Recalling (3.2.8) which defines the covariant derivative of a generic tensor field according to the tensor calculus of Ricci and Levi Civita, we discover the interpretation of (3.7.13). It just states that the covariant derivative of the metric tensor should be zero:

Hence the Levi Civita connection is that affine torsionless connection with respect to which the metric tensor is covariantly constant.

3.8 Geodesics

Once we have an affine connection we can answer the question that was at the root of the whole development of differential geometry, namely which lines are straight in a curved space? To use a car driving analogy, the straight lines are obviously those that imply no turning of the steering wheel. In geometric terms steering the wheel corresponds to changing one’s direction while proceeding along the curve and such a change is precisely measured by the parallel transport of the tangent vector to the curve along itself.

Let C (λ) be a curve [0, 1] → M in a manifold of dimension m and let t be its tangent vector. In each coordinate patch the considered curve is represented as xμ = xμ(λ) and the tangent vector has the following components:

According to the above discussion we can rightly say that a curve is straight if we have:

The above condition immediately translates into a set of m differential equations of the second order for the functions xμ(λ). Observing that:

we conclude that (3.8.1) just coincides with:

which is named the geodesic equation. The solutions of these differential equations are the straight lines of the considered manifold and are named the geodesics. A solution is completely determined by the initial conditions which, given the order of the differential system, are 2m. These correspond to giving the values xμ(0), namely the initial point of the curve, and the values of dxdλρ (0), namely the initial tangent