Levi-Civita connection

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

{{Cite journal|author-link=Tullio Levi-Civita|year=1917|title=Nozione di parallelismo in una varietà qualunque|trans-title=The notion of parallelism on any manifold|url=https://zenodo.org/record/1428456|journal=Rendiconti del Circolo Matematico di Palermo|language=it|volume=42|pages=173–205|doi=10.1007/BF03014898|jfm=46.1125.02|author-first=Tullio|author-last=Levi-Civita|s2cid=122088291}}

along with Gregorio Ricci-Curbastro, used Christoffel's symbols{{cite journal |title=Ueber die Transformation der homogenen Differentialausdrücke zweiten Grades|last=Christoffel|first=Elwin B.|author-link=Elwin Bruno Christoffel|journal=Journal für die reine und angewandte Mathematik|volume=1869|issue=70|pages=46–70|year=1869|url=http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002153882&IDDOC=266356|doi=10.1515/crll.1869.70.46|s2cid=122999847}} to define the notion of parallel transport and explore the relationship of parallel transport with the curvature, thus developing the modern notion of holonomy.See {{cite book|first=Michael|last=Spivak|author-link=Michael Spivak | title=A Comprehensive introduction to differential geometry (Volume II)|publisher=Publish or Perish Press|year=1999|isbn=0-914098-71-3 |page=238 }}

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.

{{Cite journal|author-link=L. E. J. Brouwer|year=1906|title=Het krachtveld der niet-Euclidische, negatief gekromde ruimten|journal=Koninklijke Akademie van Wetenschappen. Verslagen|volume=15|pages=75–94|author-first=L. E. J.|author-last=Brouwer}}

{{Cite journal|year=1906|title=The force field of the non-Euclidean spaces with negative curvature|journal=Koninklijke Akademie van Wetenschappen. Proceedings|volume=9|pages=116–133|author-first=L. E. J.|author-last=Brouwer|bibcode=1906KNAB....9..116B}}

In 1917, Tullio 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. 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 $M^n\; \backslash subset\; \backslash mathbf\{R\}^\{n(n+1)/2\}.$

In 1918, independently of Levi-Civita, Jan Arnoldus Schouten obtained analogous results.

{{cite journal

|author-first=Jan Arnoldus

|author-last=Schouten

|author-link =Jan Arnoldus Schouten

|title =Die direkte Analysis zur neueren Relativiteitstheorie

|journal =Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam

|volume =12

|issue=6

|pages =95

|year =1918

}}

In the same year, Hermann Weyl generalized

Levi-Civita's results.

{{cite journal

|author-first=Hermann

|author-last=Weyl

|title=Gravitation und Elektrizität

|author-link =Hermann Weyl

|journal =Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften

|pages =465–480

|year =1918

|url=https://www.biodiversitylibrary.org/item/126468#page/537

|bibcode=1918SPAW.......465W |doi=

}} Reprinted in Das Relativitätsprinzip: Eine Sammlung von Abhandlungen (1923), pp. 147–159, {{doi|10.1007/978-3-663-19510-8_11}}, {{isbn|978-3-663-19372-2}}.

{{cite journal

|author-first=Hermann

|author-last=Weyl

|title =Reine Infinitesimal geometrie

|journal =Mathematische Zeitschrift

|volume =2

|issue=3–4

|pages =384–411

|year =1918

|doi =10.1007/bf01199420

|bibcode=1918MatZ....2..384W

|s2cid=186232500

|url=https://zenodo.org/record/1447411

}}

• {{math|(M, g)}} denotes a pseudo-Riemannian manifold.
• {{math|TM}} is the tangent bundle of {{math|M}}.
• {{math|g}} is the pseudo-Riemannian metric of {{math|M}}.
• {{math|X, Y, Z}} are smooth vector fields on {{math|M}}, i. e. smooth sections of {{math|TM}}.
• {{math|[X, Y]}} is the Lie bracket of {{math|X}} and {{math|Y}}. It is again a smooth vector field.

The metric {{math|g}} can take up to two vectors or vector fields {{math|X, Y}} as arguments. In the former case the output is a number, the (pseudo-)inner product of {{math|X}} and {{math|Y}}. In the latter case, the inner product of {{math|X_p, Y_p}} is taken at all points {{math|p}} on the manifold so that {{math|g(X, Y)}} defines a smooth function on {{math|M}}. Vector fields act (by definition) as differential operators on smooth functions. In local coordinates $(x\_1,\backslash ldots,\; x\_n)$, the action reads

: $X(f)\; =\; X^i\backslash frac\{\backslash partial\}\{\backslash partial\; x^i\}f\; =\; X^i\backslash partial\_i\; f$

where Einstein's summation convention is used.

An affine connection $\backslash nabla$ is called a Levi-Civita connection if

1. it preserves the metric, i.e., $\backslash nabla\; g\; =\; 0$.
2. it is torsion-free, i.e., for any vector fields $X$ and $Y$ we have $\backslash nabla\_X\; Y\; -\; \backslash nabla\_Y\; X\; =\; [X,Y]$, 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. Do Carmo's text.{{Cite book |last=Carmo |first=Manfredo Perdigão do |url=https://www.worldcat.org/oclc/24667701 |title=Riemannian geometry |date=1992 |publisher=Birkhäuser |others=Francis J. Flaherty |isbn=0-8176-3490-8 |location=Boston |oclc=24667701}}

{{main|Fundamental theorem of Riemannian geometry}}

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

Proof:{{cite book|title=Introduction to Riemannian manifolds|author=John M Lee|publisher=Springer-Verlag|year=2018|page=22}}{{cite book|author=Barrett O'Neill|title=Semi-Riemannian geometry with Applications to relativity|publisher=Academic Press|year=1983|page=61}}

To prove uniqueness, unravel the definition of the action of a connection on tensors to find

: $X\backslash bigl(g(Y,Z)\backslash bigr)\; =\; (\backslash nabla\_X\; g)(Y,\; Z)\; +\; g(\backslash nabla\_X\; Y,\; Z)\; +\; g(\; Y,\; \backslash nabla\_X\; Z)$.

Hence one can write the condition that $\backslash nabla$ preserves the metric as

: $X\backslash bigl(g(Y,Z)\backslash bigr)\; =\; g(\backslash nabla\_X\; Y,\; Z)\; +\; g(\; Y,\; \backslash nabla\_X\; Z)$.

By the symmetry of $g$,

: $X\; \backslash bigl(g(Y,Z)\backslash bigr)\; +\; Y\; \backslash bigl(g(Z,X)\backslash bigr)\; -\; Z\; \backslash bigl(g(Y,X)\backslash bigr)\; =\; g(\backslash nabla\_X\; Y\; +\; \backslash nabla\_Y\; X,\; Z)\; +\; g(\backslash nabla\_X\; Z\; -\; \backslash nabla\_Z\; X,\; Y)\; +\; g(\backslash nabla\_Y\; Z\; -\; \backslash nabla\_Z\; Y,\; X)$.

By torsion-freeness, the right hand side is therefore equal to

: $2g(\backslash nabla\_X\; Y,\; Z)\; -\; g([X,Y],\; Z)\; +\; g([X,Z],\; Y)\; +\; g([Y,Z],\; X)$.

Thus, the Koszul formula

: $g(\backslash nabla\_X\; Y,\; Z)\; =\; \backslash tfrac\{1\}\{2\}\; \backslash Big\backslash \{\; X\; \backslash bigl(g(Y,Z)\backslash bigr)\; +\; Y\; \backslash bigl(g(Z,X)\backslash bigr)\; -\; Z\; \backslash bigl(g(X,Y)\backslash bigr)\; +\; g([X,Y],Z)\; -\; g([Y,Z],\; X)\; -\; g([X,Z],\; Y)\; \backslash Big\backslash \}$

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

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

: $g(\backslash nabla\_X\; (Y\_1\; +\; Y\_2),\; Z)\; =\; g(\backslash nabla\_X\; Y\_1,\; Z)\; +\; g(\backslash nabla\_X\; Y\_2\; ,\; Z)$

: $g(\backslash nabla\_X\; (f\; Y),\; Z)\; =\; X(f)\; g(Y,\; Z)\; +\; f\; g(\backslash nabla\_X\; Y,Z)$

: $g(\backslash nabla\_X\; Y,\; Z)\; +\; g(\backslash nabla\_X\; Z,\; Y)\; =\; X\backslash bigl(g(Y,Z)\backslash bigr)$

: $g(\backslash nabla\_X\; Y,\; Z)\; -\; g(\backslash nabla\_Y\; X,\; Z)\; =\; g([X,Y],\; Z).$

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 Levi-Civita connection.

With minor variation, the same proof shows that there is a unique connection that is compatible with the metric and has prescribed torsion.

Let $\backslash nabla$ be an affine connection on the tangent bundle. Choose local coordinates $x^1,\; \backslash ldots,\; x^n$ with coordinate basis vector fields $\backslash partial\_1,\; \backslash ldots,\; \backslash partial\_n$ and write $\backslash nabla\_j$ for $\backslash nabla\_\{\backslash partial\_j\}$. The Christoffel symbols $\backslash Gamma^l\_\{jk\}$ of $\backslash nabla$ with respect to these coordinates are defined as

: $\backslash nabla\_j\backslash partial\_k\; =\; \backslash Gamma^l\_\{jk\}\; \backslash partial\_l$

The Christoffel symbols conversely define the connection $\backslash nabla$ on the coordinate neighbourhood because

: $$

\begin{align}

\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^l_{jk}\partial_l\bigr)

\\ &= X^j\bigl(\partial_j(Y^l) + Y^k\Gamma^l_{jk}\bigr)\partial_l

\end{align}

that is,

: $(\backslash nabla\_j\; Y)^l\; =\; \backslash partial\_jY^l\; +\; \backslash Gamma^l\_\{jk\}\; Y^k$

An affine connection $\backslash nabla$ is compatible with a metric iff

: $\backslash partial\_i\; \backslash bigl(g(\backslash partial\_j,\; \backslash partial\_k)\; \backslash bigr)$

= g(\nabla_i\partial_j, \partial_k) + g(\partial_j, \nabla_i\partial_k)

= g(\Gamma^l_{ij}\partial_l, \partial_k) + g(\partial_j, \Gamma_{ik}^l\partial_l)

i.e., if and only if

: $\backslash partial\_i\; g\_\{jk\}\; =\; \backslash Gamma^l\_\{ij\}g\_\{lk\}\; +\; \backslash Gamma^l\_\{ik\}g\_\{jl\}.$

An affine connection {{math|∇}} is torsion free iff

: $\backslash nabla\_j\backslash partial\_k\; -\; \backslash nabla\_k\; \backslash partial\_j\; =\; (\backslash Gamma^l\_\{jk\}\; -\; \backslash Gamma^l\_\{kj\})\backslash partial\_l\; =\; [\backslash partial\_j,\; \backslash partial\_k]=\; 0.$

i.e., if and only if

: $\backslash Gamma^l\_\{jk\}\; =\; \backslash Gamma^l\_\{kj\}$

is symmetric in its lower two indices.

As one checks by taking for $X,\; Y,\; Z$, coordinate vector fields $\backslash partial\_j,\; \backslash partial\_k,\; \backslash partial\_l$ (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

: $\backslash Gamma^l\_\{jk\}\; =\; \backslash tfrac\{1\}\{2\}\; g^\{lr\}\; \backslash left(\; \backslash partial\; \_k\; g\_\{rj\}\; +\; \backslash partial\; \_j\; g\_\{rk\}\; -\; \backslash partial\; \_r\; g\_\{jk\}\; \backslash right)$

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

The Levi-Civita connection (like any affine connection) also defines a derivative along curves, sometimes denoted by {{math|D}}.

Given a smooth curve {{math|γ}} on {{math|(M, g)}} and a vector field {{math|V}} along {{math|γ}} its derivative is defined by

: $D\_tV=\backslash nabla\_\{\backslash dot\backslash gamma(t)\}V.$

Formally, {{math|D}} is the pullback connection {{math|γ*∇}} on the pullback bundle {{math|γ*TM}}.

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

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

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.

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 induced by the Levi-Civita connection associated to two different Riemannian metrics on the punctured plane $\backslash mathbf\; R^2\; \backslash backslash\; \backslash \{0,0\backslash \}$. The curve the parallel transport is done along is the unit circle. In polar coordinates, the metric on the left is the standard Euclidean metric $ds^2\; =\; dx^2\; +\; dy^2\; =\; dr^2\; +\; r^2\; d\backslash theta^2$, while the metric on the right is $ds^2\; =\; dr^2\; +\; d\backslash theta^2$. The first metric extends to the entire plane, but the second metric has a singularity at the origin:

: $dr\; =\; \backslash frac\{xdx\; +\; ydy\}\{\backslash sqrt\{x^2\; +\; y^2\}\}$

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

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

{{multiple image

| align = center

| direction = horizontal

| caption_align = center

| width = 200

| header = Parallel transports on the punctured plane under Levi-Civita connections

| header_align = center

| header_background =

| footer =

| footer_align =

| footer_background =

| background color =

|image1=Cartesian_transport.gif

|width1=200

|caption1=This transport is given by the metric $ds^2\; =\; dr^2\; +\; r^2\; d\backslash theta^2$.

|alt1=Cartesian transport

|image2=Circle_transport.gif

|width2=200

|caption2=This transport is given by the metric $ds^2\; =\; dr^2\; +\; d\backslash theta^2$.

|alt2=Polar transport

}}

Warning: This is parallel transport on the punctured plane along the unit circle, not parallel transport on the unit circle. Indeed, in the first image, the vectors fall outside of the tangent space to the unit circle.

Let {{math|⟨ , ⟩}} be the usual scalar product on {{math|R³}}. Let {{math|S²}} be the unit sphere in {{math|R³}}. The tangent space to {{math|S²}} at a point {{math|m}} is naturally identified with the vector subspace of {{math|R³}} consisting of all vectors orthogonal to {{math|m}}. It follows that a vector field {{math|Y}} on {{math|S²}} can be seen as a map {{math|Y : S² → R³}}, which satisfies

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

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

{{math theorem|name=Lemma|math_statement= The formula

$$\backslash left(\backslash nabla\_X\; Y\backslash right)(m)\; =\; d\_mY(X(m))\; +\; \backslash langle\; X(m),Y(m)\backslash rangle\; m$$

defines an affine connection on {{math|S²}} with vanishing torsion.}}

{{math proof|proof= It is straightforward to prove that {{math|∇}} satisfies the Leibniz identity and is {{math|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 {{math|S²}}. That is, we need to prove that for all {{math|m}} in {{math|S²}}

$$\backslash bigl\backslash langle\backslash left(\backslash nabla\_X\; Y\backslash right)(m),m\backslash bigr\backslash rangle\; =\; 0\backslash qquad\; (1).$$

Consider the map {{math|f}} that sends every {{math|m}} in {{math|S²}} to {{math|⟨Y(m), m⟩}}, which is always 0. The map {{math|f}} is constant, hence its differential vanishes. In particular

$$d\_mf(X)\; =\; \backslash bigl\backslash langle\; d\_m\; Y(X),m\backslash bigr\backslash rangle\; +\; \backslash bigl\backslash langle\; Y(m),\; X(m)\backslash bigr\backslash rangle\; =\; 0.$$

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

In fact, this connection is the Levi-Civita connection for the metric on {{math|S²}} inherited from {{math|R³}}. Indeed, one can check that this connection preserves the metric.

If the metric $g$ in a conformal class is replaced by the conformally rescaled metric of the same class $\backslash hat\; g=e^\{2\backslash gamma\}g$, then the Levi-Civita connection transforms according to the rule{{cite book|author=Arthur Besse|title=Einstein manifolds|year=1987|publisher=Springer|page=58}}

$$\backslash widehat\backslash nabla\_X\; Y\; =\; \backslash nabla\_XY\; +\; X(\backslash gamma)Y\; +\; Y(\backslash gamma)X\; -\; g(X,Y)\backslash mathrm\{grad\}\_g(\backslash gamma).$$

where $\backslash mathrm\{grad\}\_g(\backslash gamma)$ is the gradient vector field of $\backslash gamma$ i.e. the vector field $g$-dual to $d\backslash gamma$, in local coordinates given by $g^\{ik\}(\backslash partial\_i\; \backslash gamma)\backslash partial\_k$. Indeed, it is trivial to verify that $\backslash widehat\backslash nabla$ is torsion-free. To verify metricity, assume that $g(Y,Y)$ is constant. In that case,

$$\backslash hat\; g(\backslash widehat\backslash nabla\_XY,Y)\; =\; X(\backslash gamma)\backslash hat\; g(Y,Y)\; =\; \backslash frac12\; X(\backslash hat\; g(Y,Y)).$$

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

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

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

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

With the Euclidean gradient $\backslash mathrm\{grad\}\_\{Euc\}(\backslash gamma)\; =\; -(1+z\backslash bar\; z)^\{-1\}(\backslash bar\; z\backslash partial\_z\; +\; z\backslash partial\_\{\backslash bar\; z\})$, we have

$$\backslash widehat\backslash nabla\_\{\backslash partial\_z\}\backslash partial\_\{\backslash bar\; z\}\; =\; 0.$$

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

• Weitzenböck connection

{{reflist}}

• {{cite book|first=William M.|last=Boothby|author-link=William M. Boothby|title=An introduction to differentiable manifolds and Riemannian geometry |url=https://archive.org/details/introductiontodi0000boot|url-access=registration|publisher=Academic Press|year=1986|isbn=0-12-116052-1}}
• {{cite book|last1=Kobayashi|first1= Shoshichi|author-link1=Shoshichi Kobayashi |last2=Nomizu| first2=Katsumi |author-link2= Katsumi Nomizu |title=Foundations of differential geometry|publisher=John Wiley & Sons|year=1963|isbn=0-470-49647-9}} See Volume I pag. 158

• {{springer|title=Levi-Civita connection|id=p/l058230}}
• [http://mathworld.wolfram.com/Levi-CivitaConnection.html MathWorld: Levi-Civita Connection]
• [https://planetmath.org/levicivitaconnection PlanetMath: Levi-Civita Connection]
• [http://www.map.mpim-bonn.mpg.de/Levi-Civita_connection Levi-Civita connection] at the Manifold Atlas

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