測地線
整理すると、
g a i ( x ̈ i − x ̇ i s ̈ s ̇ ) + ∂ g a i ∂ x j x ̇ i x ̇ j − 1 2 ∂ g i j ∂ x a x ̇ i x ̇ j = g a i ( x ̈ i − x ̇ i s ̈ s ̇ ) + 1 2 ( ∂ g a i ∂ x j + ∂ g a j ∂ x i − ∂ g i j ∂ x a ) x ̇ i x ̇ j = 0 {\displaystyle g_{ai}\left({\ddot {x}}^{i}-{\dot {x}}^{i}{\frac {\ddot {s}}{\dot {s}}}\right)+{\frac {\partial g_{ai}}{\partial x^{j}}}{\dot {x}}^{i}{\dot {x}}^{j}-{\frac {1}{2}}{\frac {\partial g_{ij}}{\partial x^{a}}}{\dot {x}}^{i}{\dot {x}}^{j}=g_{ai}\left({\ddot {x}}^{i}-{\dot {x}}^{i}{\frac {\ddot {s}}{\dot {s}}}\right)+{\frac {1}{2}}\left({\frac {\partial g_{ai}}{\partial x^{j}}}+{\frac {\partial g_{aj}}{\partial x^{i}}}-{\frac {\partial g_{ij}}{\partial x^{a}}}\right){\dot {x}}^{i}{\dot {x}}^{j}=0}