Part 1
Chapter 3 Definition: Levi-Civita connection
Definition On a Riemannian manifold $(M, g)$, the Levi-Civita connection is the unique affine connection, denoted by $\nabla$, that satisfies two fundamental properties: it is torsion-free and it is compatible with the metric $g$.
Let $X$, $Y$, and $Z$ be smooth vector fields on $M$. The two defining properties are detailed below.
Torsion-Free (Symmetry)
A connection is torsion-free if the order of the lower arguments does not introduce a torsion term. This property relates the connection to the Lie bracket of vector fields.
\[\nabla_X Y - \nabla_Y X = [X, Y]\]Metric Compatibility
A connection is compatible with the metric $g$ if the metric tensor is covariantly constant ($\nabla g = 0$). This ensures that lengths and angles are preserved under parallel transport. The condition is expressed via the product rule:
\[X(g(Y, Z)) = g(\nabla_X Y, Z) + g(Y, \nabla_X Z)\]The Koszul Formula
The existence and uniqueness of the Levi-Civita connection are guaranteed by the Fundamental Theorem of Riemannian Geometry. This theorem also provides an explicit formula for the connection, known as the Koszul formula, which defines $\nabla_X Y$ by its inner product with an arbitrary vector field $Z$.
\[2g(\nabla_X Y, Z) = X(g(Y, Z)) + Y(g(Z, X)) - Z(g(X, Y)) - g(X, [Y, Z]) - g(Y, [X, Z]) + g(Z, [X, Y])\]This formula is obtained by summing the folloing three formulas … and using the torsion-free condition.