Generalized Riemannian Geometry

Geometry in a generalized 3D Riemannian space is described by a symmetric metric tensor, g_ij=g_ji, that relates the geometry in a non-orthogonal coordinate system, $\{x_1,x_2,x_3\}$, to an underlying Cartesian mesh, $\{\xi_1,\xi_2,\xi_3\}$ Guggenheimer (1977). In matrix form, the metric tensor is written,  

$$\left[g_{ij}\right] = \left[\begin{array} {ccc} g_{11} & g... ...g_{22} & g_{23} \ g_{13} & g_{23} & g_{33} \end{array}\right],$$ (1)

where g₁₁, g₁₂, g₂₂, g₁₃, g₂₃ and $\AA$ are functions linking the two coordinate systems through,

$$\begin{eqnarray} g_{11}=\frac{\partial x_k}{\partial \xi_1}\frac{\partial x_k}{\... ...\partial x_k}{\partial \xi_3}\frac{\partial x_k}{\partial \xi_3}. \end{eqnarray}$$ (2)

(Summation notation - g_ii = g₁₁+g₂₂+g₃₃ - is used in equations throughout this paper.) The associated (or inverse) metric tensor, g^ij, is defined by $g_{ij}=\,\vert\mathbf{g}\vert\,g^{ij}$, where $\vert\mathbf{g}\vert$ is metric tensor matrix determinant. The associated metric tensor is given by,  

$$\left[g^{ij}\right] =\frac{1}{\left\vert\mathbf{g}\right\ve... ..._{13}-g_{11}g_{23}& g_{11}g_{22}-g_{12}^2 \end{array}\right],$$ (3)

and has the following metric determinant,  

$$\vert\mathbf{g}\vert = \AA\,(g_{11}g_{22}-g_{12}^2)\, \left... ...13}^2-2g_{12}g_{23}g_{13}}{\AA(g_{11}g_{22}-g_{12}^2)} \right].$$ (4)

Weighted metric tensor, $m^{ij}=\sqrt{\left\vert \mathbf{g} \right\vert}\, g^{ij}$, is a useful definition for the following development.