The map takes one covector argument and returns two vector slots, so

S

has two contravariant and one covariant index. Its tensor type is

(1 2),

or type

(2, 1)

, with components

S^{ab}_{c}

ii

Solution

The covariant derivative adds one covariant index, so

\nabla S

has type

(2, 2)

. The outer product of two tensors adds their index counts. Hence

S \otimes \nabla S has type (4, 3), written (3 4) .

Let

L (x, \overset{x}{˙}) = g_{μν} \overset{x}{˙}^{μ} \overset{x}{˙}^{ν} = \frac{d s}{d λ} .

\frac{d}{d λ} (\frac{g _{β ν} x ˙ ^{ν}}{L}) - \frac{1}{2 L} \partial_{β} g_{μν} \overset{x}{˙}^{μ} \overset{x}{˙}^{ν} = 0.

Since

\overset{x}{˙}^{μ} = L d x^{μ} / d s

and

d / d λ = L d / d s

, this becomes

\frac{d}{d s} (g_{β ν} \frac{d x ^{ν}}{d s}) - \frac{1}{2} \partial_{β} g_{μν} \frac{d x ^{μ}}{d s} \frac{d x ^{ν}}{d s} = 0.

Expanding the derivative, multiplying by the inverse metric, and using the Christoffel symbol

Γ_{μν}^{α} = \frac{1}{2} g^{α β} (\partial_{μ} g_{β ν} + \partial_{ν} g_{β μ} - \partial_{β} g_{μν})

\frac{d ^{2} x ^{α}}{d s ^{2}} + Γ_{μν}^{α} \frac{d x ^{μ}}{d s} \frac{d x ^{ν}}{d s} = 0 .

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