Consider Ricci flow on a compact smooth Riemannian manifold $(M,g(t))$,the Ricci tensor is $Ric(t)$. Then , they meets $$ \partial_tg_{ij}=-2R_{ij} \\ \partial_tR_{ik}=\Delta R_{ik}+2g^{pr}g^{qs}R_{piqk}R_{rs}-2g^{pq}R_{pi}R_{qk} $$ If at $t=0$, the metric and Ricci tensor is diagonal, namely , $g_{ij}(0)=0 ~R_{ij}(0)=0~,~i\ne j$ , whether the metric and Ricci tensor keep diagonal under Ricci flow ?

The following system of ODEs arises when studying Ricci flow on 3-manifolds: $$ \frac{dm_1}{dt} = m_1^2+m_2m_3 \\ \frac{dm_2}{dt} = m_2^2+m_1m_3 \\ \frac{dm_3}{dt} = m_3^2+m_1m_2 \\ $$ Going back over Hamilton’s 1986 paper I realised I didn’t understand the first step of his reasoning: Note that $$\frac{d}{dt}(m_2 – m_1) = (m_2-m_1)(m_2+m_1-m_3)$$ so that if $m_1 \le […]

As my question 1 and 2, I still have many problems. First, the hyperbolic manifold is the manifold $(\mathbb R^n , g)$ given by one chart $\mathbb R^n$, where in spherical coordinates $(\theta^0= s, \theta^1, \cdots, \theta^{n-1})$, the metric is given by $$\tag{1} g = ds^2 + \frac1M \sinh^2(\sqrt M s) d\Omega^2.$$ I want to […]

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