Levi-Civita Connection

Differential Geometry · Axiom Academy

LESSON The Levi-Civita Connection The unique metric-compatible, torsion-free connection on Riemannian manifolds 1. The Problem: Differentiating Vector Fields on Manifolds On Euclidean space R n , we can differentiate a vector field V component-wise. But on a curved manifold M , vectors at different points live in different tangent spaces. We cannot simply subtract vectors at different points. We need a way to "connect" nearby tangent spaces so we can differentiate. This leads to the concept of a connection . 2. Definition of a Connection (Covariant Derivative) Linearity in X: ∇ fX+gY Z = f∇ X Z + g∇ Y Z for functions f, g Leibniz rule in Y: ∇ X (fY) = (Xf)Y + f∇ X Y Linearity in Y: ∇ X (Y + Z) = ∇ X Y + ∇ X Z These axioms ensure that ∇ X Y behaves like a directional derivative, but there are infinitely many connections on any manifold. Which one should we use? Torsion-free: ∇ X Y - ∇ Y X = [X,Y] Metric compatibility means the connection preserves the metric: parallel transport preserves inner products. Torsion-free means the connection is symmetric: the order of differentiation doesn't matter except for the Lie bracket. In local coordinates (x 1 , ..., x n ), the Levi-Civita connection is completely determined by the Christoffel symbols Γ k ij , defined by: where ∂ i = ∂/∂x i are the coordinate vector fields. 5. Metric Compatibility: ∇g = 0 Metric compatibility means that the covariant derivative of the metric tensor vanishes: In local coordinates, this expands to:

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