Curvature of Coupled Planes
Consider the product of two planes \(\mathbb R^2\times\mathbb R^2\), with the following Riemannian metric \[ ds^2=dx^2+dy^2+e^{2f(x,y)}(du^2+dv^2), \] where \(f(x,y)\) is a smooth function. We compute the sectional/Ricci/scalar curvature of the 'coupled' planes using orthonormal frames. Basics about moving frames are referred to Loring W. Tu.
Consider the following orthonormal bases \[ e_1=\partial_x,e_2=\partial_y,e_3=e^{-f}\partial_u,e_4=e^{-f}\partial_v. \] The dual 1-forms are \[ \theta^1=dx,\theta^2=dy,\theta^3=e^{f}du,\theta^4=e^fdv. \] There exists a unique skew-symmetric matrix of 1-forms \([\omega^i_j]\) such that \[ d\theta^i+\omega^i_j\wedge\theta^j=0. \] The \(\omega^i_j\)'s are called the connection 1-forms, and the above equation is called the first structure equation. Suppose \(\omega^i_j=a^i_jdx+b^i_jdy+c^i_jdu+d^i_jdv\). Substituting into the first structure equation, we solve all the connection 1-forms \[ \begin{aligned} &\omega^1_2=\omega^3_4=0,\\ &\omega^1_3=-e^ff_xdu,\omega^1_4=-e^ff_xdv,\\ &\omega^2_3=-e^ff_ydu,\omega^2_4=-e^ff_ydv. \end{aligned} \] The curvature 2-forms are defined by the second structure equation \[ \Omega^i_j=d\omega^i_j+\omega^i_k\wedge\omega^k_j. \] Therefore, we write out all the curvature forms \[ \begin{aligned} &\Omega^1_2=0,\\ &\Omega^1_3=-e^f(f_{xx}+f_x^2)dx\wedge du-e^f(f_{xy}+f_xf_y)dy\wedge du,\\ &\Omega^1_4=-e^f(f_{xx}+f_x^2)dx\wedge dv-e^f(f_{xy}+f_xf_y)dy\wedge dv,\\ &\Omega^2_3=-e^f(f_{xy}+f_xf_y)dx\wedge du-e^f(f_{yy}+f_y^2)dy\wedge du,\\ &\Omega^2_4=-e^f(f_{xy}+f_xf_y)dx\wedge dv-e^f(f_{yy}+f_y^2)dy\wedge dv,\\ &\Omega^3_4=-e^{2f}(f_x^2+f_y^2)du\wedge dv. \end{aligned} \] By the definition of sectional curvature, for any plane spanned by \(e_i,e_j\), \[ K_{ij}=\langle R(e_i,e_j)e_j,e_i\rangle=\Omega_j^i(e_i,e_j). \] Hence, we can compute all the sectional curvature using curvature forms, \[ \begin{aligned} &K_{12}=0,\\ &K_{13}=K_{14}=-(f_{xx}+f_x^2),\\ &K_{23}=K_{24}=-(f_{yy}+f_y^2),\\ &K_{34}=-(f_x^2+f_y^2). \end{aligned} \] Ricci curvature is the sum of sectional curvatures, thus, \[ \begin{aligned} &Ric_{11}=K_{12}+K_{13}+K_{14}=-2(f_{xx}+f_x^2),\\ &Ric_{22}=K_{12}+K_{23}+K_{24}=-2(f_{yy}+f_y^2),\\ &Ric_{33}=K_{13}+K_{23}+K_{34}=-(f_{xx}+f_{yy}+2f_x^2+2f_y^2),\\ &Ric_{44}=K_{14}+K_{24}+K_{34}=-(f_{xx}+f_{yy}+2f_x^2+2f_y^2),\\ \end{aligned} \] Scalar curvature is the sum of Ricci curvature, thus, \[ S=Ric_{11}+Ric_{22}+Ric_{33}+Ric_{44}. \]
Curvature of 4 Dimensional Warped Product Spaces
More generally, let \(M\) and \(N\) be two Riemannian manifolds with metric \(g_M\) and \(g_N\), respectively. Consider the product space \(M\times N\) with the following metric \[ g=g_M+e^{2f}g_N, \] where \(f\) is a smooth function on \(M\). This is called the warped products of \(M\) and \(N\), and often denoted by \(M\times_{e^f}N\) (see John Lee, Example 2.24). Let \(\theta^1,\theta^2\) be the orthonormal coframe on \(M\) and \(\omega^1_2\) be the corresponding connection form. Similarly, let \(\theta^3,\theta^4,\omega^3_4\) be the orthonormal coframe and connection form on \(N\). For the product space \(M\times N\), we have the orthonormal coframe \[ \bar{\theta^1}=\theta^1,\bar{\theta^2}=\theta^2,\bar{\theta^3}=e^f\theta^3,\bar{\theta^4}=e^f\theta^4. \] Assume that the connection 1-form for \(M\times N\) is \[ \bar{\omega^i_j}=a^i_j\bar{\theta_1}+b_j^i\bar{\theta_2}+c^i_j\bar{\theta_3}+d^i_j\bar{\theta_4}. \] Substituting the connection 1-forms into the first structure equation, we have \[ \begin{aligned} &\bar{\omega^1_2}=\omega^1_2,\\ &\bar{\omega^1_3}=-e^{-f}\frac{de^f}{\theta^1}\bar{\theta^3},\\ &\bar{\omega^1_4}=-e^{-f}\frac{de^f}{\theta^1}\bar{\theta^4},\\ &\bar{\omega^2_3}=-e^{-f}\frac{de^f}{\theta^2}\bar{\theta^3},\\ &\bar{\omega^2_4}=-e^{-f}\frac{de^f}{\theta^2}\bar{\theta^4},\\ &\bar{\omega^3_4}=\omega^3_4. \end{aligned} \] Since \(de^f\) is a 1-form on \(M\), it can be written as a linear combination of \(\theta^1\) and \(\theta^2\). Let \(de^f/\theta^1\) and \(de^f/\theta^2\) be the coefficients. Similarly, let \(\omega^1_2=a\theta^1+b\theta^2\), by structure equation, \(d\theta^1=-\omega^1_2\wedge\theta^2=-a\theta^1\wedge\theta^2\), \(d\theta^2=-\omega^2_1\wedge\theta^1=-b\theta^1\wedge\theta^2\). We can write \[ \begin{aligned} &\omega^1_2=-\frac{d\theta^1}{\theta^1\wedge\theta^2}\theta^1-\frac{d\theta^2}{\theta^1\wedge\theta^2}\theta^2,\\ &\omega^3_4=-\frac{d\theta^3}{\theta^3\wedge\theta^4}\theta^3-\frac{d\theta^4}{\theta^3\wedge\theta^4}\theta^4. \end{aligned} \] With these notations, we can list all the curvature 2-forms \[ \begin{aligned} &\bar{\Omega^1_2}=d\omega^1_2=\Omega^1_2,\\ &\bar{\Omega^1_3}=(\frac{de^f}{\theta^2}\frac{d\theta^1}{\theta^1\wedge\theta^2}-\frac{d^2e^f}{(\theta^1)^2})\theta^1\wedge\theta^3+(\frac{de^f}{\theta^2}\frac{d\theta^2}{\theta^1\wedge\theta^2}-\frac{d^2e^f}{\theta^1\theta^2})\theta^2\wedge\theta^3,\\ &\bar{\Omega^1_4}=(\frac{de^f}{\theta^2}\frac{d\theta^1}{\theta^1\wedge\theta^2}-\frac{d^2e^f}{(\theta^1)^2})\theta^1\wedge\theta^4+(\frac{de^f}{\theta^2}\frac{d\theta^2}{\theta^1\wedge\theta^2}-\frac{d^2e^f}{\theta^1\theta^2})\theta^2\wedge\theta^4,\\ &\bar{\Omega^2_3}=(-\frac{de^f}{\theta^1}\frac{d\theta^1}{\theta^1\wedge\theta^2}-\frac{d^2e^f}{\theta^2\theta^1})\theta^1\wedge\theta^3+(-\frac{de^f}{\theta^1}\frac{d\theta^2}{\theta^1\wedge\theta^2}-\frac{d^2e^f}{(\theta^2)^2})\theta^2\wedge\theta^3,\\ &\bar{\Omega^2_4}=(-\frac{de^f}{\theta^1}\frac{d\theta^1}{\theta^1\wedge\theta^2}-\frac{d^2e^f}{\theta^2\theta^1})\theta^1\wedge\theta^4+(-\frac{de^f}{\theta^1}\frac{d\theta^2}{\theta^1\wedge\theta^2}-\frac{d^2e^f}{(\theta^2)^2})\theta^2\wedge\theta^4,\\ &\bar{\Omega^3_4}=\Omega^3_4-((\frac{de^f}{\theta^1})^2+(\frac{de^f}{\theta^2})^2)\theta^3\wedge\theta^4. \end{aligned} \] Hence, the sectional curvatures are \[ \begin{aligned} &\bar{K}_{12}=K_{12},\\ &\bar{K}_{13}=\bar{K}_{14}=e^{-f}(\frac{de^f}{\theta^2}\frac{d\theta^1}{\theta^1\wedge\theta^2}-\frac{d^2e^f}{(\theta^1)^2}),\\ &\bar{K}_{23}=\bar{K}_{24}=e^{-f}(-\frac{de^f}{\theta^1}\frac{d\theta^2}{\theta^1\wedge\theta^2}-\frac{d^2e^f}{(\theta^2)^2}),\\ &\bar{K}_{34}=e^{-2f}(K_{34}-(\frac{de^f}{\theta^1})^2-(\frac{de^f}{\theta^2})^2). \end{aligned} \]
More Examples
Let \(\mathbb S^2\) be the unit sphere in \(\mathbb R^3\). Using stereographic projection, the round metric in local coordinate \((x,y)\) is \[ g_{\mathbb S^2}=\frac{4}{(1+x^2+y^2)^2}(dx^2+dy^2). \] Let \(\theta^1=2/(1+x^2+y^2)dx\) and \(\theta^2=2/(1+x^2+y^2)dy\) be the orthonormal coframe. We compute that the connection 1-form is \(\omega^1_2=-y\theta^1+x\theta^2\). Therefore, \[ \frac{d\theta^1}{\theta^1\wedge\theta^2}=y,\frac{d\theta^2}{\theta^1\wedge\theta^2}=-x. \] Denote \((1+x^2+y^2)/2\) by \(J\). Then, \[ \begin{aligned} &\frac{de^f}{\theta^1}=\frac{e^ff_xdx+e^ff_ydy}{\theta^1}=e^ff_xJ,\\ &\frac{d^2e^f}{(\theta^1)^2}=\frac{d(e^ff_xJ)}{\theta^1}=e^f(f_x)^2J^2+e^ff_{xx}J^2+xe^ff_xJ,\\ &\frac{d^2e^f}{\theta^1\theta^2}=\frac{d(e^ff_xJ)}{\theta^2}=e^ff_xf_yJ^2+e^ff_{xy}J^2+ye^ff_xJ,\\ \end{aligned} \] Similarly, we have \[ \begin{aligned} &\frac{de^f}{\theta^2}=\frac{e^ff_xdx+e^ff_ydy}{\theta^2}=e^ff_yJ,\\ &\frac{d^2e^f}{(\theta^2)^2}=\frac{d(e^ff_xJ)}{\theta^2}=e^f(f_y)^2J^2+e^ff_{yy}J^2+ye^ff_yJ,\\ &\frac{d^2e^f}{\theta^2\theta^1}=\frac{d(e^ff_xJ)}{\theta^1}=e^ff_xf_yJ^2+e^ff_{xy}J^2+xe^ff_yJ,\\ \end{aligned} \] Note that \(\frac{d^2e^f}{\theta^1\theta^2}\neq\frac{d^2e^f}{\theta^2\theta^1}\).
Let \(\mathbb H^2\) be the upper plane with hyperbolic metric
\[
g_{\mathbb H^2}=\frac{1}{v^2}(du^2+dv^2).
\] Under this metric \(\mathbb H^2\) will be a Riemannian manifold with constant curvature \(-1\). Consider the warped product \(\mathbb S^2\times_f \mathbb H^2\). The sectional curvatures are \[
\begin{aligned}&\bar{K}_{12}=1,\\&\bar{K}_{13}=\bar{K}_{14}=yf_yJ-(f_x)^2J^2-f_{xx}J^2-xf_xJ,\\&\bar{K}_{23}=\bar{K}_{24}=xf_xJ-(f_y)^2J^2-f_{yy}J^2-yf_yJ,\\&\bar{K}_{34}=e^{-2f}(-1-e^{2f}(f_x)^2J^2-e^{2f}(f_y)^2J^2).\end{aligned}
\]