I have some questions about Sasaki -and Sasaki-Einstein geometry. I hope, you can answer me some of these questions.
1.) Let $(M, g)$ a Riemannian manifold and $(C(M)=\mathbb{R}_{>0} \times M, \overline{g}=dr^{2}+r^{2}g)$ denote its metric
cone. In the paper "Sasaki-Einstein Manifolds" from Lames Sparks, I find that the Sasaki manifold $(M, g)$ is naturally isometrically embedded into the cone via the inclusion
\begin{equation} M=\lbrace r=1\rbrace=\lbrace1 \times M\rbrace \subset C(M). \end{equation}
How can I see this?
2.) Let $\overline{\nabla}$ the Levi-Civita connection of $\overline{g}$. Is then $\overline{\nabla}_{\partial_{r}}\partial_{r}=0$ or
$\overline{\nabla}_{\partial_{r}}\partial_{r}=\partial_{r}$ with $\partial_{r}:=\frac{\partial}{\partial r}$?
3.) How can I show, that the Lie-derivative of $J$ in direction from $r \partial_{r}$ is equal to zero?
4.) If I define $\xi=r \partial_{r}$, how can I show that $\mathcal{L}_{\xi} \overline{g}=0$, i.e. $\xi$ is a Killing vectorfield. With $\mathcal{L}$ I mean the Lie-derivative.
5.) Let $\eta=d^{c} \log r=i(\partial-\overline{\partial}) \log r$, where $\partial$ and $\overline{\partial}$ the usual Dolbeault operators. How can I show that
\begin{equation} \eta(X)=\frac{1}{r^{2}} \overline{g}(J(r\partial_{r}), X) \end{equation}
and the Kähler 2-form on $C(M)$ is $\omega=\frac{1}{2}d(r^{2}\eta)=\frac{1}{2} i \partial \overline{\partial} r^{2}$?
6.) We define $\Phi(X)$ as $\Phi(X)=\nabla_{X} \xi$. How can I show that $(\nabla_{X}\Phi) Y=g(\xi, Y)X-g(X, Y)\xi$?
7.) http://arxiv.org/PS_cache/hep-th/pdf/9810/9810250v1.pdf, Proposition 1.1.2, page number 4. Can someone proof the equivalence (i) $\Leftrightarrow$ (ii)?
8.) Let $M=S^{2m+1} \cong \mathbb{C}^{m+1}$. This is a Sasaki manifold. But what is $\xi$ oder $\Phi$?
9.) Analogous: Let $M=\mathbb{R}^{2m+1}$, it is also Sasaki, but what is $\xi$ or $\Phi$?
10.) I have read the following about 3-Sasaki-manifolds. The definition of 3-Sasaki-manifolds (cone is hyperkähler) implies
$\text{Hol}(\overline{g}) \subset Sp(k) \subset SU(2k)$ and then that 3-Sasaki manifolds are also Sasaki-Einstein.
How can I prove this?
And the last question:
11.) Let $(M, g)$ a Sasaki-manifold of dimension $2m+1$. Then the following are equivalent:
i) $(M, g)$ is Sasaki with $\text{Ric}_{g}=2m \cdot g$. ii) $(C(M), \overline{g})$ is Ricci flat.
What is the sketch of proof?
Thanks a lot and best regards Ronald
