Levi-Civita connection

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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.

History[edit]

The Levi-Civita connection is named after Tullio Levi-Civita, although originally "discovered" by Elwin Bruno Christoffel. 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.[3]

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

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 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.

Notation[edit]

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

Xf = X^i\frac{\partial}{\partial x^i}f = X^i\partial_i f,

where Einstein's summation convention is used.

Formal definition[edit]

An affine connection is called a Levi-Civita connection if

  1. it preserves the metric, i.e., g = 0.
  2. 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:

 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 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.

Christoffel symbols[edit]

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 Levi-Civita connection derived above is equivalent to a definition of the Christoffel symbols in terms of the metric as

 \Gamma^l{}_{jk} = \tfrac{1}{2}\sum_r 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[edit]

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

D_tV=\nabla_{\dot\gamma(t)}V.

(Formally D is the pullback connection 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. 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[edit]

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[edit]

Let \langle \cdot,\cdot \rangle 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: S2R3, which satisfies

\langle Y(m), m\rangle = 0, \qquad \forall m\in \mathbf{S}^2.

Denote by dY the differential of such a map. 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 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

\langle\left(\nabla_X Y\right)(m),m\rangle = 0\qquad (1).

Consider the map

\begin{cases}
f: \mathbf{S}^2 \to  \mathbf{R} \\
     m \mapsto    \langle Y(m), m\rangle.
\end{cases}

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 Levi-Civita connection for the metric on S2 inherited from R3. Indeed, one can check that this connection preserves the metric.

See also[edit]

Notes[edit]

  1. ^ See Levi-Civita (1917)
  2. ^ See Christoffel (1869)
  3. ^ See Spivak (1999) Volume II, page 238

References[edit]

Primary historical references[edit]

  • Christoffel, Elwin Bruno (1869), "Über die Transformation der homogenen Differentialausdrücke zweiten Grades", J. für die Reine und Angew. Math. 70: 46–70 
  • Levi-Civita, 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[edit]

  • Boothby, William M. (1986). An introduction to differentiable manifolds and Riemannian geometry. Academic Press. ISBN 0-12-116052-1. 
  • Kobayashi, S., and Nomizu, K. (1963). Foundations of differential geometry. John Wiley & Sons. ISBN 0-470-49647-9.  See Volume I pag. 158
  • Spivak, Michael (1999). A Comprehensive introduction to differential geometry (Volume II). Publish or Perish Press. ISBN 0-914098-71-3. 

External links[edit]