May 27, 2026
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 sum of a rotationally Killing field and a mixed Killing radial field. Finally, we establish that the dimension of vector space of complete mixed Killing fields of the Cigar Ricci–Bourguignon soliton is \(5\). Moreover, we explicitly find its basis. Thus Cigar Ricci–Bourguignon soliton depicts 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.
M. S. C. 2020: 53B20, 53B25, 53C18, 53C35.
Keywords: Ricci-Bourguignon Soliton; Conformal Vector Field; Mixed Killing Field; Cigar Soliton, Killing Field
Geometric evolution equations have become central tools in differential geometry since Hamilton introduced the Ricci flow \[\frac{\partial g}{\partial t} = -2\operatorname{Ric}(g)\] [1]. Self-similar solutions of Ricci flow, known as Ricci solitons satisfy \[\operatorname{Ric} + \frac{1}{2} L_{V} g = \lambda g,\]
where \(V\) is a potential vector field and \(L_V\) denotes the Lie derivative with respect to \(V\), and play a fundamental role in singularity analysis and
geometric classification. A natural generalisation of Ricci flow is the Ricci–Bourguignon flow \[\frac{\partial g}{\partial t}
= -2(\operatorname{Ric} - \rho R g),\] introduced in [2]. Its self-similar solutions (whenever they exist) Ricci–Bourguignon solitons, interpolate between
the Ricci flow (\(\rho=0\)) and the other curvature-driven evolutions. Allowing the soliton constant to vary leads to almost Ricci–Bourguignon solitons [3], which generalise almost Ricci solitons and arise naturally in warped product and Einstein-type geometries. Mathematically, Ricci-Bourguignon almost solitons are solutions of \[\operatorname{Ric} + \frac{1}{2}\mathcal{L}_\xi g = (\lambda + \rho R)g,\] with a potential vector field \(\xi\), soliton function \(\lambda\), \(\rho\) an arbitrary constant and \(R\) scalar curvature.
One of the prime examples of Ricci solitons is two dimensional Hamilton’s Cigar soliton. This is the unique complete, non-compact, steady gradient Ricci soliton with positive curvature [1]. The soliton data is \[\left(\mathbb{R}^2,\;
g_{\mathrm{Cigar}} =\frac{dx^2+dy^2}{1+x^2+y^2},\; \xi=-2\Bigl(x\frac{\partial}{\partial x} +y\frac{\partial}{\partial y}\Bigr),\; \lambda=0\right).\] Here \[\xi := -2 \left(x\frac{\partial}{\partial x}
+y\frac{\partial}{\partial y}\right) = \nabla f, \\ \nonumber \; where \; f(x,y) := - \log\bigl(1+x^2+y^2\bigr).\] It is conformally flat, asymptotically cylindrical, and has Gaussian curvature decaying exponentially at infinity.
The Cigar Ricci-Bourguignon soliton was described in [4], and the soliton data is described as follows: \[\left(\mathbb{R}^2,\;
g_{\mathrm{Cigar-RB}}(t) =\frac{dx^2+dy^2}{E(t)+x^2+y^2},\; \xi=-2 (1-2 \rho) \Bigl(x\frac{\partial}{\partial x} +y\frac{\partial}{\partial y}\Bigr),\; \lambda=0\right).\] Here \[\label{xi-Cigar} \xi := -2 (1-2\rho) \left(x\frac{\partial}{\partial x} +y\frac{\partial}{\partial y}\right) = \nabla f, \\ \nonumber \; where \; f(x,y, t) := - (1-2 \rho)\log\bigl(E(t)+x^2+y^2\bigr)\tag{1}\] and \(E(t)=e^{4(1-2\rho)t}\). It satisfies \[\operatorname{Ric} + \nabla^2 f - \rho R g = 0\] for all \(t\). Thus \((\mathbb{R}^2,
g_{\mathrm{Cigar-RB}}(t),\xi,\lambda\equiv0,\rho)\) is a steady gradient almost Ricci–Bourguignon soliton. See Section 3 for more details on Cigar Ricci-Bourguignon almost solitons.
Cigar Ricci-Bourguignon almost solitons are characterised by our Theorem 4: Any complete surface admitting a steady almost gradient Ricci Bourguignon soliton with
positive Gaussian curvature is isometric, up to scaling, to Hamilton’s Cigar soliton. The potential vector fields (the family of potential vector fields depending on \(\rho\)) of Cigar gradient almost Ricci-Bourguignon
solitons satisfy a remarkable property, an iterated Lie-derivative relation: its second Lie derivative of the metric is proportional to the first. See Proposition 5 for the derivation. This observation motivates the study of a class of vector fields, so-called mixed Killing vector fields.
A vector field \(V\) on a Riemannian manifold \((M,g)\) is called mixed Killing if, \[L_V L_V g = f\, L_V g\] for some smooth function \(f\). This notion, introduced in [5], generalizes 2-Killing vector fields (where \(f
\equiv 0\), see [6]) and interpolates between Killing vector fields (\(L_V g = 0\)) and conformal vector fields
(\(L_V g = \lambda g\), \(\lambda\) is a function). Every conformal vector field is mixed Killing, whereas in flat and product geometries, the mixed Killing class is infinite-dimensional.
See subsection 2.1 for further discussion. It is therefore natural to ask whether curvature imposes rigidity on mixed Killing fields.
This paper aims to investigate mixed Killing vector fields on the complete Cigar Ricci–Bourguignon solitons and to establish rigidity phenomena specific to this geometry. We establish a complete characterisation of mixed Killing vector fields. In
particular, we show that every smooth angular mixed Killing field is a rotational Killing field. See Corollary 3 for more details. The main result about complete mixed Killing vector fields
on Cigar Ricci–Bourguignon solitons is that this space \(\mathfrak{MK}(g)\) has dimension \(5\). See Theorem 10.
We also provide an explicit description of the geodesics of the Cigar Ricci–Bourguignon soliton and show that all geodesics (except those starting at the tip) escape to infinity. This behaviour of the geodesics reflects the fact that the Cigar
Ricci–Bourguignon soliton is a complete non-compact surface. See Section 4 for more details.
The article is divided into four sections. In Section 2, we describe mixed Killing vector fields in flat and product manifolds. In Section 3, we obtain rigidity of Cigar almost
Ricci-Bourguignon solitons (Theorem 4). We also show that the potential vector field of Cigar almost Ricci-Bourguignon solitons is a mixed Killing vector field. In this
section, we also obtain a basis and dimension for complete mixed Killing vector fields in Cigar almost Ricci-Bourguignon solitons, Theorem 10. In the final section, viz., Section 4, we analyse geodesics of Cigar almost Ricci-Bourguignon solitons.
In this section, we first describe mixed Killing vector fields on a Riemannian manifold. This was introduced by Ghosh [5] as a natural generalisation of Killing and conformal vector fields. In subsection 2.1, we show that mixed Killing vector fields in an open cylinder with base as manifolds, in particular flat manifolds, have infinite dimension.
Definition 1 (Mixed Killing Vector Field). A vector field \(V\) on a (semi) Riemannian manifold \((M,g)\) is said to be a mixed Killing vector field, if there exists a smooth function \(f\) on \(M\) such that \[\label{e1} L_V L_V g = f\,L_V g.\tag{2}\] We call \(f\) the mixed Killing factor of \(V\).
Note that if \(L_V g \equiv 0\) (i.e., \(V\) is a Killing field), then \(V\) is trivially mixed Killing with any function \(f\). The interesting case is when \(L_V g \neq 0\), in which case condition 2 states that the second Lie derivative of the metric along \(V\) is proportional to the first. This condition is weaker than being conformal but stronger than having no geometric constraint.
It is easy to see that conformal fields form a subclass of mixed Killing fields.
Lemma 1 (Conformal fields are mixed Killing). Let \((M,g)\) be a (semi) Riemannian manifold and \(V\) a conformal vector field with \(L_V g = 2\lambda\,g\) for some smooth function \(\lambda\). Then on any open set where \(L_V g \neq 0\), \(V\) is a mixed Killing vector field with mixed Killing factor \[f = \frac{V(\lambda)}{\lambda} + 2\lambda.\] Consequently, (i) if \(\lambda\) is constant (i.e., \(V\) is homothetic), then \(f \equiv 2\lambda\). And (ii) if \(V(\lambda) = 0\) (i.e. \(\lambda\) is constant along flow lines of \(V\)), then \(f = 2\lambda\).
Proof. From 2 we have, \[L_V L_V g = L_V(2\lambda g) = 2(V\lambda)\,g + 2\lambda\,L_V g = 2(V\lambda)\,g + 4\lambda^2\,g = (2V\lambda + 4\lambda^2)\,g.\] On the other hand, \(L_V g = 2\lambda g\), so on the set \(\{\lambda\neq 0\}\) we can write \[L_V L_V g = \frac{2V\lambda + 4\lambda^2}{2\lambda}\,L_V g = \Bigl(\frac{V(\lambda)}{\lambda} + 2\lambda\Bigr)\,L_V g.\] Thus \(V\) is mixed Killing in the sense of 2 , with mixed Killing factor \(f\) as in the statement. ◻
Corollary 1. If \(V\) is homothetic, i.e.\(L_V g = 2c\,g\) for some constant \(c\in\mathbb{R}\), then \(V\) is mixed Killing with constant mixed Killing factor \(f = 2c\).
While conformal fields are always mixed Killing, the converse is false in general.
Note: It should be noted that the concept of a strictly nontrivial mixed Killing vector field \(V\), that is, one which is not Killing makes sense on \(M \setminus \{
L_{V} g
\neq 0 \}\).
In this subsection, we now explore mixed Killing vector fields in the aforementioned product manifolds. A key observation is that in one-dimensional directions, the mixed Killing condition reduces to an ordinary differential equation.
Proposition 1 (One-dimensional reduction). Let \((M,g)\) be a Riemannian manifold and suppose that, in suitable coordinates, a vector field has the form \(V = v(x)\,\partial_x\), and satisfies \(L_V g = A(x)\,T, L_V L_V g = B(x)\,T\), for some symmetric \((0,2)\)-tensor \(T\). Then \(V\) is a mixed Killing vector field, \(L_V L_V g = f\,L_V g\), if and only if \(f = \frac{B}{A}\), at every point where \(A \neq 0\) and \(T\) is nonzero. In particular, the mixed Killing condition reduces to an ordinary differential equation for \(v\).
Proof. By assumption, \[\label{form}
L_V g = A(x)\,T, \qquad L_V L_V g = B(x)\,T.\tag{3}\] At any point where \(A \neq 0\) and \(T\) is nonzero, we may write \[L_V L_V g =
\frac{B(x)}{A(x)}\,L_V g,\] thus the mixed Killing condition holds with \(f(x) = \frac{B(x)}{A(x)}\).
Conversely, if \(L_V L_V g = f\,L_V g\), then 3 implies that \[B(x)\,T = f(x)\,A(x)\,T.\] Since \(T\) is nonzero at the point in
question, it follows that \(f(x) = \frac{B(x)}{A(x)}\). ◻
Now we study a particular class of mixed Killing fields in manifolds of the type \(I\times N\), in particular in flat manifolds
Proposition 2 (Mixed Killing fields on product manifolds). Let \((M,g) = (I\times N,\,dx^2 + ds^2)\) be a Riemannian product, where \(I\subset\mathbb{R}\) is an open interval. For any \(v\in C^\infty(I)\), consider the vector field \(V = v(x)\,\partial_x\). Then \[L_V g = 2v'(x)\,dx^2, \qquad L_V L_V g = \bigl(2v v'' + 4(v')^2\bigr)\,dx^2.\] Consequently, (i) \(V\) is mixed Killing on \(\{v'\neq 0\}\) with factor \(f = v\,\frac{v''}{v'} + 2v'\) and (ii) \(V\) is conformally Killing if and only if \(v'\equiv 0\)
Proof. Clearly, \((L_V g)(\partial_x, \partial_x) = 2v'(x)\), and all other components vanish. Applying \(L_V\) again yields \[L_V L_V g = \bigl(2v v'' + 4(v')^2\bigr)\,dx^2,\] because the only nontrivial contribution comes from differentiating \(v'\) along \(V\). On the set \(\{v'\neq 0\}\), Proposition 1 applies with \(T = dx^2\), giving \[f = \frac{2vv'' + 4(v')^2}{2v'} = v\frac{v''}{v'} + 2v'.\] If \(v'\equiv 0\), then \(L_V g=0\) and hence \(V\) is Killing. Since \(L_V g\) has only a \(dx^2\) component, \(L_V g\) cannot be proportional to \(g\) unless it vanishes identically; thus,\(V\) is conformal if and only if \(v'\equiv 0\). ◻
Remark 1. In particular, on Euclidean space \((\mathbb{R}^2,dx^2+dy^2)\), any vector field of the form \(V=v(x)\partial_x\) with \(v'\not\equiv 0\) is mixed Killing but neither Killing nor conformal. Hence, in flat or product geometries of type \(I \times N\), the mixed Killing class is strictly much larger than the conformal class. But we will see that Cigar almost Ricci-Bourguignon solitons behave in a dramatically different way. We will see this in the next section.
In this section, we first characterise Cigar almost Ricci-Bourguignon solitons up to homothety. It is interesting to see that the potential vector field of Cigar almost Ricci-Bourguignon solitons is a mixed Killing vector field. Hence, it is natural to ask whether we can classify mixed Killing vector fields on Cigar almost Ricci-Bourguignon solitons? We answer this affirmatively by categorising all the mixed Killing vector Fields on Cigar almost Ricci-Bourguignon solitons. In fact, we find the basis of all mixed Killing vector fields and show that its dimension is \(5\).
Recall that the Cigar almost RB-soliton described in the introduction has soliton data as: \[\left(\mathbb{R}^2,\; g_{\mathrm{Cigar-RB}}(t) =\frac{dx^2+dy^2}{E(t)+x^2+y^2},\; \xi=-2 (1-2\rho) \Bigl(x\frac{\partial}{\partial x} +y\frac{\partial}{\partial y}\Bigr),\; \lambda=0, \; \rho \right),\] where \(E(t)=e^{4(1-2\rho)t}\) and the vector field \(\xi = -2(1-2\rho)\left(x\frac{\partial}{\partial x} +y\frac{\partial}{\partial y}\right) = \nabla f,\) where \[f(x,y,t) := -(1-2\rho)\,\log D(t,x,y) = -(1-2\rho)\,\log\bigl(E(t)+x^2+y^2\bigr).\]
Geodesic polar coordinate transformation of Cigar Ricci–Bourguignon metric: Consider the transformation \(x = r \cos \theta, y = r \sin \theta\). Then, \[\label{Cigar1} g_{\mathrm{Cigar-RB}}(t) = \frac{dr^2 + r^2 d\theta^2}{E(t) + r^2}, \qquad E(t)=e^{4(1-2\rho)t}.\tag{4}\] If we consider the transformation in geodesic polar coordinates \((s,\theta)\) defined by \[r = \sqrt{E}\sinh s,\] then \[\label{iso1} g_{\mathrm{Cigar-RB}}(t) = ds^2 + \psi(s)^2\,d\theta^2, \qquad \psi(s)=\tanh s.\tag{5}\]
Behaviour of Cigar metric near zeros of \(\tanh s\) : Note that as \(s \rightarrow 0\), \(\tanh s \sim s\). Hence, the metric looks like
\(g_{\mathrm{Cigar-RB}}(t) = ds^2 + s^2 d\theta^2\) which is a flat metric. As \(s \rightarrow \infty, \tanh s \rightarrow 1\), so it approaches a cylinder of radius \(1\). Observe that zeros of \(\tanh s\) are the same as zeros of \(\sinh s\), and it vanishes when \(s = n \pi i\). Near these
points, the warped product metric behaves like \(\tanh s \sim (s - s_{0})\), where \(s_{0} = n \pi i,\) for some \(n \in \mathbb{Z}\). And as \({\tanh}' s_{0} =1\), the metric becomes \(g_{\mathrm{Cigar-RB}}(t) = ds^2 + (s - s_{0})^2 d\theta^2\). This is just a flat polar metric on \({\mathbb{R}}^2\). So near each zero, the Cigar metric becomes locally Euclidean. And the Cigar has no conical singularity there; it is smooth and locally \({\mathbb{R}}^2\).
From our discussion earlier, we see that the Cigar Ricci-Bourguignon metric 5 satisfies the equation \[\operatorname{Ric} + \nabla^2 f - \rho R g = 0,\] for all \(t\). Thus \((\mathbb{R}^2,g(t),\xi,\lambda\equiv 0,\rho)\) is a steady gradient almost Ricci–Bourguignon soliton. Moreover:
At \(t=0\) and \(\rho=0\), we recover Hamilton’s steady Cigar Ricci soliton.
For \(\rho < 1/2\), the metric expands; for \(\rho > 1/2\), it shrinks.
The soliton is self-similar under the one-parameter group of diffeomorphisms \[\varphi_t(x,y) = (a(t)x, a(t)y), \qquad a(t) = e^{-2(1-2\rho)t}.\]
We establish a classification result showing that the Cigar is the unique complete surface with positive curvature admitting a steady, almost gradient Ricci-Bourguignon soliton structure. First, we show that the Cigar Ricci-Bourguignon soliton has positive sectional curvature.
Proposition 3. The metric given by \[g = \frac{dr^2 + r^2 d\theta^2}{E(t) + r^2}, \qquad E(t)=e^{4(1-2\rho)t}.\] has Gaussian curvature \[K= \frac{2E}{(E+r^2)}.\] Thus the Cigar Ricci -Bourguignon soliton has positive curvature.
Proof. For a conformal metric \[g=u(r)(dr^2+r^2d\theta^2),\] the Gaussian curvature satisfies \[K = -\frac{1}{2u} \Delta_{\mathbb{R}^2}(\log u).\]
Therefore \[K = \frac{2E}{(E+r^2)}.\] Thus the Cigar Ricci-Bourguignon soliton has positive curvature. ◻
Theorem 4 (2D rigidity). Let \((\Sigma^2,g,f)\) be a complete steady almost gradient Ricci-Bourguignon soliton with \(\rho\neq \tfrac12\). Suppose that \(\nabla f\) has a zero. If the Gauss curvature of \(g\) is positive, then \((\Sigma^2,g)\) is isometric, up to homothety, to Hamilton’s Cigar Ricci-Bourguignon soliton.
Proof. Let \((\Sigma^2,g,f)\) be a complete steady gradient Ricci–Bourguignon soliton with \(\rho\neq \tfrac12\) and positive Gauss curvature. Thus
\[\label{RB}
\operatorname{Ric} + \nabla^2 f = \rho R g ,\tag{6}\] where \(\rho\) is a constant. On a surface, \(\operatorname{Ric} = \tfrac{R}{2}g\), hence the above equation reduces to
\[\label{Hessian}
\nabla^2 f = \left(\rho-\tfrac12\right) R\, g .\tag{7}\] Therefore, \(\nabla f\) is a gradient conformal vector field. Since \((\Sigma^2,g)\) is complete with \(K>0\), equation 7 implies that the only possible critical point of \(f\) is either a minimum or a maximum. By a standard result [7] on complete surface admitting a nontrivial gradient conformal vector field with positive curvature, \((\Sigma^2,g)\) is rotationally
symmetric about this point.
Consequently, in the neighbourhood of this point, \(g\) can be written in polar coordinates as \[g = dr^2 + h(r)^2\, d\theta^2,\] where \(h(0)=0\) and \(h'(0)=1\). For such a metric, the Gauss curvature and scalar curvature are \[K = -\frac{h''}{h}, \qquad R = 2K = -\frac{2h''}{h}.\] From (7 ), it
follows that \(f\) is also a radial function. The Hessian of \(f\) is given by \[(\nabla^2 f)_{rr} = f'', \qquad
(\nabla^2 f)_{\theta\theta} = h h' f'.\] Substituting into 7 yields \[\label{system}
\begin{align}
f'' &= \left(\rho-\tfrac12\right) R, \\
\frac{h'}{h} f' &= \left(\rho-\tfrac12\right) R .
\end{align}\tag{8}\] Subtracting the two equations gives \(f'' - \frac{h'}{h}f' = 0\) which integrates to \(f' = a\, h\) for some constant \(a\). Substituting \(f'=ah\) and \(R=-2h''/h\) into 8 , we obtain \[(1-2\rho)h'' - a h
h' = 0.\] Since \(\rho\neq \tfrac12\), this can be written as \[h'' = k\, h h', \qquad k =\frac{a}{1-2\rho}.\] Set \(u=h'(r)\).
Then \(h'' = u \frac{du}{dh}\), and the equation becomes \[u\frac{du}{dh} = k h u.\] Since \(K>0\), we have \(h>0\) and \(h'>0\) for all \(r>0\), so dividing by \(u\) and integrating yields \[u(h) =
\frac{k}{2}h^2 + C.\] The smoothness conditions at \(r=0\) imply \(C=1\), hence \[h' = 1 + \frac{k}{2}h^2.\]
Positive curvature implies \(h''<0\) for \(r>0\), and since \(h,h'>0\), this forces \(k<0\). Writing \(A =-\frac{k}{2}>0\), we obtain \[h' = 1 - A h^2.\] Separating variables and integrating gives \[\frac{1}{\sqrt A}\operatorname{arctanh}(\sqrt A\, h)= r,\] so that \[h(r)=\frac{1}{\sqrt A}\tanh(\sqrt A\, r).\]
Therefore, \[g = dr^2 + \frac{1}{A}\tanh^2(\sqrt A\, r)\, d\theta^2.\] This metric is complete, has positive curvature, and coincides (up to scaling) with Hamilton’s Cigar soliton. See (5 ) for more details. Hence \((\Sigma^2,g)\) is the Cigar Ricci-Bourguignon soliton. ◻
Now we start our discussion of mixed Killing vector field on Cigar Ricci-Bourguignon soliton.
To understand the mixed Killing vector fields of the Cigar soliton, we first show that its potential vector field is indeed a mixed Killing vector field. Then we determine its conformal algebra. Note that the Lie algebra of conformal vector fields on a Riemannian manifold \(M\) is called the conformal algebra.
Proposition 5. The potential vector field of the Cigar almost Ricci–Bourguignon soliton \((\mathbb{R}^2,g(t),\xi,\lambda\equiv 0,\rho)\) is a mixed Killing vector field.
Proof. The potential vector field is \[\xi = -2(1-2\rho)\left(x\frac{\partial}{\partial x} +y\frac{\partial}{\partial y}\right).\] Set \(r^2=x^2+y^2\) and \(D(t,x,y):=E(t)+r^2\) where \(E(t)=e^{4(1-2\rho)t}.\) \[L_\xi g = \psi\,g \;\;\; where \; \qquad \psi(x,y,t) = -\frac{4(1-2\rho)\,E(t)}{D(t,x,y)},\] and \[L_\xi L_\xi g = \alpha \,L_\xi g\] holds with mixed Killing factor \[\label{eq:mixed-killing-factor-Cigar} \alpha(x,y,t) = -4(1-2\rho)\,\frac{E(t) - (x^2+y^2)}{E(t) + (x^2+y^2)}.\tag{9}\] In particular, \(\xi\) is mixed Killing and \(\alpha\) is not identically zero on \(\mathbb{R}^2\). Note that \(f \neq 0\) on \(\mathbb{R}^2 \setminus \{r = \sqrt{E}\}\), confirming that \(\xi\) is a genuinely non-trivial example of a mixed Killing field. ◻
Theorem 6 (Conformal algebra of the Cigar RB metric). Let \(g\) be a Cigar-type Ricci–Bourguignon metric on \(\mathbb{R}^2\), \[g = \phi(r)\,(dx^2 + dy^2),\qquad \phi(r) := \frac{1}{E + r^2},\quad r^2 = x^2 + y^2, \; E(t)=e^{4(1-2\rho)t}\] and let \(\delta = dx^2 + dy^2\) be the Euclidean metric. Then:
A smooth vector field \(V\) is conformal for \(g\) if and only if it is conformal for \(\delta\).
The Lie algebra of complete conformal vector fields on \((\mathbb{R}^2,g)\) is \(4\)–dimensional and is spanned by \[\partial_x,\quad \partial_y,\quad -y\partial_x + x\partial_y,\quad x\partial_x + y\partial_y.\]
Proof. (i) Since \(g=\phi\,\delta\) with \(\phi>0\) smooth, we have, for any vector field \(V\), \[L_V g = L_V(\phi\delta) = (V\phi)\,\delta + \phi\,L_V\delta.\] If \(V\) is conformal for \(\delta\), i.e.\(L_V\delta = 2\lambda_\delta\,\delta\), then \[L_V g = (V\phi)\,\delta + 2\phi\lambda_\delta\,\delta = 2\Bigl(\lambda_\delta + \frac{V(\log\phi)}{2}\Bigr)\,\phi\delta = 2\lambda_g\,g,\] so \(V\) is conformal for \(g\) with \[\lambda_g = \lambda_\delta + \frac{V(\log\phi)}{2}.\] Conversely, if \(V\) is conformal for \(g\), \(L_V g = 2\lambda_g g\), then \[(V\phi)\,\delta + \phi\,L_V\delta = 2\lambda_g\,\phi\,\delta.\] Dividing by \(\phi\) gives \[L_V\delta = 2\Bigl(\lambda_g - \frac{V(\log\phi)}{2}\Bigr)\,\delta,\] so \(V\) is conformal for \(\delta\). This proves (i).
(ii) By (i), it suffices to determine the Lie algebra of complete conformal vector fields on \((\mathbb{R}^2,\delta)\).
Identify \((\mathbb{R}^2,\delta)\) with the complex plane \((\mathbb{C},|\cdot|^2)\) via \(z=x+iy\). The global conformal diffeomorphisms of \(\mathbb{C}\) are precisely the affine maps \[\Phi(z) = a z + b,\qquad a\in\mathbb{C}^\ast,\;b\in\mathbb{C},\] (see [8]. Thus \[\mathrm{Aut}(\mathbb{C}) = \{z\mapsto az+b\}\] is a \(4\)–dimensional real Lie group. A complete conformal vector field \(V\) on \((\mathbb{R}^2,\delta)\cong\mathbb{C}\) generates a one-parameter group of conformal diffeomorphisms \(\{\Phi_t\}_{t\in\mathbb{R}}\subset\mathrm{Aut}(\mathbb{C})\): \[\frac{d}{dt}\Phi_t(z) = V(\Phi_t(z)),\qquad \Phi_0=\mathrm{id}_{\mathbb{C}}.\] Since each \(\Phi_t\) is affine, we can write \(\Phi_t(z)=a(t)z + b(t)\), with \(a(0)=1\), \(b(0)=0\). Differentiating at \(t=0\) yields \[V(z) = \left.\frac{d}{dt}\right|_{t=0}\Phi_t(z) = \alpha + \beta z\] for some \(\alpha,\beta\in\mathbb{C}\). Conversely, any vector field of the form \(V(z)=\alpha + \beta z\) generates a one-parameter subgroup of affine conformal automorphisms and is therefore a complete conformal vector field. Writing \(\alpha = a_1+ia_2\), \(\beta=b_1+ib_2\) and separating real and imaginary parts, the corresponding real vector field on \(\mathbb{R}^2\) is \[V = a_1\,\partial_x + a_2\,\partial_y + b_1\,(x\partial_x + y\partial_y) + b_2\,(-y\partial_x + x\partial_y),\] which is a linear combination of \[\partial_x,\;\partial_y,\;x\partial_x + y\partial_y,\;-y\partial_x + x\partial_y.\] These four vector fields are complete and conformal for \(\delta\), and hence, by (i), conformal for \(g\) as well. They generate all complete conformal vector fields on \((\mathbb{R}^2,\delta)\) and thus on \((\mathbb{R}^2,g)\). ◻
Remark 2. The restriction to complete conformal vector fields is essential. On \(\mathbb{C}\), the vector field \(V(z) = z^2 \partial_z\) is holomorphic and therefore conformal, but generates the flow \(z(t) = z_0/(1-z_0 t)\), which has finite maximal existence time for \(z_0 \neq 0\). Such fields generate an infinite-dimensional Lie algebra. Moreover, completeness is an intrinsic property of the vector field (independent of the metric), so the four-dimensional conformal algebra is the same for \(g\) and \(\delta\).
In this subsection, we classify strict mixed Killing fields in the Cigar-type Ricci–Bourguignon metric. Unless otherwise stated, while working with a mixed Killing field \(V\), we will work on the complement of the set where \(L_{V} g = 0\) (see note after Corollary 1).
Theorem 7 (Angular rigidity). Let \[V = v(r, \theta)\,\partial_\theta\] be a smooth local vector field on the Cigar Ricci–Bourguignon soliton. Then \(V\) is a mixed Killing vector field if and only if \(v\) is constant. In this case, \(V\) is a rotational Killing field.
Proof. Write the Cigar metric 4 in polar coordinates as \[g_{rr}=\frac{1}{E+r^2}=:u(r),\qquad g_{\theta\theta}=u(r)\,r^2,\qquad g_{r\theta}=0.\] Let \(V=v(r,
\theta)\partial_\theta\), so \(V^r=0\), \(V^\theta=v(r, \theta)\). Using \[(L_V g)_{ij}=V^k\partial_k g_{ij}+g_{kj}\partial_i V^k+g_{ik}\partial_j
V^k,\] one computes that the components of \(L_V g\) are: \[(L_V g)_{r\theta}=(L_V g)_{\theta r}=g_{\theta\theta}\,\partial_r V^\theta
= u(r)\,r^2 \frac{{\partial}}{{\partial r}} v(r, \theta).\] Thus, \[(L_V g)_{r\theta}=(L_V g)_{\theta r} = \frac{r^2}{E+r^2} \frac{{\partial}}{{\partial r}} v(r, \theta).\] And \((L_V
g)_{rr}= 0\), \[(L_V g)_{\theta\theta} = \frac{r^2}{E+r^2} \frac{{\partial}}{{\partial \theta}} v(r, \theta).\] \[(L_VL_Vg)_{rr}= 2 \frac{r^2}{E+r^2} (\frac{{\partial}}{{\partial \theta}}
v(r, \theta))^2.\] If \(V\) is mixed-Killing, then \[(L_VL_Vg)_{rr} = f (L_V g)_{rr}= 0,\] This implies that \(v(r, \theta) = v(r).\)
Now, to see that \(v(r)\) is constant, we proceed as follows. Applying \(L_V\) again, one finds that \(L_VL_Vg\) has no \(r\theta\)-component, as \(v(r, \theta)\) does not depend on \(\theta\). That is \((L_VL_Vg)_{r \theta} = 0\), but then mixed
Killing condition implies that \((L_V g)_{r\theta} = 0\). Consequently, \(v(r, \theta)\) also does not depend on \(r\) and thus \(v(r, \theta)\) is constant.
Thus, \(V = C \frac{{\partial}}{{\partial \theta}}\) is mixed Killing. But, then the above considerations show that all the components of \(L_V g\) vanish and this gives \(V = C \frac{{\partial}}{{\partial \theta}}\) is rotational Killing. ◻
Remark 3. Unlike the product case, angular symmetry exhibits strong rigidity here; mixed Killing implies genuine Killing.
Theorem 8 (Radial mixed Killing fields). Let \[V = w(s)\,\partial_s,\] be a complete smooth local vector field. Then \(V\) is a mixed Killing vector field if and only if \[\label{eq:radial-classification} w(s)^2 = A\,\psi(s)^2 + B,\tag{10}\] for constants \(A,B\in\mathbb{R}\). Moreover:
\(V\) is conformal if and only if \(B=0\),
if \(B\neq 0\), then \(V\) is mixed Killing but not conformal,
at tip of the Cigar soliton and near zeros \(s_{0}\) of \(\psi(s)\), \(V\) extends smoothly as \[V = w(s)\,\partial_s,\] where \(w(s)^2 = A\, s^2 + B,\) and \(w(s)^2 = A\, (s - s_{0}) ^2 + B,\) respectively.
Proof. Here we will use the Cigar Ricci-Bourguignon metric (4 ) given by \[g_{\mathrm{Cigar-RB}}(t) = ds^2 + \psi(s)^2\,d\theta^2, \qquad \psi(s)=\tanh s.\] For the simplicity of
computations, here we work in this metric. In the sequel, we denote by \('\) derivative with respect to \(s\) and note that \(\psi'\) never vanishes.
A direct computation yields \[L_V g = 2w'\,ds^2 + 2w\psi\psi'\,d\theta^2,\] and \[L_V L_V g = (2ww'' + 4(w')^2)\,ds^2 + \bigl(2ww'\psi\psi' + 2w^2((\psi')^2 +
\psi\psi'')\bigr)\,d\theta^2.\] The mixed Killing condition \(L_V L_V g = f\,L_V g\) is equivalent to solving the system \[\begin{cases}
2ww'' + 4(w')^2 = f \cdot 2w' ,\\
2ww'\psi\psi' + 2w^2((\psi')^2 + \psi\psi'') = f \cdot 2w\psi\psi'.
\end{cases}\] Without loss of generality we may assume that \(\psi(s) \neq 0\), because if \(\psi(s_{0}) = 0\), then from our earlier discussion (see subsection 3.1) We know that in the neighbourhood of \(s_{0}\) we have \(\psi(s) = s - s_{0}\) and \(\psi'(s) = 1\). Then we can work on
this neighbourhood.
Eliminating \(f\) from these equations and simplifying (using \((\psi\psi')' = (\psi')^2 + \psi\psi''\)), we obtain: \[\frac{d}{ds}\log(w'w) = \frac{d}{ds}\log(\psi\psi').\] Note that the above expression makes sense, as in the complement of \(L_{V}g = 0\), \(w, w'\)
don’t vanish. Integrating, \[w'w = C\,\psi\psi',\] and hence \[\frac{1}{2}(w^2)' = C\,\psi\psi',\] and this again implies that \[w^2 = A\,\psi^2
+ B\] for constants \(A,B\in\mathbb{R}\).
Conversely, substituting \(w^2 = A\psi^2 + B\) into the expressions for \(L_V g\) and \(L_V L_V g\) shows that the proportionality relation \(L_V L_V g = f\,L_V g\) holds on the open set under consideration.
Finally, the conformality condition \(L_V g = 2\lambda g\) requires \(w' = \lambda\) and \(w\psi\psi' = \lambda\psi^2\), which forces \(w = c\psi\) and therefore \(B = 0\). ◻
Corollary 2. Let \[V = w(s, \theta)\,\partial_s,\] be a complete smooth local vector field on the complete Cigar Ricci–Bourguignon soliton. Then \(V\) is a mixed Killing vector field if and only if \(w(s, \theta)\) is a function of \(w(s)\) only.
Proof. Comparing the Lie derivative components and using the mixed Killing condition, the argument, similar to the proof of the Theorem 7, implies that \(w\) does not depend on \(\theta\). ◻
Corollary 3. (Local characterization of complete mixed Killing vector fields in Cigar metric): Let \[V = v(r, \theta)\,\partial_\theta + w(s, \theta)\,\partial_s,\] be any complete smooth local vector field on the complete Cigar Ricci–Bourguignon soliton. Then \(V\) is a mixed Killing vector field if and only if \(v(r, \theta) = C,\) constant and \(w(s, \theta) = \sqrt{(A\,\psi(s)^2 + B)}\). Thus, \[\begin{align} \label{mkvf} V = C \partial_{\theta} + \sqrt{(A\,\psi(s)^2 + B)} \partial_s. \end{align}\tag{11}\] Moreover:
\(V\) is conformal if and only if \(B=0\),
if \(B\neq 0\), then \(V\) is mixed Killing but not conformal,
at the tip of the Cigar soliton and near zeros \(s_{0}\) of \(\psi(s)\), \(V\) extends smoothly as \[V = C \partial_{\theta} + \sqrt{(A\, (s-s_{0})^2 + B)} \partial_s,\] respectively.
Let \(U_1 = \left((S^1 \setminus (0, 1)) \times (0, \infty)\right)\), \(U_2 = \left((S^1 \setminus (0, -1)) \times (0, \infty)\right)\) be the covering of Cigar \(\Sigma^2 = S^1 \times (0, \infty)\). From the above corollary, we conclude that:
Theorem 9. (Global characterization of complete mixed Killing vector fields in Cigar metric): Let \(V\) be a complete mixed Killing vector field on the Cigar Ricci–Bourguignon soliton \((\Sigma^2 = S^1 \times (0, \infty), g, f)\). Then \[\begin{align} V & = & \sum_{i=1}^{2} C_i \partial_{{\theta}_i} + \sqrt{(A_i\,\psi(s_{i})^2 + B_i)} \partial_{s_i}, \\ \nonumber & = & V_1 + V_2, \end{align}\] where \(\left(\Omega_1, \partial_{{\theta}_1}, \partial_{s_1} \right)\), \(\left(\Omega_2, \partial_{{\theta}_2}, \partial_{s_2}\right)\) denote the covering by charts of the complement in \(\Sigma^2\) of the set \(L_{V}g \neq 0\). Moreover,
The constants \(C_i, A_i, B_i\) are unique upto diffeomorphisms of \(\Omega_1, \Omega_2\) respectively. And \(V_1, V_2\) coincide on \(\Omega_1 \cap \Omega_2\).
\(V\) is conformal if and only if \(B_i=0\) for at least one \(i =1,2\).
if \(B_i\neq 0\) for any one of \(i\), then \(V\) is mixed Killing but not conformal,
at the tip of the Cigar soliton and near zeros \(s_{0}\) of \({\psi}(s_i)\), \(V\) extends smoothly as \[V =\sum_{{i=1}^{2}} C_{i} \partial_{{\theta}_{i}} + \sqrt{(A\, (s_{i}-s_{0})^2 + B)} \partial_{s_{i}},\] respectively.
Remark 4. In view of the characterisation of mixed Killing fields obtained here, it is natural to ask: Can we characterise the zero set of mixed Killing fields? Note that it is known that the zeros of closed conformal vector fields are discrete [9].
Let \(\mathfrak{MK}(g), \mathfrak{conf}(g)\) respectively denote the vector space of complete mixed Killing vector fields and complete conformal vector fields, respectively. From Theorem 6, we know that the Lie algebra of complete conformal vector fields on \((\mathbb{R}^2,g)\) is \(4\)–dimensional and is spanned by \[\partial_x,\quad \partial_y,\quad -y\partial_x + x\partial_y,\quad x\partial_x + y\partial_y.\]
Theorem 10. (Basis for mixed Killing vector fields):Let \((\mathbb{R}^2,g)\) be the complete Cigar Ricci–Bourguignon soliton. Then \(\mathfrak{MK}(g)\) is spanned by \[\mathfrak{conf}(g), \; \sqrt{1+ \frac{E}{x^2+y^2} } \; \left(x \partial_x + y \partial_y\right) .\] Hence, \(\dim\mathfrak{MK}(g)=5\).
Proof. Recall that from 11 , any complete mixed Killing vector field is locally represented as: \[V = C \partial_{\theta} + \sqrt{(A\,\psi(s)^2 + B)} \partial_s.\] Clearly, note that \[\partial_{\theta} = - y \partial_{x} + x \partial_{y}.\] And \[\partial_{s} = \coth s \; (x \partial_{x} + y \partial_{y}) = \sqrt{1+ \frac{E}{x^2+y^2} } \; \left(x \partial_x + y \partial_y\right).\] Hence, the result follows as stated. ◻
In this section, we completely characterise the geodesics on the Cigar RB soliton.
In what follows, it is convenient to work with geodesic polar coordinates \((s,\theta)\) as discussed in Subsection 3.1. Recall that
\[\label{eq:geodesic-polar-coord} r = \sqrt{E}\,\sinh s, \qquad s\in[0,\infty),\tag{12}\] in which the metric takes the warped product form \[\label{eq:warped-form} g = ds^2 + \psi(s)^2\,d\theta^2, \qquad \psi(s)=\tanh s.\tag{13}\]
The metric components are diagonal: \[g_{ss} = 1, \qquad g_{\theta\theta} = \psi(s)^2, \qquad g_{s\theta} = 0.\]
For the warped product metric 13 , the nonzero Christoffel symbols are \[\Gamma^s_{\theta\theta} = -\psi(s)\psi'(s), \qquad \Gamma^\theta_{s\theta} = \Gamma^\theta_{\theta s} = \frac{\psi'(s)}{\psi(s)}.\]
A geodesic \(\gamma(\sigma) = (s(\sigma), \theta(\sigma))\) satisfies \[\label{eq:geod-s-main} \ddot{s} - \psi(s)\psi'(s)\dot{\theta}^2 = 0,\tag{14}\] \[\label{eq:geod-theta-main} \ddot{\theta} + 2\frac{\psi'(s)}{\psi(s)}\dot{s}\dot{\theta} = 0,\tag{15}\] where \(\sigma\) is an affine parameter and dots denote \(d/d\sigma\).
The geodesic equations admit two conserved quantities.
Proposition 11 (Angular momentum). Since the metric is independent of \(\theta\), the quantity \[\label{eq:angular-momentum-geo} \ell := \psi(s)^2\dot{\theta} = \tanh^2(s)\,\dot{\theta}\qquad{(1)}\] is constant along geodesics.
Proof. Equation 15 can be rewritten as \[\frac{d}{d\sigma}\left(\psi^2\dot{\theta}\right) = \psi^2\ddot{\theta} + 2\psi\psi'\dot{s}\dot{\theta} = 0.\] ◻
Proposition 12 (Speed parameter). For an affinely parameterised geodesic, the speed \[\label{eq:speed-geo} k := \dot{s}^2 + \psi(s)^2 \dot{\theta}^2 = \dot{s}^2 + \ell^2\coth^2 s\qquad{(2)}\] is constant. For unit-speed geodesics, \(k=1\).
From these conservation laws, we obtain the radial equation \[\label{eq:radial-equation-s} \dot{s}^2 = k - \ell^2\coth^2 s.\tag{16}\]
Theorem 13 (Geodesic classification). Geodesics on the Cigar RB soliton are of two types:
Radial geodesics (\(\ell=0\)): These satisfy \(\theta = \text{const}\) and \[s(\sigma) = \pm\sqrt{k}\,\sigma + s_0, \qquad r(\sigma) = \sqrt{E}\,\sinh(\pm\sqrt{k}\,\sigma + s_0).\] Every radial geodesic passes through the origin \(r=0\).
Non-radial geodesics (\(\ell\neq 0\), with \(k>\ell^2\)): These satisfy \[\label{eq:s-solution-main} \cosh s(\sigma) = \sqrt{\frac{k}{k-\ell^2}}\, \cosh\!\bigl(\sqrt{k-\ell^2}\,\sigma\bigr),\tag{17}\] with angular coordinate \[\dot{\theta} = \ell\coth^2 s.\] These geodesics have a unique turning point at \(s_{\min}\) where \(\dot{s}=0\), given by \[\coth^2 s_{\min} = \frac{k}{\ell^2}, \qquad r_{\min} = \sqrt{E}\sinh s_{\min} = \sqrt{\frac{E(k-\ell^2)}{\ell^2}}.\] They escape to infinity in both directions: \(r(\sigma) \to \infty\) as \(\sigma \to \pm\infty\).
Proof. For \(\ell=0\), equation 16 gives \(\dot{s}^2 = k\), yielding the radial case immediately.
For \(\ell \neq 0\), using \(\coth^2 s = 1 + \mathop{\mathrm{csch}}^2 s\) in 16 gives \[\dot{s}^2 = (k-\ell^2) - \ell^2 \mathop{\mathrm{csch}}^2 s.\] The substitution \(u = \cosh s\) leads to the standard integral \[\int \frac{du}{\sqrt{(k-\ell^2)u^2 - k}} = \pm \sigma + C,\] which yields 17 after choosing the origin of \(\sigma\) appropriately. The turning point corresponds to \(\dot{s}=0\). ◻
Key Observations: The geodesic structure of the Cigar reflects its key geometric features:
Complete non-compactness: All geodesics (except those starting at the tip) escape to infinity.
Positive curvature: The Gaussian curvature \(K = \frac{2E}{(E+r^2)^2}\) is everywhere positive but decays exponentially, preventing spiralling behaviour.
Scattering geometry: Non-radial geodesics approach a minimum radius \(r_{\min}\), then recede to infinity, characteristic of scattering in a repulsive potential.
These properties distinguish the Cigar from both flat space (where geodesics are arbitrary straight lines with no special point) and compact positively curved spaces like the sphere (where geodesics are closed).
Question: In view of the results obtained, here we can ask: Do the results obtained here about mixed Killing vector fields in dimension \(2\) extend to higher dimensions? More generally, what is the
structure of mixed Killing fields, higher-dimensional gradient Ricci solitons? In particular, the Cigar RB soliton in higher dimension given by rotationally symmetric metric: \[g_n = ds^2 + \tanh^2(s) \, g_{S^{n-1}},\]
where \(g_{S^{n-1}}\) is the round metric on \(S^{n-1}\)