# 11 questions about Sasaki -and Sasaki-Einstein geometry [closed]

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

$$M=\lbrace r=1\rbrace=\lbrace1 \times M\rbrace \subset C(M).$$

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

$$\eta(X)=\frac{1}{r^{2}} \overline{g}(J(r\partial_{r}), X)$$

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

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@ Ronald Muller (with an umlaut): I don't want to nitpick, but I think you should try to ask only one or a couple of questions at the time on this website. If you break up this question, I think you have more and, most importantly, more useful answers ;). –  Max Muller Nov 19 '10 at 14:10
Ah okay, thank you for the information. Then I will do so :) –  Ronald Müller Nov 19 '10 at 14:37
+1 to Max, and I don't think it's a nitpick. Asking $11$ questions at once is poor etiquette on most Q&A sites and in any event not a good strategy: it is very unlikely that all $11$ of your questions will get addressed. –  Pete L. Clark Nov 20 '10 at 3:33
FWIW, I considered voting to close this question, but none of the reasons seemed fitting. There's no choice for "please don't ask $11$ questions at once"! –  Pete L. Clark Nov 20 '10 at 3:34
I will close this as not constructive, for lack of a better reason. Please Ronald —or anyone—, if you come back and are interested in asking this, resk the questions one or two at a time. –  Mariano Suárez-Alvarez May 9 '12 at 6:43