In Riemannian geometry, the Levi-Civita connection is a specific connection on the tangent bundle of a manifold. More specifically, it is the torsion-free metric connection, i.e., the torsion-free connection on the tangent bundle (an affine connection) preserving a given (pseudo-)Riemannian metric.
The fundamental theorem of Riemannian geometry states that there is a unique connection which 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 of this connection with respect to a system of local coordinates are called Christoffel symbols.
The Levi-Civita connection is named after Tullio Levi-Civita, although originally "discovered" by Elwin Bruno Christoffel. Levi-Civita, along with Gregorio Ricci-Curbastro, used Christoffel's symbols to define the notion of parallel transport and explore the relationship of parallel transport with the curvature, thus developing the modern notion of holonomy.
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
since the definition of the Christoffel symbols make sense in any Riemannian manifold. In 1869, Christoffel discovered that the components of the intrinsic derivative of a vector transform as the components of a contravariant vector. This discovery was the real beginning of tensor analysis. It was not until 1917 that Levi-Civita interpreted the intrinsic derivative in the case of an embedded surface as the tangential component of the usual derivative in the ambient affine space.
- (M, g) denotes a Riemannian or pseudo-Riemannian manifold.
- TM is the tangent bundle of M.
- g is the Riemannian or pseudo-Riemannian metric of M.
- X, Y, Z are smooth vector fields on M, i. e. smooth sections of TM.
- [X, Y] is the Lie bracket of X and Y. It is again a smooth vector field.
The metric g can take up to two vectors or vector fields X, Y as arguments. In the former case the output is a number, the (pseudo-)inner product of X and Y. 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 as differential operators on smooth functions. In a basis, the action reads
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 = [X,Y], where [X,Y] 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.
Let ∇ be the connection of the Riemannian metric. Choose local coordinates and let be the Christoffel symbols with respect to these coordinates. The torsion freeness condition 2 is then equivalent to the symmetry
The definition of the Levi-Civita connection derived above is equivalent to a definition of the Christoffel symbols in terms of the metric as
Derivative along curve
The Levi-Civita connection (like any affine connection) also defines a derivative along curves, sometimes denoted by D.
Given a smooth curve γ on (M,g) and a vector field V along γ its derivative is defined by
In particular, is a vector field along the curve γ itself. If 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.
In general, parallel transport along a curve with respect to a connection defines isomorphisms between the tangent spaces at the points of the curve. If the connection is a Levi-Civita connection, then these isomorphisms are orthogonal – that is, they preserve the inner products on the various tangent spaces.
Example: The unit sphere in R3
Let be the usual scalar product on R3. Let S2 be the unit sphere in R3. The tangent space to S2 at a point m is naturally identified with the vector sub-space of R3 consisting of all vectors orthogonal to m. It follows that a vector field Y on S2 can be seen as a map Y: S2 → R3, which satisfies
Denote by dmY(X) the covariant derivative of the map Y in the direction of the vector X. Then we have:
Lemma: The formula
defines an affine connection on S2 with vanishing torsion.
Proof: 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 does indeed define a vector field. That is, we need to prove that for all m in S2
Consider the map f that sends every m in S2 to <Y(m), m>, which is always 0. The map f is constant, hence its differential vanishes. In particular
In fact, this connection is the Levi-Civita connection for the metric on S2 inherited from R3. Indeed, one can check that this connection preserves the metric.
- See Levi-Civita (1917)
- See Christoffel (1869)
- See Spivak (1999) Volume II, page 238
Primary historical references
|CitationClass=book }} See Volume I pag. 158
- MathWorld: Levi-Civita Connection
- PlanetMath: Levi-Civita Connection
- Levi-Civita connection at the Manifold Atlas