Questions: Levi-Civita Connection

Question 1

The Christoffel symbols of the Levi-Civita connection can be computed from the metric alone using the formula Γᵏᵢⱼ = ½gᵏˡ(∂ᵢgⱼₗ + ∂ⱼgᵢₗ - ∂ₗgᵢⱼ). What two properties force this specific formula?

• A. Symmetry in i,j (torsion-free) and positive-definiteness of g
• B. Torsion-free (Γᵏᵢⱼ = Γᵏⱼᵢ) and metric compatibility (∂ₖgᵢⱼ = Γˡₖᵢgₗⱼ + Γˡₖⱼgᵢₗ)
• C. Antisymmetry in i,j and the Bianchi identity
• D. Metric compatibility and the Jacobi identity

Answer: (B)

The two defining properties are torsion-free (which makes Christoffel symbols symmetric: Γᵏᵢⱼ = Γᵏⱼᵢ) and metric compatibility (which says ∇g = 0, expressed in coordinates as ∂ₖgᵢⱼ = Γˡₖᵢgₗⱼ + Γˡₖⱼgᵢₗ). Writing the metric-compatibility equation three times with cyclically permuted indices and using symmetry of the Christoffel symbols, you can solve uniquely for Γ — this derivation is called the Koszul trick.

Question 2

Metric compatibility (∇g = 0) has the geometric meaning that parallel transport preserves lengths and angles.

• T. True
• F. False

Answer: True

If V and W are parallel along a curve γ (∇_{γ'} V = 0 and ∇_{γ'} W = 0), then d/dt g(V,W) = g(∇_{γ'} V, W) + g(V, ∇_{γ'} W) = 0. So the inner product is constant along the curve. Since length |V| = √g(V,V) and angle cos θ = g(V,W)/(|V||W|), both are preserved. This is physically natural: moving a ruler along a path should not change the ruler's length. Non-metric connections exist but lack this property.

Question 3

Why is the torsion-free condition (rather than allowing torsion) the natural choice for Riemannian geometry?

Model answer: The torsion-free condition means ∇_X Y - ∇_Y X = [X,Y], so the antisymmetric part of the connection carries no independent information beyond the Lie bracket. This ensures that geodesics depend only on the metric (not on extra torsion data), that the Christoffel symbols are symmetric, and that the connection is fully determined by the metric. Allowing torsion would introduce additional geometric degrees of freedom that the metric alone does not control. The uniqueness of the Levi-Civita connection — that the metric determines the connection — is only possible with the torsion-free condition.

In physics, torsion appears in Einstein-Cartan theory (where spinning matter sources torsion) and in string theory. But in pure Riemannian geometry, the torsion-free condition is universally adopted because it gives the most economical theory: one piece of input data (the metric) determines everything. If you allowed torsion, you would need to specify the metric AND the torsion tensor independently.

Question 4

In normal coordinates centered at a point p, the Christoffel symbols of the Levi-Civita connection vanish at p: Γᵏᵢⱼ(p) = 0.

• T. True
• F. False

Answer: True

Normal coordinates (or geodesic coordinates) at p are constructed using the exponential map: geodesics through p become straight lines in these coordinates. The Christoffel symbols vanish at p (the center) because the geodesic equation d²xᵏ/dt² + Γᵏᵢⱼ dxⁱ/dt dxʲ/dt = 0 must be satisfied by all straight lines through the origin, which forces Γᵏᵢⱼ(0) = 0. However, the derivatives of the Christoffel symbols at p are generally nonzero — they encode the curvature. Normal coordinates make the connection look flat to first order but reveal curvature at second order.