On complete Kählerian manifolds endowed with closed conformal vector fields

Let M¯2n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{M}}^{2n}$$\end{document}, n>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n>1$$\end{document}, be a complete, noncompact Kählerian manifold, endowed with a nontrivial closed conformal vector field ξ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi $$\end{document} having at least one singular point. Under a reasonable set of conditions, we show that ξ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi $$\end{document} has just one singular point p and that M¯\{p}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{M}}{\setminus }\{p\}$$\end{document} is isometric to a one dimensional cone over a simply connected Sasakian manifold N diffeomorphic to S2n-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {S}}^{2n-1}$$\end{document}.As a straightforward consequence, we conclude that if the addition of a single point to the Kählerian cone of a (2n-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(2n-1)$$\end{document}-dimensional Sasakian manifold N has the structure of a complete, noncompact, 2n-dimensional Kählerian manifold whose metric extends that of the cone, and such that the canonical vector field of the cone extends to it having a singularity at the extra point, then N is isometric to S2n-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb S^{2n-1}$$\end{document}, endowed with an appropriate Sasakian structure.


Introduction and preliminaries
Given an m-dimensional Riemannian manifold (M m , g) with Levi-Civita connection ∇, we recall that a conformal vector field ξ on M is said to be closed if the 1-form ξ is closed. This is easily seen to be equivalent to the existence of a smooth function ψ : M → R (called the B Luis J. Alías ljalias@um.es Antonio Caminha caminha@mat.ufc.br F. Yure do Nascimento yure@ufc.br conformal factor of ξ ) such that for all X ∈ X(M). In turn, this readily yields Also in this setting, item 1 of Lemma 1 of [9] shows that, for every nontrivial closed and conformal vector field ξ , where ∇ψ stands for the gradient of ψ and for the normalized Ricci curvature of (M m , g, ∇) in the direction of ξ . Moreover, if m > 2, then item 3 of that result shows that ξ −1 (0), the set of singular points of ξ , is a set of isolated points and ψ( p) = 0 for every p ∈ ξ −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 ⊂ 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 × N m−1 and let π I : M → I and π N : M → N denote the projections. If h : I → (0, +∞) is a smooth function andh = h • π I : M → (0, +∞), then ·, · = π * I dt 2 +h 2 π * N g N 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 Ifh = h • π I , ∂ t denotes the canonical vector field on I and∂ t its horizontal lift to M, then it is a standard fact that the vector fieldh∂ t is a closed conformal vector field on M with no singular points, with conformal factorh = h • π I , where h is the derivative of h.
In particular, letting I = (0, +∞) and h(t) = t for every t > 0, we obtain the one dimensional cone M m = (0, +∞) × t N m−1 , with closed conformal vector field t∂ 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 M = (0, +∞) × t N 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 ξ(t, p) = t∂ t the closed conformal vector field, then it can be proved (cf. [11], for instance) that: (a) Z := J ξ is a unit Killing vector field on N ≈ {1} × N .
for all X , Y ∈ X(N ).
Conversely, let (N , g N ) be an odd dimensional Riemannian manifold with Levi-Civita connection D and M = (0, +∞)× t N 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 ∈ End(T N), given by (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 T N satisfies (3).
With notations as in the previous paragraph, warped product geometry (cf. Corollary 7.43 of [7]) readily shows that 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 In this paper, we aim at proving the following The fact that the conformal factor ψ ≡ 1 on M implies that the Lie derivative of the metric tensor g with respect to ξ satisfies L ξ g = 2g. Geometrically, this means that the flow {ϕ t } t∈R of the vector field ξ consists of hometheties of positive coefficient, since (ϕ * t g) p = e 2t g p for all p ∈ M and t ∈ R. For that reason ξ 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, +∞) × t N has a removable singularity if the following condition is satisfied: for some symbol p not in M, the space M := M ∪ { 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∂ t of M likewise extends to ξ ∈ X(M). Corollary 1.2 Let N be a (2n − 1)-dimensional Sasakian manifold whose Kählerian cone M := (0, +∞) × t N has a removable singularity. With notations as above, if p is a singular point of ξ , then N is isometric to S 2n−1 , endowed with an appropriate Sasakian structure.

Proof of Theorem 1.1
First of all, as observed in the second paragraph of Sect. 1, since 2n > 2 and ξ is nontrivial, it has isolated zeros. Recall also that we are assuming Ric M (ξ ) ≤ 0 on M. We divide the subsequent analysis in several steps.
Back to the proof of Claim 1, let η = ψξ . It follows from (1) and (2) that, at every nonsingular point of ξ , we have By continuity, div M η ≥ 0 on M. Assume, for the sake of contradiction, that ψ is not identically 1 on M. Setting f = ψ −1, we have f ≥ 0 and f ≡ 0 on M. Also from (2), we get, at the nonsingular points of ξ , Again, by continuity, ∇ f , η ≥ 0 on M.
Once we know that ψ ≡ 1, Eq. (2) shows that Ric M (ξ ) ≡ 0 on M\ξ −1 (0), and, trivially, on ξ −1 (0). Anyway, as we have already noticed, this will also follow once we show that M\{ p} is isometric to a one dimensional cone over a simply connected Sasakian manifold. Claim 2. ξ has exactly one singular point. Arguing once more by contradiction, assume that ξ( p) = 0 and ξ(q) = 0, for some distinct points p, q ∈ M. Thanks to the completeness of M, we can take a normalized geodesic γ : [0, ] → M from p to q. Letting ξ(t) denote the restriction of ξ to γ and ϕ(t) := ξ(t), γ (t) , we get ϕ(0) = ϕ( ) = 0 and However, this contradicts the fact that ϕ( ) = 0. It thus follows from the previous claim and our hypotheses that ξ has exactly one singular point.

Claim 3.
If p is the singular point of ξ , then, for each q = p, there is just one normalized geodesic γ from p to q. Moreover, letting ξ(t) denote the restriction of ξ to γ , we have ξ(t) = tγ (t).
Therefore, [Y , Z ] ∈ E, as we wished to show. Claim 6. If N = exp p (S 2n−1 ), then ξ |N is a unit normal vector field along N .
It follows in particular from the previous claim that N is a leaf of the distribution E and N is diffeomorphic to S 2n−1 .
To this end, first note that, from Claim 3, the integral curve of ξ through any point q ∈ M is a pregeodesic of M. Actually, a simple computation shows that the flow of ξ |M is Finally, consider the parametrized surface Since ψ(t, s) = ϕ(log t, s), the above computations translate into Therefore, ((0, +∞) × N , g 0 ) is isometric to (0, +∞) × t N , 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 ·, · , and D its corresponding Levi-Civita connection.
Since : (0, +∞) × t N → M is an isometry from Claim 7, it follows that (0, +∞) × t N is naturally a Kählerian manifold: one just has to use the isometry to import, to (0, +∞)× t N , 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 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 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 S 5 . article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.