Formal Definition
Let (M,g) be a Riemannian manifold (or pseudo-Riemannian manifold). Then an affine connection ∇ is called a Levi-Civita connection if
- it preserves the metric, i.e., ∇g = 0.
- it is torsion-free, i.e., for any vector fields X and Y we have ∇XY − ∇YX =, where is the Lie bracket of the vector fields X and Y.
Condition 1 above is sometimes referred to as compatibility with the metric, and condition 2 is sometimes called symmetry, cf. DoCarmo's text.
Assuming a Levi-Civita connection exists it is uniquely determined. Using conditions 1 and the symmetry of the metric tensor g we find:
By condition 2 the right hand side is equal to
so we find
Since Z is arbitrary, this uniquely determines ∇XY. Conversely, using the last line as a definition one shows that the expression so defined is a connection compatible with the metric, i.e. is a Levi-Civita connection.
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