共性几何中的曲率计算

It is customary to call the function function defined on $U$ by $\nabla_{\frac{\partial}{\partial x_i}}\frac{\partial}{\partial x_j}=\sum\limits_k\Gamma_{ij}^k\frac{\partial}{\partial x_k}$, the coefficients of the connection $\nabla$ on $U$ or the Christoffel symbols of the connection.

 

It is convenient to express what was see above in a coordinate system $(U,x)$ based at the point $p\in M$. Let us indicate as usual, $\frac{\partial}{\partial x_i}=X_i$. We put
\[R(X_i,X_j)X_k=\sum\limits_{l}=R_{ijk}^lX_l.\]

To express $R_{ijk}^l$ in terms of the coefficients $\Gamma_{ij}^k$ of the Riemannian connection, we write
\begin{align*}
R(X_i,X_j)X_k&=\nabla_{X_i}\nabla_{X_j}X_k-\nabla_{X_j}\nabla_{X_i}X_k-\nabla _{[X_i,X_j]}X_k\\
&=\nabla_{X_i}(\sum\limits_l\Gamma_{jk}^lX_l)-\nabla_{X_j}(\sum\limits_l\Gamma_{ik}^lX_l),
\end{align*}
which by a direct calculation yieds
\begin{align*}
R_{ijk}^s&=\frac{\partial}{\partial x_i}\Gamma_{jk}^s-\frac{\partial}{\partial x_j}\Gamma_{ik}^s+\sum\limits_p\Gamma_{jk}^p\Gamma_{ip}^s -\sum\limits_p\Gamma_{ik}^p\Gamma_{jp}^s,
\end{align*}
and the Ricci curvature is given by ((1,4) contraction)
\begin{align*}
R_{jk}=R_{ijk}^i=\frac{\partial}{\partial x_i}\Gamma_{jk}^i-\frac{\partial}{\partial x_j}\Gamma_{ik}^i+\sum\limits_p\Gamma_{jk}^p\Gamma_{ip}^i- \sum\limits_p\Gamma_{ik}^p\Gamma_{jp}^i
\end{align*}

Rearrange the subscript indeces, we get
\begin{align*}
R_{ij}=\frac{\partial}{\partial x_s}\Gamma_{ij}^s-\frac{\partial}{\partial x_i}\Gamma_{sj}^s+\sum\limits_t\Gamma_{ij}^t\Gamma_{st}^s -\sum\limits_t\Gamma_{sj}^t\Gamma_{it}^s.
\end{align*}

Notice that
\begin{equation}
\Gamma_{ij}^k=\frac{1}{2}g^{kl}\{\frac{\partial g_{il}}{\partial x_j}+\frac{\partial g_{lj}}{\partial x_i}-\frac{\partial g_{ji}}{\partial x_l}\}.
\end{equation}

If $\tilde{g}=\rho g$ and $\rho>0$, then
\begin{align*}
\tilde{\Gamma}_{ij}^k&=\frac{1}{2}\rho^{-1}g^{kl}\{\frac{\partial \tilde{g}_{il}}{\partial x_j}+\frac{\partial \tilde{g}_{lj}}{\partial x_i}-\frac{\partial \tilde{g}_{ji}}{\partial x_l}\}\\
&=\Gamma_{ij}^k+\frac{1}{2}\rho^{-1}\{\delta^k_i\frac{\partial\rho}{\partial x_j}+\delta^k_j\frac{\partial \rho}{\partial x_i}-g_{ji}g^{kl}\frac{\partial \rho}{\partial x_l}\}\\
&=\Gamma_{ij}^k+\frac{1}{2}\{\delta^k_i\frac{\partial\log\rho}{\partial x_j}+\delta^k_j\frac{\partial \log\rho}{\partial x_i}-g_{ji}g^{kl}\frac{\partial\log \rho}{\partial x_l}\}
\end{align*}

Let $\rho=e^f$ or $f=\log\rho$, and $f^i=\frac{\partial f}{\partial x_j}g^{ij}$, $df=\frac{\partial f}{\partial x_i}dx^i$. Then we get

\begin{align*}
\tilde{\Gamma}_{ij}^k
&=\Gamma_{ij}^k+\frac{1}{2}\{\delta^k_i\frac{\partial\log\rho}{\partial x_j}+\delta^k_j\frac{\partial \log\rho}{\partial x_i}-g_{ji}g^{kl}\frac{\partial \log\rho}{\partial x_l}\}\\
&=\Gamma_{ij}^k+\frac{1}{2}\{\delta^k_i\frac{\partial f}{\partial x_j}+\delta^k_j\frac{\partial f}{\partial x_i}-g_{ji}g^{kl}\frac{\partial f}{\partial x_l}\}\\
&=\Gamma_{ij}^k+\frac{1}{2}\{\delta^k_i\frac{\partial f}{\partial x_j}+\delta^k_j\frac{\partial f}{\partial x_i}-g_{ij}f^k\}=\Gamma_{ij}^k+L_{ij}^k.
\end{align*}

Note that
\begin{align*}
R_{ij}=\frac{\partial}{\partial x_s}\Gamma_{ij}^s-\frac{\partial}{\partial x_i}\Gamma_{sj}^s+\sum\limits_t\Gamma_{ij}^t\Gamma_{st}^s -\sum\limits_t\Gamma_{sj}^t\Gamma_{it}^s,\\
\tilde{R}_{ij}=\frac{\partial}{\partial x_s}\tilde{\Gamma}_{ij}^s-\frac{\partial}{\partial x_i}\tilde{\Gamma}_{sj}^s+\sum\limits_t\tilde{\Gamma}_{ij}^t\tilde{\Gamma}_{st}^s -\sum\limits_t\tilde{\Gamma}_{sj}^t\tilde{\Gamma}_{it}^s,
\end{align*}

we have (for simplicity, we drop $\Sigma$)
\begin{align*}
\tilde{R}_{ij}=&R_{ij}+\frac{\partial}{\partial x_s}L_{ij}^s-\frac{\partial}{\partial x_i}L_{sj}^s+\Gamma_{ij}^tL_{st}^s+L_{ij}^t\Gamma_{st}^s+L_{ij}^tL_{st}^s\\
& -\Gamma_{sj}^tL_{it}^s-L_{sj}^t\Gamma_{it}^s-L_{sj}^tL_{it}^s,
\end{align*}

Now will calculate the expresion term by term.
\begin{align*}
\frac{\partial}{\partial x_s}L_{ij}^s=&\frac{1}{2}(\delta_i^s\frac{\partial^2 f}{\partial x_jx_s}+\delta_j^s\frac{\partial^2 f}{\partial x_ix_s}-\frac{\partial g_{ij}}{\partial x_s}f^s-g_{ij}\frac{\partial f^s}{\partial x_s})\\
&=\frac{1}{2}(2\frac{\partial^2 f}{\partial x_ix_j}-\frac{\partial g_{ij}}{\partial x_s}f^s-g_{ij}\frac{\partial f^s}{\partial x_s}),
\end{align*}

\begin{align*}
\frac{\partial}{\partial x_i}L_{sj}^s&=\frac{1}{2}(\delta_s^s\frac{\partial^2 f}{\partial x_jx_i}+\delta_j^s\frac{\partial^2 f}{\partial x_sx_i}-\frac{\partial (g_{sj}f^j)}{\partial x_i})\\
&=\frac{n}{2}\frac{\partial^2 f}{\partial x_ix_j},
\end{align*}

\begin{align*}
\Gamma_{ij}^tL_{st}^s&=\frac{1}{2}\Gamma_{ij}^t(\delta_s^sf_t+\delta_t^sf_s-g_{st}f^s)\\
&=\frac{1}{2}\Gamma_{ij}^t((n+1)f_t-f_t)\\
&=\frac{n}{2}\Gamma_{ij}^tf_t,
\end{align*}

\begin{align*}
L_{ij}^t\Gamma_{st}^s&=\frac{1}{2}(\delta_i^tf_j+\delta_j^tf_i-g_{ij}f^t)\Gamma_{st}^s\\
&=\frac{1}{2}(\Gamma_{si}^sf_j+\Gamma_{sj}^sf_i-g_{ij}\Gamma_{st}^sf^t),
\end{align*}

\begin{align*}
L_{ij}^tL_{st}^s=&\frac{1}{4}\{(\delta_i^tf_j+\delta_j^tf_i-g_{ij}f^t)(\delta_s^sf_t+\delta_t^sf_s-g_{st}f^s)\}\\
=&\frac{1}{4}\{(\delta_i^tf_j\delta_s^sf_t+\delta_i^tf_j\delta_t^sf_s-\delta_i^tf_jg_{st}f^s)+(\delta_j^tf_i\delta_s^sf_t+\delta_j^tf_i\delta_t^sf_s-\delta_j^tf_ig_{st}f^s)\\
&-(g_{ij}f^t\delta_s^sf_t+g_{ij}f^t\delta_t^sf_s-g_{ij}f^tg_{st}f^s)\}\\
=&\frac{n}{4}(2f_if_j-|\nabla f|^2g_{ij}),
\end{align*}

\begin{align*}
\Gamma_{sj}^tL_{it}^s&=\frac{1}{2}\Gamma_{sj}^t(\delta_i^sf_t+\delta_t^sf_i-g_{it}f^s)\\
&=\frac{1}{2}(\Gamma_{ij}^tf_t+\Gamma_{tj}^tf_i-g_{it}\Gamma_{sj}^tf^s),
\end{align*}

\begin{align*}
L_{sj}^t\Gamma_{it}^s&=\frac{1}{2}\Gamma_{it}^s(\delta_s^tf_j+\delta_j^tf_s-g_{sj}f^t)\\
&=\frac{1}{2}(\Gamma_{it}^tf_j+\Gamma_{ij}^sf_s-g_{sj}\Gamma_{it}^sf^t),
\end{align*}

\begin{align*}
L_{sj}^tL_{it}^s=&\frac{1}{4}(\delta_s^tf_j+\delta_j^tf_s-g_{sj}f^t)(\delta_i^sf_t+\delta_t^sf_i-g_{it}f^s)\\
=&\frac{1}{4}\{(\delta_s^tf_j\delta_i^sf_t+\delta_s^tf_j\delta_t^sf_i-\delta_s^tf_jg_{it}f^s)+(\delta_j^tf_s\delta_i^sf_t+\delta_j^tf_s\delta_t^sf_i-\delta_j^tf_sg_{it}f^s)\\
&-(g_{sj}f^t\delta_i^sf_t+g_{sj}f^t\delta_t^sf_i-g_{sj}f^tg_{it}f^s)\}\\
=&\frac{1}{4}((n+2)f_if_j-2|\nabla f|^2g_{ij}).
\end{align*}

Combine the above identities, we have

\begin{align*}
\tilde{R}_{ij}=&R_{ij}+\frac{1}{2}(2\frac{\partial^2 f}{\partial x_ix_j}-\frac{\partial g_{ij}}{\partial x_s}f^s-g_{ij}\frac{\partial f^s}{\partial x_s})-\frac{n}{2}\frac{\partial^2 f}{\partial x_ix_j}\\
&+\frac{n}{2}\Gamma_{ij}^tf_t+\frac{1}{2}(\Gamma_{si}^sf_j+\Gamma_{sj}^sf_i-g_{ij}\Gamma_{st}^sf^t)+\frac{n}{4}(2f_if_j-|\nabla f|^2g_{ij})\\
&-\frac{1}{2}(\Gamma_{ij}^tf_t+\Gamma_{tj}^tf_i-g_{it}\Gamma_{sj}^tf^s)-\frac{1}{2}(\Gamma_{it}^tf_j+\Gamma_{ij}^sf_s-g_{sj}\Gamma_{it}^sf^t)-\frac{1}{4}((n+2)f_if_j-2|\nabla f|^2g_{ij}).
\end{align*}

Note that by the compatibility of the connection and metric, we have
\begin{align*}
\frac{\partial g_{ij} }{\partial x_s}=\Gamma_{si}^tg_{tj}+\Gamma_{sj}^tg_{ti}.
\end{align*}

It follows that
\begin{align*}
\tilde{R}_{ij}=&R_{ij}+\frac{1}{2}(2\frac{\partial^2 f}{\partial x_ix_j}-(\Gamma_{si}^tg_{tj}+\Gamma_{sj}^tg_{ti})f^s-g_{ij}\frac{\partial f^s}{\partial x_s})-\frac{n}{2}\frac{\partial^2 f}{\partial x_ix_j}\\
&+\frac{n}{2}\Gamma_{ij}^tf_t+\frac{1}{2}(\Gamma_{si}^sf_j+\Gamma_{sj}^sf_i-g_{ij}\Gamma_{st}^sf^t)+\frac{n}{4}(2f_if_j-|\nabla f|^2g_{ij})\\
&-\frac{1}{2}(\Gamma_{ij}^tf_t+\Gamma_{tj}^tf_i-g_{it}\Gamma_{sj}^tf^s)-\frac{1}{2}(\Gamma_{it}^tf_j+\Gamma_{ij}^sf_s-g_{sj}\Gamma_{it}^sf^t)-\frac{1}{4}((n+2)f_if_j-2|\nabla f|^2g_{ij})\\
=&R_{ij}-\frac{n-2}{2}\frac{\partial^2 f}{\partial x_ix_j}-\frac{1}{2}g_{ij}\left(\frac{\partial f^s}{\partial x_s}+\Gamma_{st}^sf^t\right)+\frac{n-2}{2}\Gamma_{ij}^tf_t-\frac{n-2}{4}|\nabla f|^2g_{ij}+\frac{n-2}{4}f_if_j\\
=&R_{ij}-\frac{n-2}{2}(\frac{\partial^2 f}{\partial x_ix_j}-\Gamma_{ij}^tf_t)-\frac{1}{2}g_{ij}\Delta f-\frac{n-2}{4}|\nabla f|^2g_{ij}+\frac{n-2}{4}f_if_j\\
=&R_{ij}-\frac{n-2}{2}f_{,ij}+\frac{n-2}{4}f_{,i}f_{,j}-(\frac{1}{2}\Delta f+\frac{n-2}{4}|\nabla f|^2)g_{ij}\\
=&R_{ij}-\frac{n-2}{2}(\log\rho)_{,ij}+\frac{n-2}{4}(\log\rho)_{,i}(\log\rho)_{,j}-\frac{1}{2}\Big(\Delta\log\rho+\frac{n-2}{2}|\nabla\log\rho|^2\Big)g_{ij}.
\end{align*}

Recall that if $X=X^i\frac{\partial}{\partial x_j}$, then
\[div(X)=\frac{\partial X^i}{\partial x_i}+\Gamma_{ik}^iX^k.\]
Especially, we have
\[\Delta f=div(\nabla f)=\frac{\partial f^i}{\partial x_i}+\Gamma_{ik}^if^k.\]
This formula has been used in the above simplification.

Let's do some computation at first.
\begin{align}
\Delta (\phi(f))&=g^{ij}\Big(\partial_{ij}(\phi(f))-\Gamma_{ij}^k\partial_k(\phi(f))\Big)\nonumber\\
&=g^{ij}\Big(\phi'(f)\partial_{ij}f+\phi''(f)\partial_{i}f\partial_{j}f-\Gamma_{ij}^k\phi'(f)\partial_k(f)\Big)\nonumber\\
&=\phi'(f)\Delta f+\phi''(f)|\nabla f|^2.
\end{align}

Thus we get
\begin{align} \Delta(\log\rho)=\frac{1}{\rho}\Delta\rho-\frac{1}{\rho^2}|\nabla\rho|^2.
\end{align}

As in Yau and Schoen's book or the above computation, we must arrive at
\begin{align}
\tilde{R}_{ij}=&R_{ij}-\frac{n-2}{2}(\log\rho)_{ij}+\frac{n-2}{4}(\log\rho)_{i}(\log\rho)_{j}\nonumber\\
&-\frac{1}{2}\Big(\Delta\log\rho+\frac{n-2}{2}|\nabla\log\rho|^2\Big)g_{ij}.
\end{align}

It follows from $\tilde{R}=\tilde{g}^{ij}\tilde{R}_{ij}=\frac{1}{\rho}g^{ij}\tilde{R}_{ij}$ that
\begin{align}
\tilde{R}=\rho^{-1}R-(n-1)\rho^{-1}\Delta\log\rho-\frac{(n-1)(n-2)}{4}\rho^{-1}|\nabla\log\rho|^2.
\end{align}
Or
\begin{align}
\tilde{R}=\rho^{-1}R-(n-1)\rho^{-2}\Delta\rho-\frac{(n-1)(n-6)}{4}\rho^{-3}|\nabla\rho|^2.
\end{align}

Now we consider two case:

Case 1. $n=2$. Let $\rho=e^{2u}$, then we have
\[\tilde{R}=e^{-2u}(R-2\Delta u).\]
Note that Gauss curvature $\tilde{K}=\frac{1}{2}\tilde{R}$, $K=\frac{1}{2}R$, then
\begin{equation}
-\Delta u+K(x)=\tilde{K}(x)e^{2u}.
\end{equation}

Case 2. $n=3$. Let $\rho=u^{\frac{4}{n-2}}$. Then we get
\[\Delta\rho=\Delta u^{\frac{4}{n-2}}=\frac{4}{n-2}u^{\frac{4}{n-2}-1}\Delta u+\frac{4}{n-2}(\frac{4}{n-2}-1)u^{\frac{4}{n-2}-2}|\nabla u|^2,\]
and
\[\nabla\rho=\frac{4}{n-2}u^{\frac{4}{n-2}-1}\nabla u.\]

Then we can eliminate the first order term, and get
\begin{align}
-\Delta u+\frac{n-2}{4(n-1)}R(x)u=\frac{n-2}{4(n-1)}\tilde{R}(x)u^\frac{n+2}{n-2}.
\end{align}

Equations are the famous \textbf{Yamabe Equations}.

In the special case, when $n\geq3$ and $M=\mathbb{S}^n$ with $g$ is the standard metric, then we can transfer the equation to $\mathbb{R}^n$.

 

In this case, $R(x)=n(n-1)K(x)=n(n-1)$. Then the equation becomes
\begin{align}
-\Delta_g u+\frac{n(n-2)}{4}u=\frac{n-2}{4(n-1)}\tilde{R}(x)u^\frac{n+2}{n-2}.
\end{align}

Using the stereographic projection with the local coordinate for $\mathbb{S}^n\setminus\{1\}$ by $(U,y^i)$,

We  calculate the coordinate transformation firstly. For $X\in \mathbb{R}^{n+1}$
\[|X|^2=\sum_{i=1}^{n+1}x_i^2=1, \frac{y_i-x_i}{y_i}=x_{n+1}\]
i.e.
\[|X|^2=\sum_{i=1}^{n+1}x_i^2=1, x_i=(1-x_{n+1})y_i,\]

Calculatingas below,
\begin{align*}
1-x_{n+1}^2&=|y|^2(1-x_{n+1})^2,\\
1+x_{n+1}&=|y|^2(1-x_{n+1}),\\
x_{n+1}&=\frac{|y|^2-1}{|y|^2+1},\\
x_i&=(1-x_{n+1})y_i=\frac{2y_i}{|y|^2+1}
\end{align*}
A more direct computation is to let $X=\lambda((y,0)-(0,..,0,1))+(0,...,0,1)$ and substitute it into $|X|^2=1$ and solve $\lambda$. The idea is the same.

Now in this local coordinate, we get the reduced metric from $\mathbb{R}^{n+1}$.
\begin{align*}
\frac{\partial X}{\partial y_i} =(...,2\delta_{ki}(1+|y|^2-4y_ky_i),...,\frac{4y_i}{1+|y|^2}).
\end{align*}
Then
\begin{align}
g_{ij}=\frac{1}{(1+|y|^2)^2}\{(2\delta_{ki}(1+|y|^2)-4y_ky_i)(2\delta_{kj}(1+|y|^2)-4y_ky_j)+16y_iy_j\}.
\end{align}

i.e.,
\begin{align}
g_{ij}=\frac{4}{(1+|y|^2)^2}\delta_{ij}, \ g^{ij}=\frac{(1+|y|^2)^2}{4}\delta_{ij}.
\end{align}

It is easy to check that
\begin{equation}
\frac{\partial g_{ij}}{\partial x_k}=-\frac{16y_k}{(1+|y|^2)^3}\delta_{ij}.
\end{equation}

By composition, we can denote $u$ as a function of $y$, we have
\begin{equation}
\Delta u=g^{ij}\frac{\partial^2 u}{\partial y_i\partial y_j}-g^{ij}\Gamma_{ij}^k\frac{\partial u}{\partial y_k}.
\end{equation}

Note that
\begin{align*}
g^{ij}\Gamma_{ij}^k&=g^{ij}\frac{1}{2}g^{kl}(\frac{\partial g_{lj}}{\partial y_i}+\frac{\partial g_{il}}{\partial y_j}-\frac{\partial g_{ij}}{\partial y_l})\\
&=\frac{(1+|y|^2)^2}{4}\delta^{ij}\frac{1}{2}\frac{(1+|y|^2)^2}{4}\delta^{kl}(-\frac{16}{(1+|y|^2)^3}(\delta_{j}^ly_i+\delta_{i}^ly_j-\delta_{ij}y_l)\\
&=(1+|y|^2)\frac{n-2}{2}y_k,
\end{align*}
\begin{align*}
g^{ij}\frac{\partial^2 u}{\partial x_i\partial x_j}=\frac{(1+|y|^2)^2}{4}\Delta _{\mathbb{R}^n}u.
\end{align*}

It follows that
\begin{align*}
\Delta_gu=(\frac{1+|y|^2}{2})^2\Delta u-(n-2)(\frac{1+|y|^2}{2})y_k\frac{\partial u}{\partial y_k}.
\end{align*}

Then
\begin{align}
-(\frac{1+|y|^2}{2})^2\Delta u+(n-2)(\frac{1+|y|^2}{2})y_k\frac{\partial u}{\partial y_k}+\frac{n(n-2)}{4}u=\frac{(n-2)}{4(n-1)}\tilde{R}(x)u^\frac{n+2}{n-2}.
\end{align}

In order to eliminate the lower order term, Let $v=\Big(\frac{1+|y|^2}{2}\Big)^{\frac{2-n}{2}}u=\phi u$, we will derive the equation for $v$.

Note that
\begin{align*}
\Delta v=&u\Delta\phi+2\nabla\phi\nabla u+\phi\Delta u\\
=&\{\frac{2-n}{2}(\frac{1+|y|^2}{2})^{-\frac{n}{2}}n+\frac{2-n}{2}(-\frac{n}{2})(\frac{1+|y|^2}{2})^{-\frac{n}{2}-1}|y|^2\}u\\
&+2\{\frac{2-n}{2}(\frac{1+|y|^2}{2})^{-\frac{n}{2}}y_i\frac{\partial u}{\partial y_i}+(\frac{1+|y|^2}{2})^{\frac{2-n}{2}}\Delta u\\
&=-\frac{n(n-2)}{4}(\frac{1+|y|^2}{2})^{-\frac{n}{2}-1}u-(n-2)(\frac{1+|y|^2}{2})^{-\frac{n}{2}}y_i\frac{\partial u}{\partial y_i}+(\frac{1+|y|^2}{2})^{\frac{2-n}{2}}\Delta u\\
&=(\frac{1+|y|^2}{2})^{-\frac{n}{2}-1}\left(-\frac{n(n-2)}{4}u-(n-2)(\frac{1+|y|^2}{2})y_i\frac{\partial u}{\partial y_i}+(\frac{1+|y|^2}{2})^2\Delta u\right)\\
&=-(\frac{1+|y|^2}{2})^{-\frac{n}{2}-1}\frac{(n-2)}{4(n-1)}\tilde{R}(x)u^\frac{n+2}{n-2}
\end{align*}

Thus we get
\begin{align*}
-\Delta_{\mathbb{R}^n} v =[(\frac{1+|y|^2}{2})^{\frac{2-n}{2}}]^{\frac{n+2}{n-2}}\frac{(n-2)}{4(n-1)}\tilde{R}(x)u^\frac{n+2}{n-2}=\frac{(n-2)}{4(n-1)}\tilde{R}(x)v^\frac{n+2}{n-2}
\end{align*}

For simplicity, we get
\begin{align*}
-\Delta_{\mathbb{R}^n}v=T(y)v^\frac{n+2}{n-2},
\end{align*}
where
\[T(y)=\frac{(n-2)}{4(n-1)}\tilde{R}(x(y)).\]

It also should note that
\begin{equation}
v=O(|y|^{2-n})\ \ as\ \ |y|\rightarrow \infty.
\end{equation}

When $T\equiv 1$ , the positive solution of the equation was extensively studied by Gidas,Spruck,Caffarelli,Obata,Wenxiong Chen, Congming Li, etc.