1 Introduction and preliminaries

Given an m-dimensional Riemannian manifold \((M^m,g)\) with Levi–Civita connection \(\nabla \), we recall that a conformal vector field \(\xi \) on M is said to be closed if the 1-form \(\xi ^{\flat }\) is closed. This is easily seen to be equivalent to the existence of a smooth function \(\psi :M\rightarrow {\mathbb {R}}\) (called the conformal factor of \(\xi \)) such that

$$\begin{aligned} \nabla _X\xi =\psi X, \end{aligned}$$

for all \(X\in {\mathfrak {X}}(M)\). In turn, this readily yields

$$\begin{aligned} \psi =\frac{1}{m}\textrm{div}_M\xi . \end{aligned}$$
(1)

Also in this setting, item 1 of Lemma 1 of [9] shows that, for every nontrivial closed and conformal vector field \(\xi \),

$$\begin{aligned} |\xi |^2\nabla \psi =-{{\,\textrm{Ric}\,}}_M(\xi )\xi , \end{aligned}$$
(2)

where \(\nabla \psi \) stands for the gradient of \(\psi \) and

$$\begin{aligned} {{\,\textrm{Ric}\,}}_M(\xi )=\frac{1}{m-1}{{\,\textrm{Ric}\,}}_M(\xi ,\xi ) \end{aligned}$$

for the normalized Ricci curvature of \((M^m,g,\nabla )\) in the direction of \(\xi \). Moreover, if \(m>2\), then item 3 of that result shows that \(\xi ^{-1}(0)\), the set of singular points of \(\xi \), is a set of isolated points and \(\psi (p)\ne 0\) for every \(p\in \xi ^{-1}(0)\). See also Lemma 1 in [5], which summarizes some of the known results about Riemannian manifolds which admit closed and conformal vector fields. We also refer the reader to [6] and [10] for some recent results on the structure and geometric properties of Riemannian manifolds endowed with closed conformal vector fields.

The standard class of examples of Riemannian manifolds equipped with closed conformal vector fields is that of Riemannian warped products with one dimensional fibers, as we now recall. Let \(I\subset \mathbb R\) be an open interval with its standard metric \(dt^2\) and \(N^{m-1}\) be an \((m-1)\)-dimensional Riemannian manifold with metric \(g_N\). We set \(M^m=I\times N^{m-1}\) and let \(\pi _I:M\rightarrow I\) and \(\pi _N:M\rightarrow N\) denote the projections. If \(h:I\rightarrow (0,+\infty )\) is a smooth function and \({\tilde{h}}=h\circ \pi _I:M\rightarrow (0,+\infty )\), then

$$\begin{aligned} \langle \cdot ,\cdot \rangle =\pi _I^*dt^2+{\tilde{h}}^2\pi _N^*g_N \end{aligned}$$

is a metric tensor on M, with respect to which M is said to be the warped product of I and N, with warping function h. We summarize this by writing

$$\begin{aligned} M^m=I\times _hN^{m-1}. \end{aligned}$$

If \({\tilde{h}}=h\circ \pi _I\), \(\partial _t\) denotes the canonical vector field on I and \(\tilde{\partial _t}\) its horizontal lift to M, then it is a standard fact that the vector field \(\tilde{h}\tilde{\partial _t}\) is a closed conformal vector field on M with no singular points, with conformal factor \(\tilde{h'}=h'\circ \pi _I\), where \(h'\) is the derivative of h.

In particular, letting \(I=(0,+\infty )\) and \(h(t)=t\) for every \(t>0\), we obtain the one dimensional cone \(M^m=(0,+\infty )\times _tN^{m-1}\), with closed conformal vector field \(t\tilde{\partial _t}\) of conformal factor 1.

A Sasakian manifold is a Riemannian manifold \((N,g_N)\) with Levi–Civita connection D such that the one dimensional cone

$$\begin{aligned} M=(0,+\infty )\times _tN \end{aligned}$$

is a Kählerian manifold; in particular, N is odd dimensional. In such a case, if we let J denote the complex structure of M and \(\xi (t,p)=t\tilde{\partial _t}\) the closed conformal vector field, then it can be proved (cf. [11], for instance) that:

  1. (a)

    \( Z:=J\xi \) is a unit Killing vector field on \(N\approx \{1\}\times N\).

  2. (b)

    For \(X\in {\mathfrak {X}}(N)\), one has

    $$\begin{aligned} JX=-\langle X, Z\rangle \xi +D_X Z. \end{aligned}$$
    (3)
  3. (c)

    If \(\Phi \in {{\,\textrm{End}\,}}(TN)\) is given by \(\Phi (X)=D_X Z\), then

    $$\begin{aligned} \big (D_X\Phi \big )(Y)=\langle Y, Z\rangle X-\langle X,Y\rangle Z, \end{aligned}$$
    (4)

    for all \(X,Y\in {\mathfrak {X}}(N)\).

Conversely, let \((N,g_N)\) be an odd dimensional Riemannian manifold with Levi–Civita connection D and \(M=(0,+\infty )\times _tN\) be the one dimensional cone over N. Assume that there exists a unit Killing vector field Z on N for which the field of endomorphisms \(\Phi \in {{\,\textrm{End}\,}}(TN)\), given by \(\Phi (X)=D_X Z\), satisfies (4). Then (see also [11]), N is a Sasakian manifold and the restriction of the complex structure J of M to TN satisfies (3).

With notations as in the previous paragraph, warped product geometry (cf. Corollary 7.43 of [7]) readily shows that

$$\begin{aligned} {{\,\textrm{Ric}\,}}_M(\xi )=0. \end{aligned}$$
(5)

Actually, if the Sasakian manifold N is \((2n-1)\)-dimensional, then, computing as suggested above with the aid of (4), it can be shown that \((N,g_N)\) is Einstein if, and only if, \(M=(0,+\infty )\times _tN\) is Ricci flat.

In this paper, we aim at proving the following

Theorem 1.1

Let \(({\overline{M}}^{2n},g=\langle \cdot ,\cdot \rangle ,J)\), \(n>1\), be a complete, noncompact Kählerian manifold, endowed with a nontrivial closed conformal vector field \(\xi \), of conformal factor \(\psi \) and having at least one singular point. Assume that \(\psi \ge 1\) on \({\overline{M}}\) and \(\psi \rightarrow 1\) at infinity. If the Ricci curvature of \({\overline{M}}\) in the direction of \(\xi \) is nonpositive, then:

  1. (a)

    \(\psi \equiv 1\) on \({\overline{M}}\) and \(\xi \) has just one singular point, say p.

  2. (b)

    \({\overline{M}}{\setminus }\{p\}\) is isometric to a one dimensional cone over a Sasakian manifold diffeomorphic to \({\mathbb {S}}^{2n-1}\).

The fact that the conformal factor \(\psi \equiv 1\) on \({\overline{M}}\) implies that the Lie derivative of the metric tensor g with respect to \(\xi \) satisfies \({\mathcal {L}}_\xi g=2g\). Geometrically, this means that the flow \(\{\varphi _t\}_{t\in {\mathbb {R}}}\) of the vector field \(\xi \) consists of hometheties of positive coefficient, since \((\varphi _t^*g)_p=e^{2t}g_p\) for all \(p\in {\overline{M}}\) and \(t\in {\mathbb {R}}\). For that reason \(\xi \) is also said to be a homothetic vector field (see, for instance, Chapter 5 in [8]).

It is worth pointing out that this result lies in the complementary setting of the one dealt with by the second author in [4].

For the coming corollary, given a \((2n-1)\)-dimensional Sasakian manifold N, we say that the Kählerian cone \(M:=(0,+\infty )\times _tN\) has a removable singularity if the following condition is satisfied: for some symbol p not in M, the space \({\overline{M}}:=M\cup \{p\}\) has the structure of a complete, noncompact, 2n-dimensional Kählerian manifold whose metric and complex structure extend that of M, and such that the closed conformal vector field \(t\partial _t\) of M likewise extends to \(\xi \in {\mathfrak {X}}({\overline{M}})\).

Corollary 1.2

Let N be a \((2n-1)\)-dimensional Sasakian manifold whose Kählerian cone \(M:=(0,+\infty )\times _tN\) has a removable singularity. With notations as above, if p is a singular point of \(\xi \), then N is isometric to \({\mathbb {S}}^{2n-1}\), endowed with an appropriate Sasakian structure.

2 Proof of Theorem 1.1

First of all, as observed in the second paragraph of Sect. 1, since \(2n>2\) and \(\xi \) is nontrivial, it has isolated zeros. Recall also that we are assuming \({{\,\textrm{Ric}\,}}_{{\overline{M}}}(\xi )\le 0\) on \({\overline{M}}\). We divide the subsequent analysis in several steps.

Claim 1. \(\psi =1\) on \({\overline{M}}\) and \({{\,\textrm{Ric}\,}}_{{\overline{M}}}(\xi )=0\) on \({\overline{M}}{\setminus }\xi ^{-1}(0)\).

We shall need the following result.

Theorem 2.1

(Theorem 2.2 of [1]) Let \(({\overline{M}},\langle \cdot ,\cdot \rangle )\) be a connected, oriented, complete noncompact Riemannian manifold, and let \(\eta \in {\mathfrak {X}}({\overline{M}})\) be a vector field on \(\overline{M}\). Assume that there exists a nonnegative, non identically vanishing function \(f\in {\mathcal {C}}^\infty ({\overline{M}})\), converging to zero at infinity and such that \(\langle \nabla f,\eta \rangle \ge 0\) on \({\overline{M}}\). If \(\textrm{div}_{\overline{M}}\eta \ge 0\) on \(\overline{M}\), then:

  1. (a)

    \(\langle \nabla f,\eta \rangle \equiv 0\) on \({\overline{M}}\).

  2. (b)

    \(\textrm{div}_{\overline{M}}\eta \equiv 0\) on \({\overline{M}}{\setminus } f^{-1}(0)\).

Back to the proof of Claim 1, let \(\eta =\psi \xi \). It follows from (1) and (2) that, at every nonsingular point of \(\xi \), we have

$$\begin{aligned} \textrm{div}_{\overline{M}}\eta =\langle \nabla \psi ,\xi \rangle +\psi \textrm{div}_{\overline{M}}\xi =-{{\,\textrm{Ric}\,}}_{{\overline{M}}}(\xi )+2n\psi ^2\ge 0. \end{aligned}$$
(6)

By continuity, \(\textrm{div}_{\overline{M}}\eta \ge 0\) on \({\overline{M}}\).

Assume, for the sake of contradiction, that \(\psi \) is not identically 1 on \({\overline{M}}\). Setting \(f=\psi -1\), we have \(f\ge 0\) and \(f\not \equiv 0\) on \({\overline{M}}\). Also from (2), we get, at the nonsingular points of \(\xi \),

$$\begin{aligned} \langle \nabla f,\eta \rangle =\langle \nabla \psi ,\psi \xi \rangle =-{{\,\textrm{Ric}\,}}_{{\overline{M}}}(\xi )\psi \ge 0. \end{aligned}$$

Again, by continuity, \(\langle \nabla f,\eta \rangle \ge 0\) on \({\overline{M}}\).

Item (b) of Theorem 2.1 gives \(\textrm{div}_{\overline{M}}\eta \equiv 0\) on \({\overline{M}}{\setminus } f^{-1}(0)={\overline{M}}{\setminus }\psi ^{-1}(1)\). Back to (6), this shows that \(\psi \equiv 0\) on \({\overline{M}}{\setminus }\psi ^{-1}(1)\), thus contradicting the fact that \(\psi \ge 1\) on \({\overline{M}}\). Therefore, \(\psi \equiv 1\) on \({\overline{M}}\), which means that

$$\begin{aligned} \nabla _X\xi =X \end{aligned}$$
(7)

for all \(X\in {\mathfrak {X}}({\overline{M}})\).

Once we know that \(\psi \equiv 1\), Eq. (2) shows that \({{\,\textrm{Ric}\,}}_{{\overline{M}}}(\xi )\equiv 0\) on \({\overline{M}}{\setminus }\xi ^{-1}(0)\), and, trivially, on \(\xi ^{-1}(0)\). Anyway, as we have already noticed, this will also follow once we show that \({\overline{M}}{\setminus }\{p\}\) is isometric to a one dimensional cone over a simply connected Sasakian manifold.

Claim 2. \(\xi \) has exactly one singular point.

Arguing once more by contradiction, assume that \(\xi (p)=0\) and \(\xi (q)=0\), for some distinct points \(p,q\in {\overline{M}}\). Thanks to the completeness of \({\overline{M}}\), we can take a normalized geodesic \(\gamma :[0,\ell ]\rightarrow {\overline{M}}\) from p to q. Letting \(\xi (t)\) denote the restriction of \(\xi \) to \(\gamma \) and \(\varphi (t):=\langle \xi (t),\gamma '(t)\rangle \), we get \(\varphi (0)=\varphi (\ell )=0\) and

$$\begin{aligned} \varphi '(t)=\langle \frac{D\xi }{dt}(t),\gamma '(t)\rangle = \langle \nabla _{\gamma '(t)}\xi ,\gamma '(t)\rangle = \langle \gamma '(t),\gamma '(t)\rangle =1, \end{aligned}$$

whence \(\varphi (t)=t+\varphi (0)=t\). However, this contradicts the fact that \(\varphi (\ell )=0\).

It thus follows from the previous claim and our hypotheses that \(\xi \) has exactly one singular point.

Claim 3. If p is the singular point of \(\xi \), then, for each \(q\ne p\), there is just one normalized geodesic \(\gamma \) from p to q. Moreover, letting \(\xi (t)\) denote the restriction of \(\xi \) to \(\gamma \), we have \(\xi (t)=t\gamma '(t)\).

Let \(\gamma _1,\gamma _2:[0,\ell _i]\rightarrow {\overline{M}}\) be two normalized geodesics from p to q. Since \(\xi (p)=0\), reasoning as in the proof of Claim 2, we get \(\langle \xi (\gamma _i(t)),\gamma _i'(t)\rangle =t\) for every \(t\in [0,\ell _i]\) and \(i=1,2\). On the other hand,

$$\begin{aligned} \frac{d}{dt}|\xi (\gamma _i(t))|^2=2\Bigg \langle \nabla _{\gamma _i'(t)}\xi ,\xi (\gamma _i(t))\Bigg \rangle =2\Bigg \langle \gamma _i'(t),\xi (\gamma _i(t))\Bigg \rangle =2t, \end{aligned}$$

so that \(|\xi (\gamma _i(t))|^2=t^2+|\xi (0)|^2=t^2\) for every \(t\in [0,\ell _i]\) and \(i=1,2\). Finally, Cauchy-Schwarz inequality gives

$$\begin{aligned} t=\Bigg \langle \xi (\gamma _i(t)),\gamma _i'(t)\Bigg \rangle \le |\xi (\gamma _i(t))||\gamma _i'(t)|=t, \end{aligned}$$

so that \(\xi (\gamma _i(t))=t\gamma _i'(t)\) for every \(t\in [0,\ell _i]\) and \(i=1,2\). In particular, since \(\gamma _1(\ell _1)=\gamma _2(\ell _2)=q\) and \(\ell _1\gamma '_1(\ell _1)=\xi (q)=\ell _2\gamma '_2(\ell _2)\) and they are both normalized geodesics, it must be \(\ell _1=\ell _2=|\xi (q)|=\ell >0\). Therefore, \(\gamma _1(\ell )=\gamma _2(\ell )\) with \(\gamma '_1(\ell )=\gamma '_2(\ell )\), which implies that \(\gamma _1(t)=\gamma _2(t)\) for every \(t\in [0,\ell ]\).

From now on, we let p denote the unique singular point of \(\xi \), and

$$\begin{aligned} M:={\overline{M}}{\setminus }\{p\}. \end{aligned}$$

Claim 4. The exponential map \(\exp _p:T_p{\overline{M}}\rightarrow {\overline{M}}\) is a diffeomorphism.

The previous claim accounts for the injectivity of \(\exp _p\), and the completeness of \({\overline{M}}\) for its surjectivity. On the other hand, if \(v\in T_p{\overline{M}}\) is a unit vector and \(\gamma _v:[0,+\infty )\rightarrow {\overline{M}}\) is the geodesic ray issuing from p with \(\gamma '_v(0)=v\), then, also from the previous claim, \({\gamma _v}_{|[0,t]}\) is minimizing, for every \(t>0\). Therefore, \(\exp _p\) has no conjugate points along \(\gamma _v\). Since this happens for every unit vector \(v\in T_p{\overline{M}}\), it assures that \(\exp _p\) is a local diffeomorphism. Being bijective, it is actually a global diffeomorphism.

Claim 5. If \({\mathcal {E}}=\{Y\in {\mathfrak {X}}(M);\,\langle Y,\xi \rangle =0\}\), then \({\mathcal {E}}\) is integrable.

For \(Y,Z\in {\mathcal {E}}\), we have

$$\begin{aligned} \Bigg \langle [Y,Z],\xi \Bigg \rangle&=\Bigg \langle \nabla _YZ-\nabla _ZY,\xi \Bigg \rangle \\&=Y\Bigg \langle Z,\xi \Bigg \rangle -\Bigg \langle Z,\nabla _Y\xi \Bigg \rangle -Z\langle Y,\xi \Bigg \rangle +\langle Y,\nabla _Z\xi \Bigg \rangle \\&=-\Bigg \langle Z,Y\Bigg \rangle +\Bigg \langle Y,Z\Bigg \rangle =0. \end{aligned}$$

Therefore, \([Y,Z]\in {\mathcal {E}}\), as we wished to show.

Claim 6. If \(N=\exp _p({\mathbb {S}}^{2n-1})\), then \(\xi _{|N}\) is a unit normal vector field along N.

Let \(v\in {\mathbb {S}}^{2n-1}\), \(\gamma (t)=\exp _p(tv)\), \(t\ge 0\), and \(q=\gamma (1)\in N\). Claim 3 gives \(|\xi _q|=|\gamma '(1)|=|\gamma '(0)|=|v|=1\); together with Gauss’ lemma, it shows that \(\xi _q\in T_qN^{\bot }\). The rest is immediate from the previous claim.

It follows in particular from the previous claim that N is a leaf of the distribution \({\mathcal {E}}\) and N is diffeomorphic to \({\mathbb {S}}^{2n-1}\).

Claim 7. If \(M={\overline{M}}{\setminus }\{p\}\), then M is isometric to \((0,+\infty )\times _tN\).

We already know, from Claim 4, that \(\exp _p:T_p\overline{M}{\setminus }\{0\}\rightarrow M\) is a diffeomorphism. Since \((0,+\infty )\times N\) is diffeomorphic to \(T_p{\overline{M}}{\setminus }\{0\}\) via \((t,q)\mapsto t(\exp _p)^{-1}(q)\), it follows that the map

$$\begin{aligned} \begin{array}{rccc} \Psi :&{}(0,+\infty )\times N&{}\longrightarrow &{}M\\ &{}(t,q)&{}\longmapsto &{}\exp _p(t(\exp _p)^{-1}(q)) \end{array} \end{aligned}$$
(8)

is also a diffeomorphism.

Thus, it suffices to show that the metric \(g_0\), induced on \((0,+\infty )\times N\) by such a diffeomorphism, is the warping metric of \((0,+\infty )\times _tN\).

To this end, first note that, from Claim 3, the integral curve of \(\xi \) through any point \(q\in M\) is a pregeodesic of M. Actually, a simple computation shows that the flow of \(\xi _{|M}\) is

$$\begin{aligned} \begin{array}{rccc} \Phi :&{}{\mathbb {R}}\times M&{}\longrightarrow &{}M\\ &{}(u,q)&{}\longmapsto &{}\exp _p\left( e^u(\exp _p)^{-1}(q)\right) \end{array}. \end{aligned}$$
(9)

Actually, since

$$\begin{aligned} \Phi (u,q)=\Psi (e^{u},q)=\gamma _{(\exp _p)^{-1}(q)}(e^{u}) \end{aligned}$$

we have, by Claim 3,

$$\begin{aligned} \frac{\partial \Phi }{\partial u}(u,q)=e^{u}\gamma '_{(\exp _p)^{-1}(q)}(e^{u})=\xi (\Phi (u,q)). \end{aligned}$$
(10)

Let \(\alpha :(-\varepsilon ,\varepsilon )\rightarrow N\) be an arc length parametrized curve with \(\alpha (0)=q\). We shall consider the parametrized surface in M given by the map

$$\begin{aligned} \begin{array}{rccccc} \varphi :&{} {\mathbb {R}}\times (-\varepsilon ,\varepsilon ) &{} \longrightarrow &{} {\mathbb {R}}\times N&{}\longrightarrow &{}M\\ &{} (u,s) &{}\longmapsto &{}(u,\alpha (s))&{}\longmapsto &{}\Phi (t,\alpha (s)) \end{array}. \end{aligned}$$

Since \(\xi _{|N}\in TN^{\bot }\), we have

$$\begin{aligned} \left\langle \frac{\partial \varphi }{\partial u}(u,s),\frac{\partial \varphi }{\partial s}(u,s)\right\rangle _{\varphi (u,s)}=\left\langle \xi (\varphi (u,s)),\frac{\partial \varphi }{\partial s}(u,s)\right\rangle _{\varphi (u,s)}=0. \end{aligned}$$

On the other hand, (10) gives

$$\begin{aligned} \left\langle \frac{\partial \varphi }{\partial u}(u,s),\frac{\partial \varphi }{\partial u}(u,s)\right\rangle _{\varphi (u,s)}=e^{2u}. \end{aligned}$$

Now, since \(\xi \) is a closed conformal vector field with conformal factor \(\psi \equiv 1\) (see Eq. (7)), we conclude that

$$\begin{aligned} \frac{D}{\partial s}\xi (\varphi (u,s))=\nabla _{\frac{\partial \varphi }{\partial s}}\xi =\frac{\partial \varphi }{\partial s}(u,s). \end{aligned}$$

Set

$$\begin{aligned} f(u,s):\left\langle \frac{\partial \varphi }{\partial s}(u,s),\frac{\partial \varphi }{\partial s}(u,s)\right\rangle _{\varphi (u,s)}. \end{aligned}$$

Computing pretty much as in the proof of item (b) of Theorem 3.4 of [3] (and omitting the point \(\varphi (u,s)\) from the computations, for the sake of clarity), we get

$$\begin{aligned} \frac{\partial f}{\partial u}(u,s)&=\frac{d}{du}\left\langle \frac{\partial \varphi }{\partial s}(u,s),\frac{\partial \varphi }{\partial s}(u,s)\right\rangle \\&=2\left\langle \frac{D}{\partial s}\frac{\partial \varphi }{\partial u}(u,s),\frac{\partial \varphi }{\partial s}(u,s)\right\rangle \\&=2\left\langle \frac{D}{\partial s}\xi (\varphi (u,s)),\frac{\partial \varphi }{\partial s}(u,s)\right\rangle \\&=2\left\langle \frac{\partial \varphi }{\partial s}(u,s),\frac{\partial \varphi }{\partial s}(u,s)\right\rangle \\&=2f(u,s). \end{aligned}$$

Hence, \(f(u,s)=e^{2u}f(0,s)\), which, by \(\varphi (0,s)=\alpha (s)\) and \(|\alpha '(s)|=1\), is the same as

$$\begin{aligned} \left\langle \frac{\partial \varphi }{\partial s}(u,s),\frac{\partial \varphi }{\partial s}(u,s)\right\rangle _{\varphi (u,s)}=e^{2u}\left\langle \frac{\partial \varphi }{\partial s}(0,s),\frac{\partial \varphi }{\partial s}(0,s)\right\rangle _{\varphi (0,s)}=e^{2u}. \end{aligned}$$

Finally, consider the parametrized surface

$$\begin{aligned} \begin{array}{rccccc} \psi :&{} (0,+\infty )\times (-\varepsilon ,\varepsilon ) &{}\longrightarrow &{}(0,+\infty )\times N&{}\longrightarrow &{}M\\ &{} (t,s) &{} \longmapsto &{}(t,\alpha (s))&{}\longmapsto &{}\Psi (t,\alpha (s)) \end{array}. \end{aligned}$$

Since \(\psi (t,s)=\varphi (\log t,s)\), the above computations translate into

$$\begin{aligned} \left\langle \frac{\partial \psi }{\partial t}(t,s),\frac{\partial \psi }{\partial s}(t,s)\right\rangle _{\psi (t,s)}=0,\quad \left\langle \frac{\partial \psi }{\partial t}(t,s),\frac{\partial \psi }{\partial t}(t,s)\right\rangle _{\psi (t,s)}=1 \end{aligned}$$

and

$$\begin{aligned} \left\langle \frac{\partial \psi }{\partial s}(t,s),\frac{\partial \psi }{\partial s}(t,s)\right\rangle _{\psi (t,s)}=t^2. \end{aligned}$$

Therefore, \(\left( (0,+\infty )\times N,g_0\right) \) is isometric to \((0,+\infty )\times _tN\), as we wished to show.

Claim 8. \((N,g_N,D)\) is a Sasaki manifold, where \(g_N\) is the induced metric on N, which we denote also by \(\langle \cdot ,\cdot \rangle \), and D its corresponding Levi–Civita connection.

Since \(\Psi :(0,+\infty )\times _tN\rightarrow M\) is an isometry from Claim 7, it follows that \((0,+\infty )\times _tN\) is naturally a Kählerian manifold: one just has to use the isometry to import, to \((0,+\infty )\times _tN\), the complex atlas and the complex structure of M. Therefore, according to the definition of Sasaki manifold and the discussion about that given in Sect. 1, N is a Sasaki manifold.

Remark 2.2

\({\mathbb {S}}^{2n-1}\), endowed with the canonical round metric, is the simplest example of a Sasakian manifold. Nevertheless, a sphere may, at first, be endowed with several distinct Sasakian structures. For instance, as observed at page 353 of [2], there are 63 distinct Sasakian structures on \({\mathbb {S}}^5\). Therefore, the conclusion of item (b) in Theorem 1.1 is, under our hypotheses, the best possible one. We would like to thank professor Vicente Muñoz for calling our attention to these examples of Sasakian structures on \({\mathbb {S}}^5\).