MATH590

Manifolds: metric tensor, geodesic equations

The length of a curve on an $n$ -dimensional manifold is

d s^{2} = g_{i j} d ξ^{i} d ξ^{j}

where $𝐠 = {(g_{i j})}_{1 ⩽ i, j ⩽ n}$ is the metric tensor. Whereas in flat space the scalar product of vectors $𝐮 = (u_{i}), 𝐯 = (v_{i})$ , is $𝐮 \cdot 𝐯 = u_{i} v_{i}$ , on a manifold the scalar product becomes $𝐮 \cdot 𝐯 = g_{i j} u^{i} v^{i}$ .

The connection coefficients can also be written in terms of the metric tensor

Γ_{i j}^{k} = \frac{1}{2} g^{k l} (g_{l i, j} + g_{l j, i} - g_{i j, l}) .

The geodesic equation on a manifold is

{\ddot{ξ}}^{k} + Γ_{i j}^{k} {\dot{ξ}}^{i} {\dot{ξ}}^{j} = 0