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.
Read more about this topic: Levi-Civita Connection
Famous quotes containing the words formal and/or definition:
“The bed is now as public as the dinner table and governed by the same rules of formal confrontation.”
—Angela Carter (19401992)
“It is very hard to give a just definition of love. The most we can say of it is this: that in the soul, it is a desire to rule; in the spirit, it is a sympathy; and in the body, it is but a hidden and subtle desire to possessafter many mysterieswhat one loves.”
—François, Duc De La Rochefoucauld (16131680)