1 Erratum to: Geom DedicataDOI 10.1007/s10711-017-0223-3

The main result, Theorem 3.2, of [1] is incorrect and must be replaced by the following statement:

Theorem 3.2

Let \((M, g, \xi )\) be a compact K-contact manifold of dimension \(n = 2 m + 1, m \ge 1\), carrying a closed Weyl–Einstein structure D compatible with the conformal class \(c = [g]\). Then g is Einstein and D is the Levi–Civita connection of an Einstein metric \(g _0\) in c, which, up to scaling, is equal to g, except if (M, c) is the flat conformal sphere. In the latter case, any K-contact structure is isomorphic to the standard Sasaki–Einstein structure.

Accordingly, the last sentence in the Introduction of [1] must be replaced by the following one:

We show—Theorem 3.2 and Corollary 3.1 below—that g is then Einstein and D is the Levi–Civita of an Einstein metric \(g_0\), which is actually equal to g up to scaling, except if (M, c) is the flat conformal sphere; in all cases, the K-contact structure is Sasaki–Einstein.

In the proof of Theorem 3.2 the last sentence before Case 1 must be replaced by:

Since \(\xi \) is conformal with respect to \(g _0\), from Proposition 2.2, \(\xi \) is Killing with respect to \(g _0\), or (M, c) is the flat conformal sphere.

The end of the proof of Theorem 3.2 starting with Case 2 must be removed and replaced by:

Case 2. (M, c) is the standard flat conformal sphere \((\mathbb {S} ^{2 m + 1},c_0)\). In this case we conclude the proof of Theorem 3.2 by using the following general result:

Lemma 1

For any K-contact structure \((g, \xi )\) on \(\mathbb {S} ^{2 m + 1}\) with \([g] = c_0\), the metric g has constant sectional curvature equal to 1 and the K-contact structure is then isomorphic to the standard Sasaki–Einstein structure.

Proof

We first recall that for K-contact manifold \((M, g, \xi )\), we have

$$\begin{aligned} {\mathrm{R}} _{\xi , X} \xi = X - g (\xi , X) \xi , \end{aligned}$$
(1)

for any vector field X. Indeed, since \(\varphi = \nabla \xi \) and \(\xi \) is Killing with respect to g and of norm 1, we have

$$\begin{aligned} \nabla _X \varphi = {\mathrm{R}} _{\xi , X}, \end{aligned}$$
(2)

where \({\mathrm{R}}\) denotes the curvature of g, cf. [2], so that \({\mathrm{R}} _{\xi , X} \xi = \nabla _X (\nabla _{\xi } \xi ) - \nabla _{\nabla _X \xi } \xi = - \nabla _{\nabla _X \xi } \xi = - \varphi ^2 (X) = X - g (\xi , X) \, \xi \). In the current case, when the conformal structure [g] is flat, the curvature of g is of the form

$$\begin{aligned} {\mathrm{R}} _{X, Y} = {\mathrm{S}} (X) \wedge Y + X \wedge {\mathrm{S}} (Y), \end{aligned}$$
(3)

where, in general, for any n-dimensional Riemannian manifold (M, g), the normalized Ricci tensor (or Schouten tensor) \({\mathrm{S}}\) is defined by

$$\begin{aligned} {\mathrm{S}} := \frac{1}{(n - 2)} \left( {\mathrm{Ric}} - \frac{\mathrm{Scal}}{2 (n - 1)}\right) . \end{aligned}$$

It then follows from (1), (3) and the identity

$$\begin{aligned} {\mathrm{Ric}} (\xi ) = (n - 1) \, \xi , \end{aligned}$$
(4)

cf. Proposition 3.1 in [1], that:

$$\begin{aligned} {\mathrm{S}} (X) = \frac{1}{(n - 2)} \, \left[ \left( \frac{\mathrm{Scal}}{2 (n - 1)} - 1\right) \, X + \left( n - \frac{\mathrm{Scal}}{(n - 1)}\right) \, g (\xi , X) \, \xi \right] \end{aligned}$$
(5)

[in (4), (5) and in the sequel, \({\mathrm{Ric}}\) and \({\mathrm{S}}\) are regarded as endomorphisms of the tangent bundle via the metric g]. By using the contracted Bianchi identity \(\delta {\mathrm{S}} + \frac{\mathrm{d\, Scal}}{2 (n - 1)} = 0\), we readily infer from (5) that \({\mathrm{Scal}}\) is constant, so that

$$\begin{aligned} (\nabla _X {\mathrm{S}}) (Y) = \kappa \, \left( g (\nabla _X \xi , Y) \xi + g (\xi , Y) \, \nabla _X \xi \right) , \end{aligned}$$
(6)

for any vector fields X, Y, by setting \(\kappa :=\frac{1}{(n - 2)} \left( n - \frac{\mathrm{Scal}}{(n - 1)}\right) \). Since the conformal structure is flat, \((\nabla _X {\mathrm{S}}) (Y)\) must be symmetric in X, Y, so in particular the expression \(\kappa \, g (\nabla _X \xi , Y)=g((\nabla _X {\mathrm{S}}) (Y),\xi )\) must be symmetric in X, Y. On the other hand, \(g (\nabla _X \xi , Y)\) is clearly skew-symmetric in X, Y, thus showing that \(\kappa =0\); it follows that \({\mathrm{Scal}} = n (n - 1)\), hence, by (5), that \(S = \frac{1}{2} \mathrm {Id}\), so by (3), that g is a metric of sectional curvature equal to 1.

Finally, Eq. (2) shows that \(\nabla _X\varphi =\xi \wedge X\) for every tangent vector X, meaning that the K-contact structure is Sasaki–Einstein, and it is well known that the isometry group of \(\mathbb {S} ^{2 m + 1}\) acts transitively on the set of Sasaki–Einstein structures on the sphere. \(\square \)