the Codazzi pair $(III,II)$, respectively.

In [@Gr] Grove proved that two ovaloids in $\r^3$ with the same second fundamental form and extrinsic curvature are congruent. We give a different proof, generalizing that result to Codazzi pairs. The original proof by Grove involves techniques from integration theory on surfaces.

\[gengr\] Let ${\Sigma}$ be a topological sphere and $(I_i,II)$, $i=1,2$, two Codazzi pairs on $\Sigma$ with the same extrinsic curvature $K>0$. Then $I_1=I_2$.

Since $K>0$, we can assume (changing $II$ by $-II$ if necessary) that $II$ is a Riemannian metric on ${\Sigma}$. Taking a local isothermal parameter $z$ for $II$, we can write $$\begin{aligned}
&I_i=P_i\,dz^2+2\lambda_i\,|dz|^2+\overline{P_i}\,d{\bar{z}}^2&\\
&II=2\rho\,|dz|^2,\end{aligned}$$ with $\rho>0$.

Hence, the mean and extrinsic curvatures of the pair $(I_i,II)$, $i=1,2$ can be written as $$H_i=\frac{\lambda_i\,\rho}{\lambda_i^2-|P_i|^2} \label{HK},\qquad
K=\frac{\rho^2}{\lambda_i^2-|P_i|^2},$$ and the shape operator is given by $$S_i\frac{\partial\ }{\partial z}=\frac{K}{\rho}\,\left(\lambda_i\,\frac{\partial\
}{\partial z}-P_i\,\frac{\partial\ }{\partial {\bar{z}}}\right).$$ Let us denote by $\nabla^i$ the Levi-Civita connection associated to the metric $I_i$ and put $$\nabla^i_{\frac{\partial\ }{\partial z}}\frac{\partial\ }{\partial
z}=\Gamma_{11}^{1,i}\,\frac{\partial\ }{\partial z}+
\Gamma_{11}^{2,i}\,\frac{\partial\ }{\partial {\bar{z}}},\qquad \nabla^i_{\frac{\partial\
}{\partial z}}\frac{\partial\ }{\partial {\bar{z}}}=\Gamma_{12}^{1,i}\,\frac{\partial\
}{\partial z}+ \overline{\Gamma_{12}^{1,i}}\,\frac{\partial\ }{\partial {\bar{z}}}.$$ Since $(I_i,II)$ is a Codazzi pair we have $$\begin{aligned}
0&=&I_i\left(\nabla^i_{\frac{\partial\ }{\partial
z}}S_i{\frac{\partial\ }{\partial {\bar{z}}}},{\frac{\partial\ }{\partial
z}}\right)-I_i\left(\nabla^i_{\frac{\partial\ }{\partial
{\bar{z}}}}S_i{\frac{\partial\ }{\partial z}},{\frac{\partial\ }{\partial z}}\right)\\
&=&\rho_z-I_i\left(S_i{\frac{\partial\ }{\partial
{\bar{z}}}},\nabla^i_{\frac{\partial\ }{\partial z}}\frac{\partial\
