In Riemannian or pseudo-Riemannian geometry (in particular the Lorentzian geometry of general relativity), the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold that preserves the (pseudo-)Riemannian metric and is torsion-free.
The fundamental theorem of Riemannian geometry states that there is a unique connection that satisfies these properties.
In the theory of Riemannian and pseudo-Riemannian manifolds the term covariant derivative is often used for the Levi-Civita connection.
The components (structure coefficients) of this connection with respect to a system of local coordinates are called Christoffel symbols.
Levi-Civita,[1] along with Gregorio Ricci-Curbastro, used Christoffel's symbols[2] to define the notion of parallel transport and explore the relationship of parallel transport with the curvature, thus developing the modern notion of holonomy.
In 1906, L. E. J. Brouwer was the first mathematician to consider the parallel transport of a vector for the case of a space of constant curvature.
[4][5] In 1917, Levi-Civita pointed out its importance for the case of a hypersurface immersed in a Euclidean space, i.e., for the case of a Riemannian manifold embedded in a "larger" ambient space.
The Levi-Civita notions of intrinsic derivative and parallel displacement of a vector along a curve make sense on an abstract Riemannian manifold, even though the original motivation relied on a specific embedding
In 1918, independently of Levi-Civita, Jan Arnoldus Schouten obtained analogous results.
[6] In the same year, Hermann Weyl generalized Levi-Civita's results.
In the latter case, the inner product of Xp, Yp is taken at all points p on the manifold so that g(X, Y) defines a smooth function on M. Vector fields act (by definition) as differential operators on smooth functions.
Proof:[10][11] To prove uniqueness, unravel the definition of the action of a connection on tensors to find Hence one can write the condition that
, By torsion-freeness, the right hand side is therefore equal to Thus, the Koszul formula holds.
, the right hand side of the Koszul expression is linear over smooth functions in the vector field
, the right hand side uniquely defines some new vector field, which is suggestively denoted
With minor variation, the same proof shows that there is a unique connection that is compatible with the metric and has prescribed torsion.
(or computes directly), the Koszul expression of the Levi-Civita connection derived above is equivalent to a definition of the Christoffel symbols in terms of the metric as where as usual
are the coefficients of the dual metric tensor, i.e. the entries of the inverse of the matrix
vanishes, the curve is called a geodesic of the covariant derivative.
Formally, the condition can be restated as the vanishing of the pullback connection applied to
: If the covariant derivative is the Levi-Civita connection of a certain metric, then the geodesics for the connection are precisely those geodesics of the metric that are parametrised proportionally to their arc length.
The images below show parallel transport induced by the Levi-Civita connection associated to two different Riemannian metrics on the punctured plane
The first metric extends to the entire plane, but the second metric has a singularity at the origin: Warning: This is parallel transport on the punctured plane along the unit circle, not parallel transport on the unit circle.
Indeed, in the first image, the vectors fall outside of the tangent space to the unit circle.
Denote as dmY the differential of the map Y at the point m. Then we have: Lemma — The formula
It is straightforward to prove that ∇ satisfies the Leibniz identity and is C∞(S2) linear in the first variable.
It is also a straightforward computation to show that this connection is torsion free.
So all that needs to be proved here is that the formula above produces a vector field tangent to S2.
As an application, consider again the unit sphere, but this time under stereographic projection, so that the metric (in complex Fubini–Study coordinates
These relations, together with their complex conjugates, define the Christoffel symbols for the two-sphere.