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.
Remark
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. 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. In 1918, independently of Levi-Civita, Jan Arnoldus Schouten obtained analogous results. In the same year, Hermann Weyl generalized Levi-Civita's results.
The metric can take up to two vectors or vector fields as arguments. In the former case the output is a number, the inner product of and. In the latter case, the inner product of is taken at all points on the manifold so that defines a smooth function on. Vector fields act as differential operators on smooth functions. In a basis, the action reads where Einstein's summation convention is used.
it is torsion-free, i.e., for any vector fields and we have, where is the Lie bracket of the vector fields and.
Condition 1 above is sometimes referred to as compatibility with the metric, and condition 2 is sometimes called symmetry, cf. Do Carmo's text. If a Levi-Civita connection exists, it is uniquely determined. Using conditions 1 and the symmetry of the metric tensor we find: By condition 2, the right hand side is equal to so we find the Koszul formula Since is arbitrary, this uniquely determines. 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.
Christoffel symbols
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 where as usual are the coefficients of the dual metric tensor, i.e. the entries of the inverse of the matrix.
Derivative along curve
The Levi-Civita connection also defines a derivative along curves, sometimes denoted by. Given a smooth curve on and a vector field along its derivative is defined by Formally, is the pullback connection on the pullback bundle. In particular, is a vector field along the curve 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.
Parallel transport
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. The images below show parallel transport of the Levi-Civita connection associated to two different Riemannian metrics on the plane, expressed in polar coordinates. The metric of left image corresponds to the standard Euclidean metric, while the metric on the right has standard form in polar coordinates, and thus preserves the vector tangent to the circle. This second metric has a singularity at the origin, as can be seen by expressing it in Cartesian coordinates:
Let be the usual scalar product on. Let be the unit sphere in. The tangent space to at a point is naturally identified with the vector subspace of consisting of all vectors orthogonal to. It follows that a vector field on can be seen as a map, which satisfies Denote as the covariant derivative of the map in the direction of the vector. Then we have: In fact, this connection is the Levi-Civita connection for the metric on inherited from. Indeed, one can check that this connection preserves the metric.
