explicitly.
• Combining these, every complete local mixed Killing field has the form $f \equiv 0$3.

The global statement requires care because the complement of $f \equiv 0$4 is covered by two charts; the constants $f \equiv 0$5 are unique up to diffeomorphisms of the chart domains, and the pieces agree on overlaps.

To count dimensions, the paper first establishes that the complete conformal algebra of the Cigar RB metric coincides with that of the Euclidean metric and is four-dimensional, spanned by $f \equiv 0$6, $f \equiv 0$7, $f \equiv 0$8, and $f \equiv 0$9. Completeness is crucial here: holomorphic fields such as $V$0 are conformal but incomplete, generating an infinite-dimensional algebra. Since $V$1, the extra radial generator is $V$2, giving the main result:

$V$3

so $V$4. This stands in sharp contrast to Euclidean space, where the corresponding space is infinite-dimensional—a concrete instance of curvature-induced finiteness for this symmetry class.

Geodesic structure

Working in the warped coordinates, the geodesic equations admit two conserved quantities: the angular momentum $V$5 and the speed $V$6. The resulting classification is explicit:

Type	Condition	Behaviour
Radial	$V$7	Passes through the tip; $V$8
Non-radial	$V$9, $\mathcal{L}_V g = 2\lambda g$0	Unique turning point $\mathcal{L}_V g = 2\lambda g$1; escapes to infinity in both time directions

Non-radial geodesics satisfy $\mathcal{L}_V g = 2\lambda g$2 after suitable choice of affine origin. All geodesics except those emanating from the tip escape to spatial infinity, consistent with completeness and with the repulsive character induced by the positive, exponentially decaying curvature; no closed or spiralling geodesics occur.

Limitations and open questions

Several qualifications attach to the results. The rigidity theorem assumes $\mathcal{L}_V g = 2\lambda g$3, positive curvature, and existence of a zero of $\mathcal{L}_V g = 2\lambda g$4; behaviour outside these hypotheses is not addressed. The dimension count applies to complete mixed Killing fields—incomplete ones form an infinite-dimensional class even on the Cigar, so the finiteness result is genuinely a statement about globally defined symmetries. The classification of mixed Killing fields is carried out on the complement of $\mathcal{L}_V g = 2\lambda g$5, with smoothness at the degeneracy locus checked separately rather than treated intrinsically. The paper also leaves open whether the zero set of a mixed Killing field can be characterized, analogous to the discreteness known for closed conformal fields, and poses the question of whether the two-dimensional classification extends to higher-dimensional rotationally symmetric analogues $\mathcal{L}_V g = 2\lambda g$6 and, more generally, to higher-dimensional gradient Ricci solitons.

Conclusion

The paper establishes three contributions: a homothety-rigidity classification of complete steady almost gradient Ricci–Bourguignon solitons on positively curved surfaces; a complete, explicit classification of mixed Killing fields on the Cigar Ricci–Bourguignon soliton, showing that the space of complete such fields is five-dimensional with a concrete basis; and a full description of the soliton's geodesics, all of which escape to infinity except those through the tip. Collectively, these results demonstrate that the mixed Killing condition, permissive in flat and product geometries, becomes rigid under positive curvature, and they frame the higher-dimensional extension of this rigidity as the natural outstanding problem.