Levi-Civita connection

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 (i.e. affine connection) that preserves the (pseudo-)Riemannian metric and is torsion-free.

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 (structure coefficients) of this connection with respect to a system of local coordinates are called Christoffel symbols.

History

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]

In 1869, Christoffel discovered that the components of the intrinsic derivative of a vector field, upon changing the coordinate system, transform as the components of a contravariant vector. This discovery was the real beginning of tensor analysis.

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. ^[1] He interpreted the intrinsic derivative in the case of an embedded surface as the tangential component of the usual derivative in the ambient affine 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 ${\displaystyle M^{n}\subset \mathbf {R} ^{n(n+1)/2}.}$

In 1918, independently of Levi-Civita, Jan Arnoldus Schouten obtained analogous results. ^[6] In the same year, Hermann Weyl generalized Levi-Civita's results. ^{[7] [8]}

Notation

• (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 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 (by definition) as differential operators on smooth functions. In local coordinates ${\displaystyle (x_{1},\ldots ,x_{n})}$, the action reads

${\displaystyle X(f)=X^{i}{\frac {\partial }{\partial x^{i}}}f=X^{i}\partial _{i}f}$

where Einstein's summation convention is used.

Formal definition

An affine connection ∇ is called a 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. Do Carmo's text. ^[9]

Fundamental theorem of (pseudo) Riemannian Geometry

Main article: Fundamental theorem of Riemannian geometry

Theorem Every pseudo Riemannian manifold ${\displaystyle (M,g)}$ has a unique Levi Civita connection ${\displaystyle \nabla }$.

proof: If a Levi-Civita connection exists, it must be unique. To see this, unravel the definition of the action of a connection on tensors to find

${\displaystyle X{\bigl (}g(Y,Z){\bigr )}=(\nabla _{X}g)(Y,Z)+g(\nabla _{X}Y,Z)+g(Y,\nabla _{X}Z).}$

Hence we can write condition 1 as

${\displaystyle X{\bigl (}g(Y,Z){\bigr )}=g(\nabla _{X}Y,Z)+g(Y,\nabla _{X}Z).}$

By the symmetry of the metric tensor ${\displaystyle g}$ we then find:

${\displaystyle X{\bigl (}g(Y,Z){\bigr )}+Y{\bigl (}g(Z,X){\bigr )}-Z{\bigl (}g(Y,X){\bigr )}=g(\nabla _{X}Y+\nabla _{Y}X,Z)+g(\nabla _{X}Z-\nabla _{Z}X,Y)+g(\nabla _{Y}Z-\nabla _{Z}Y,X).}$

By condition 2, the right hand side is therefore equal to

${\displaystyle 2g(\nabla _{X}Y,Z)-g([X,Y],Z)+g([X,Z],Y)+g([Y,Z],X),}$

and we find the Koszul formula

${\displaystyle g(\nabla _{X}Y,Z)={\tfrac {1}{2}}{\Big \{}X{\bigl (}g(Y,Z){\bigr )}+Y{\bigl (}g(Z,X){\bigr )}-Z{\bigl (}g(X,Y){\bigr )}+g([X,Y],Z)-g([Y,Z],X)-g([X,Z],Y){\Big \}}.}$

Hence, if a Levi-Civita connection exists, it must be unique, because ${\displaystyle Z}$ is arbitrary, ${\displaystyle g}$ is non degenerate, and the right hand side does not depend on ${\displaystyle \nabla }$.

To prove existence, note that for given vector field ${\displaystyle X}$ and ${\displaystyle Y}$, the right hand side of the Koszul expression is function-linear in the vector field ${\displaystyle Z}$, not just real linear. Hence by the non degeneracy of ${\displaystyle g}$, the right hand side uniquely defines some new vector field which we suggestively denote ${\displaystyle \nabla _{X}Y}$ as in the left hand side. By substituting the Koszul formula, one now checks that for all vector fields ${\displaystyle X,Y,Z}$, and all functions ${\displaystyle f}$

${\displaystyle g(\nabla _{X}(Y_{1}+Y_{2}),Z)=g(\nabla _{X}Y_{1},Z)+g(\nabla _{X}Y_{2},Z)}$

${\displaystyle g(\nabla _{X}(fY),Z)=X(f)g(Y,Z)+fg(\nabla _{X}Y,Z)}$

${\displaystyle g(\nabla _{X}Y,Z)+g(\nabla _{X}Z,Y)=X{\bigl (}g(Y,Z){\bigr )}}$

${\displaystyle g(\nabla _{X}Y,Z)-g(\nabla _{Y}X,Z)=g([X,Y],Z).}$

Hence the Koszul expression does, in fact, define a connection, and this connection is compatible with the metric and is torsion free, i.e. is a (hence the) Levi-Civita connection.

Note that with minor variations the same proof shows that there is a unique connection that is compatible with the metric and has prescribed torsion.

Christoffel symbols

Let ${\displaystyle \nabla }$ be an affine connection on the tangent bundle. Choose local coordinates ${\displaystyle x^{1},\ldots ,x^{n}}$ with coordinate basis vector fields ${\displaystyle \partial _{1},\ldots ,\partial _{n}}$ and write ${\displaystyle \nabla _{j}}$ for ${\displaystyle \nabla _{\partial _{j}}}$. The Christoffel symbols ${\displaystyle \Gamma _{jk}^{l}}$ of ${\displaystyle \nabla }$ with respect to these coordinates are defined as

${\displaystyle \nabla _{j}\partial _{k}=\Gamma _{jk}^{l}\partial _{l}}$

The Christoffel symbols conversely define the connection ${\displaystyle \nabla }$ on the coordinate neighbourhood because

${\displaystyle {\begin{aligned}\nabla _{X}Y&=\nabla _{X^{j}\partial _{j}}(Y^{k}\partial _{k})\\&=X^{j}\nabla _{j}(Y^{k}\partial _{k})\\&=X^{j}{\bigl (}\partial _{j}(Y^{k})\partial _{k}+Y^{k}\nabla _{j}\partial _{k}{\bigr )}\\&=X^{j}{\bigl (}\partial _{j}(Y^{k})\partial _{k}+Y^{k}\Gamma _{jk}^{l}\partial _{l}{\bigr )}\\&=X^{j}{\bigl (}\partial _{j}(Y^{l})+Y^{k}\Gamma _{jk}^{l}{\bigr )}\partial _{l}\end{aligned}}}$

that is,

${\displaystyle (\nabla _{j}Y)^{l}=\partial _{j}Y^{l}+\Gamma _{jk}^{l}Y^{k}}$

An affine connection ${\displaystyle \nabla }$ is compatible with a metric iff

${\displaystyle \partial _{i}{\bigl (}g(\partial _{j},\partial _{k}){\bigr )}=g(\nabla _{i}\partial _{j},\partial _{k})+g(\partial _{j},\nabla _{i}\partial _{k})=g(\Gamma _{ij}^{l}\partial _{l},\partial _{k})+g(\partial _{j},\Gamma _{ik}^{l}\partial _{l})}$

i.e., if and only if

${\displaystyle \partial _{i}g_{jk}=\Gamma _{ij}^{l}g_{lk}+\Gamma _{ik}^{l}g_{jl}.}$

An affine connection ∇ is torsion free iff

${\displaystyle \nabla _{j}\partial _{k}-\nabla _{k}\partial _{j}=(\Gamma _{jk}^{l}-\Gamma _{kj}^{l})\partial _{l}=[\partial _{j},\partial _{k}]=0.}$

i.e., if and only if

${\displaystyle \Gamma _{jk}^{l}=\Gamma _{kj}^{l}}$

is symmetric in its lower two indices.

As one checks by taking for ${\displaystyle X,Y,Z}$, coordinate vector fields ${\displaystyle \partial _{j},\partial _{k},\partial _{l}}$ (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

${\displaystyle \Gamma _{jk}^{l}={\tfrac {1}{2}}g^{lr}\left(\partial _{k}g_{rj}+\partial _{j}g_{rk}-\partial _{r}g_{jk}\right)}$

where as usual ${\displaystyle g^{ij}}$ are the coefficients of the dual metric tensor, i.e. the entries of the inverse of the matrix ${\displaystyle g_{kl}}$.

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

${\displaystyle D_{t}V=\nabla _{{\dot {\gamma }}(t)}V.}$

Formally, D is the pullback connection γ*∇ on the pullback bundle γ*TM.

In particular, ${\displaystyle {\dot {\gamma }}(t)}$ is a vector field along the curve γ itself. If ${\displaystyle \nabla _{{\dot {\gamma }}(t)}{\dot {\gamma }}(t)}$ 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 ${\displaystyle {\dot {\gamma }}}$:

${\displaystyle \left(\gamma ^{*}\nabla \right){\dot {\gamma }}\equiv 0.}$

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 ${\displaystyle ds^{2}=dx^{2}+dy^{2}=dr^{2}+r^{2}d\theta ^{2}}$, while the metric on the right has standard form in polar coordinates (when ${\displaystyle r=1}$), and thus preserves the vector ${\displaystyle {\partial \over \partial \theta }}$ tangent to the circle. This second metric has a singularity at the origin, as can be seen by expressing it in Cartesian coordinates:

${\displaystyle dr={\frac {xdx+ydy}{\sqrt {x^{2}+y^{2}}}}}$

${\displaystyle d\theta ={\frac {xdy-ydx}{x^{2}+y^{2}}}}$

${\displaystyle dr^{2}+d\theta ^{2}={\frac {(xdx+ydy)^{2}}{x^{2}+y^{2}}}+{\frac {(xdy-ydx)^{2}}{(x^{2}+y^{2})^{2}}}}$

Parallel transports under Levi-Civita connections

This transport is given by the metric ${\displaystyle ds^{2}=dr^{2}+r^{2}d\theta ^{2}}$.

This transport is given by the metric ${\displaystyle ds^{2}=dr^{2}+d\theta ^{2}}$.

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

Denote as d_mY the differential of the map Y at the point m. Then we have:

Lemma — The formula

$${\displaystyle \left(\nabla _{X}Y\right)(m)=d_{m}Y(X(m))+\langle X(m),Y(m)\rangle m}$$

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 produces a vector field tangent to S². That is, we need to prove that for all m in S²

$${\displaystyle {\bigl \langle }\left(\nabla _{X}Y\right)(m),m{\bigr \rangle }=0\qquad (1).}$$

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

$${\displaystyle d_{m}f(X)={\bigl \langle }d_{m}Y(X),m{\bigr \rangle }+{\bigl \langle }Y(m),X(m){\bigr \rangle }=0.}$$

The equation (1) above follows. Q.E.D.

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.

Behavior under conformal rescaling

If the metric ${\displaystyle g}$ in a conformal class is replaced by the conformally rescaled metric of the same class ${\displaystyle {\hat {g}}=e^{2\gamma }g}$, then the Levi-Civita connection transforms according to the rule ^[10]

$${\displaystyle {\widehat {\nabla }}_{X}Y=\nabla _{X}Y+X(\gamma )Y+Y(\gamma )X-g(X,Y)\nabla \gamma .}$$

Indeed, it is trivial to verify that ${\displaystyle {\widehat {\nabla }}}$ is torsion-free. To verify metricity, assume that ${\displaystyle g(Y,Y)}$ is constant. In that case,

$${\displaystyle {\hat {g}}({\widehat {\nabla }}_{X}Y,Y)=X(\gamma ){\hat {g}}(Y,Y)={\frac {1}{2}}X({\hat {g}}(Y,Y)).}$$

As an application, consider again the unit sphere, but this time under stereographic projection, so that the metric (in complex Fubini–Study coordinates ${\displaystyle z,{\bar {z}}}$) is:

$${\displaystyle g={\frac {4\,dz\,d{\bar {z}}}{(1+z{\bar {z}})^{2}}}.}$$

This exhibits the metric of the sphere as conformally flat, with the Euclidean metric ${\displaystyle dz\,d{\bar {z}}}$, with ${\displaystyle \gamma =\ln(2)-\ln(1+z{\bar {z}})}$. We have ${\displaystyle d\gamma =-(1+z{\bar {z}})^{-1}({\bar {z}}\,dz+z\,d{\bar {z}})}$, and so

$${\displaystyle {\widehat {\nabla }}_{\partial _{z}}\partial _{z}=-{\frac {2{\bar {z}}\partial _{z}}{1+z{\bar {z}}}}.}$$

With the Euclidean gradient ${\displaystyle \nabla \gamma =-(1+z{\bar {z}})^{-1}({\bar {z}}\partial _{z}+z\partial _{\bar {z}})}$, we have

$${\displaystyle {\widehat {\nabla }}_{\partial _{z}}\partial _{\bar {z}}=0.}$$

These relations, together with their complex conjugates, define the Christoffel symbols for the two-sphere.

See also

• Weitzenböck connection

Notes

1. Levi-Civita, Tullio (1917). "Nozione di parallelismo in una varietà qualunque" [The notion of parallelism on any manifold]. Rendiconti del Circolo Matematico di Palermo (in Italian). 42: 173–205. doi:10.1007/BF03014898. JFM   46.1125.02. S2CID   122088291.
2. Christoffel, Elwin B. (1869). "Ueber die Transformation der homogenen Differentialausdrücke zweiten Grades". Journal für die reine und angewandte Mathematik. 1869 (70): 46–70. doi:10.1515/crll.1869.70.46. S2CID   122999847.
3. See Spivak, Michael (1999). A Comprehensive introduction to differential geometry (Volume II). Publish or Perish Press. p. 238. ISBN   0-914098-71-3.
4. Brouwer, L. E. J. (1906). "Het krachtveld der niet-Euclidische, negatief gekromde ruimten". Koninklijke Akademie van Wetenschappen. Verslagen. 15: 75–94.
5. Brouwer, L. E. J. (1906). "The force field of the non-Euclidean spaces with negative curvature". Koninklijke Akademie van Wetenschappen. Proceedings. 9: 116–133. Bibcode:1906KNAB....9..116B.
6. Schouten, Jan Arnoldus (1918). "Die direkte Analysis zur neueren Relativiteitstheorie". Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam. 12 (6): 95.
7. Weyl, Hermann (1918). "Gravitation und Elektrizitat". Sitzungsberichte Berliner Akademie: 465–480.
8. Weyl, Hermann (1918). "Reine Infinitesimal geometrie". Mathematische Zeitschrift . 2 (3–4): 384–411. Bibcode:1918MatZ....2..384W. doi:10.1007/bf01199420. S2CID   186232500.
9. Carmo, Manfredo Perdigão do (1992). Riemannian geometry. Francis J. Flaherty. Boston: Birkhäuser. ISBN   0-8176-3490-8. OCLC   24667701.
10. Arthur Besse (1987). Einstein manifolds. Springer. p. 58.

References

• Boothby, William M. (1986). An introduction to differentiable manifolds and Riemannian geometry . Academic Press. ISBN   0-12-116052-1.
• Kobayashi, Shoshichi; Nomizu, Katsumi (1963). Foundations of differential geometry. John Wiley & Sons. ISBN   0-470-49647-9. See Volume I pag. 158

External links

• "Levi-Civita connection", Encyclopedia of Mathematics , EMS Press, 2001 [1994]
• MathWorld: Levi-Civita Connection
• PlanetMath: Levi-Civita Connection
• Levi-Civita connection at the Manifold Atlas