In Riemannian geometry, the LeviCivita connection is a specific connection on the tangent bundle of a manifold. More specifically, it is the torsionfree metric connection, i.e., the torsionfree 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 pseudoRiemannian manifolds the term covariant derivative is often used for the LeviCivita connection. The components of this connection with respect to a system of local coordinates are called Christoffel symbols.
Contents

History 1

Notation 2

Formal definition 3

Christoffel symbols 4

Derivative along curve 5

Parallel transport 6

Example: The unit sphere in R3 7

See also 8

Notes 9

References 10

Primary historical references 10.1

Secondary references 10.2

External links 11
History
The LeviCivita connection is named after Tullio LeviCivita, although originally "discovered" by Elwin Bruno Christoffel. LeviCivita,^{[1]} along with Gregorio RicciCurbastro, 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.^{[3]}
The LeviCivita 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

M^n \subset \mathbf{R}^{\frac{n(n+1)}{2}},
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 LeviCivita interpreted the intrinsic derivative in the case of an embedded surface as the tangential component of the usual derivative in the ambient affine space.
Notation
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 X_{p}, Y_{p} 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

Xf = X^i\frac{\partial}{\partial x^i}f = X^i\partial_i f,
where Einstein's summation convention is used.
Formal definition
An affine connection ∇ is called a LeviCivita connection if

it preserves the metric, i.e., ∇g = 0.

it is torsionfree, i.e., for any vector fields X and Y we have ∇_{X}Y − ∇_{Y}X = [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 LeviCivita connection exists it is uniquely determined. Using conditions 1 and the symmetry of the metric tensor g we find:

X (g(Y,Z)) + Y (g(Z,X))  Z (g(Y,X)) = g(\nabla_X Y + \nabla_Y X, Z) + g(\nabla_X Z  \nabla_Z X, Y) + g(\nabla_Y Z  \nabla_Z Y, X).
By condition 2 the right hand side is equal to

2g(\nabla_X Y, Z)  g([X,Y], Z) + g([X,Z],Y) + g([Y,Z],X)
so we find

g(\nabla_X Y, Z) = \frac{1}{2} \{ X (g(Y,Z)) + Y (g(Z,X))  Z (g(X,Y)) + g([X,Y],Z)  g([Y,Z], X)  g([X,Z], Y) \}.
Since Z is arbitrary, this uniquely determines ∇_{X}Y. 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 LeviCivita connection.
Christoffel symbols
Let ∇ be the connection of the Riemannian metric. Choose local coordinates x^1 \ldots x^n and let \Gamma^l{}_{jk} be the Christoffel symbols with respect to these coordinates. The torsion freeness condition 2 is then equivalent to the symmetry

\Gamma^l{}_{jk} = \Gamma^l{}_{kj}.
The definition of the LeviCivita connection derived above is equivalent to a definition of the Christoffel symbols in terms of the metric as

\Gamma^l{}_{jk} = \tfrac{1}{2} g^{lr} \left \{\partial _k g_{rj} + \partial _j g_{rk}  \partial _r g_{jk} \right \}
where as usual g^{ij} are the coefficients of the dual metric tensor, i.e. the entries of the inverse of the matrix (g_{kl}).
Derivative along curve
The LeviCivita 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

D_tV=\nabla_{\dot\gamma(t)}V.
Formally, D is the pullback connection \gamma^*\nabla on the pullback bundle γ*TM.
In particular, \dot{\gamma}(t) is a vector field along the curve γ itself. If \nabla_{\dot\gamma(t)}\dot\gamma(t) 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 \dot{\gamma} :

(\gamma^*\nabla) \dot{\gamma}\equiv 0.
If the covariant derivative is the LeviCivita 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 LeviCivita connection, then these isomorphisms are orthogonal – that is, they preserve the inner products on the various tangent spaces.
Parallel Transports Under LeviCivita Connections
This transport is given by the metric ds^{2} = dr^{2} + r^{2}dθ^{2}.
This transport is given by the metric ds^{2} = dr^{2} + dθ^{2}.
Example: The unit sphere in R^{3}
Let \langle \cdot,\cdot \rangle be the usual scalar product on R^{3}. Let S^{2} be the unit sphere in R^{3}. The tangent space to S^{2} at a point m is naturally identified with the vector subspace of R^{3} consisting of all vectors orthogonal to m. It follows that a vector field Y on S^{2} can be seen as a map Y: S^{2} → R^{3}, which satisfies

\langle Y(m), m\rangle = 0, \qquad \forall m\in \mathbf{S}^2.
Denote by d_{m}Y(X) the covariant derivative of the map Y in the direction of the vector X. Then we have:
Lemma: The formula

\left(\nabla_X Y\right)(m) = d_mY(X) + \langle X(m),Y(m)\rangle m
defines an affine connection on S^{2} with vanishing torsion.
Proof: It is straightforward to prove that ∇ satisfies the Leibniz identity and is C^{∞}(S^{2}) 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 S^{2}

\langle\left(\nabla_X Y\right)(m),m\rangle = 0\qquad (1).
Consider the map f that sends every m in S^{2} to <Y(m), m>, which is always 0. The map f is constant, hence its differential vanishes. In particular

d_mf(X) = \langle d_m Y(X),m\rangle + \langle Y(m), X(m)\rangle = 0.
The equation (1) above follows.\Box
In fact, this connection is the LeviCivita connection for the metric on S^{2} inherited from R^{3}. Indeed, one can check that this connection preserves the metric.
See also
Notes

^ See LeviCivita (1917)

^ See Christoffel (1869)

^ See Spivak (1999) Volume II, page 238
References
Primary historical references

Christoffel, Elwin Bruno (1869), "Über die Transformation der homogenen Differentialausdrücke zweiten Grades", J. für die Reine und Angew. Math. 70: 46–70

LeviCivita, Tullio (1917), "Nozione di parallelismo in una varietà qualunque e consequente specificazione geometrica della curvatura Riemanniana", Rend. Circ. Mat. Palermo 42: 73–205,
Secondary references

Boothby, William M. (1986). An introduction to differentiable manifolds and Riemannian geometry. Academic Press.

Kobayashi, S., and Nomizu, K. (1963). Foundations of differential geometry. John Wiley & Sons. See Volume I pag. 158

Spivak, Michael (1999). A Comprehensive introduction to differential geometry (Volume II). Publish or Perish Press.
External links

Hazewinkel, Michiel, ed. (2001), "LeviCivita connection",

MathWorld: LeviCivita Connection

PlanetMath: LeviCivita Connection

LeviCivita connection at the Manifold Atlas
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