Algebraic variety

2014-01-14T14:29:48Z

2620:AE:0:213F:CDF2:CD7B:3B31:8BE6:

In [[Riemannian geometry]], the '''Levi-Civita connection''' is a specific [[connection (mathematics)|connection]] on the tangent bundle of a [[manifold]]. More specifically, it is the [[Torsion (differential geometry)|torsion]]-free [[metric connection]], i.e., the torsion-free [[connection (mathematics)|connection]] on the [[tangent bundle]] (an [[affine connection]]) preserving a given ([[pseudo-Riemannian manifold|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 manifold|Riemannian]] and [[pseudo-Riemannian manifold]]s 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]].

The Levi-Civita connection is named after [[Tullio Levi-Civita]], although originally "discovered" by [[Elwin Bruno Christoffel]]. Levi-Civita,<ref>See Levi-Civita (1917)</ref> along with [[Gregorio Ricci-Curbastro]], used Christoffel's symbols<ref>See Christoffel (1869)</ref> to define the notion of [[parallel transport]] and explore the relationship of parallel transport with the [[Riemann curvature tensor|curvature]], thus developing the modern notion of [[holonomy]].<ref>See Spivak (1999) Volume II, page 238</ref>

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

:<math>M^n \subset \mathbf{R}^{\frac{n(n+1)}{2}},</math>

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.

==Formal definition==
Let ''(M,g)'' be a [[Riemannian manifold]] (or [[pseudo-Riemannian manifold]]). Then an [[affine connection]] ∇ is called a Levi-Civita connection if

# ''it preserves the metric'', i.e., {{nowrap|1=∇''g'' = 0}}.
# ''it is [[torsion of connection|torsion]]-free'', i.e., for any vector fields ''X'' and ''Y'' we have {{nowrap|1=∇''X''''Y'' − ∇''Y''''X'' = [''X'',''Y'']}}, where [''X'',''Y''] is the [[Lie bracket of vector fields|Lie bracket]] of the [[vector field]]s ''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:

:<math> 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). </math>

By condition 2 the right hand side is equal to

:<math> 2g(\nabla_X Y, Z) - g([X,Y], Z) + g([X,Z],Y) + g([Y,Z],X) </math>

so we find

:<math> 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) \}. </math>

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 Levi-Civita connection.

==Christoffel symbols==
Let ∇ be the connection of the Riemannian metric. Choose local coordinates <math> x^1 \ldots x^n</math> and let <math> \Gamma^l{}_{jk} </math> be the [[Christoffel symbols]] with respect to these coordinates. The torsion freeness condition 2 is then equivalent to the symmetry
:<math> \Gamma^l{}_{jk} = \Gamma^l{}_{kj}. </math>
The definition of the Levi-Civita connection derived above is equivalent to a definition of the Christoffel symbols in terms of the metric as
:<math> \Gamma^l{}_{jk} = \tfrac{1}{2}\sum_r g^{lr} \left \{\partial _k g_{rj} + \partial _j g_{rk} - \partial _r g_{jk} \right \} </math>
where as usual <math>g^{ij}</math> are the coefficients of the dual metric tensor, i.e. the entries of the inverse of the matrix <math>(g_{kl})</math>.

==Derivative along curve==
The Levi-Civita connection (like any affine connection) also defines a derivative along [[curve]]s, sometimes denoted by ''D''.

Given a smooth curve γ on ''(M,g)'' and a [[vector field]] ''V'' along γ its derivative is defined by
:<math>D_tV=\nabla_{\dot\gamma(t)}V.</math>
(Formally ''D'' is the [[pullback (differential geometry)|pullback connection]] on the [[pullback bundle]] γ*''TM''.)

In particular, <math>\dot{\gamma}(t)</math> is a vector field along the curve γ itself. If <math>\nabla_{\dot\gamma(t)}\dot\gamma(t)</math> 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 tensor|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 group|orthogonal]] – that is, they preserve the inner products on the various tangent spaces.

==Example: The unit sphere in R3==
Let <math>\langle \cdot,\cdot \rangle</math> 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 sub-space 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

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

Denote by ''dY'' the differential of such a map. Then we have:

<blockquote>'''Lemma:''' The formula

:<math>\left(\nabla_X Y\right)(m) = d_mY(X) + \langle X(m),Y(m)\rangle m</math>

defines an affine connection on '''S'''2 with vanishing torsion.</blockquote>

'''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
:<math>\langle\left(\nabla_X Y\right)(m),m\rangle = 0\qquad (1).</math>
Consider the map
:<math>\begin{cases}
f: \mathbf{S}^2 \to \mathbf{R} \\
m \mapsto \langle Y(m), m\rangle.
\end{cases}</math>
The map ''f'' is constant, hence its differential vanishes. In particular
:<math>d_mf(X) = \langle d_m Y(X),m\rangle + \langle Y(m), X(m)\rangle = 0.</math>
The equation (1) above follows.<math>\Box</math>

In fact, this connection is the Levi-Civita connection for the metric on '''S'''2 inherited from '''R'''3. Indeed, one can check that this connection preserves the metric.

==See also==
*[[Affine connection]]
*[[Weitzenböck connection]]

==Notes==
{{Reflist}}

==References==

===Primary historical references===
* {{citation | first = Elwin Bruno | last = Christoffel |title= Über die Transformation der homogenen Differentialausdrücke zweiten Grades| journal = J. für die Reine und Angew. Math.| volume = 70 | year = 1869 | pages = 46–70}}
* {{citation | first = Tullio | last = Levi-Civita |title= Nozione di parallelismo in una varietà qualunque e consequente specificazione geometrica della curvatura Riemanniana| journal = Rend. Circ. Mat. Palermo| volume = 42 | year = 1917 | pages = 73–205}}

===Secondary references===
* {{cite book|first=William M.|last=Boothby|title=An introduction to differentiable manifolds and Riemannian geometry |publisher=Academic Press|year=1986|isbn=0-12-116052-1}}
* {{cite book|author=Kobayashi, S., and Nomizu, K.|title=Foundations of differential geometry|publisher=John Wiley & Sons|year=1963|isbn=0-470-49647-9}} See Volume I pag. 158
* {{cite book|first=Michael|last=Spivak|title=A Comprehensive introduction to differential geometry (Volume II)|publisher=Publish or Perish Press|year=1999|isbn=0-914098-71-3}}

==External links==
* {{springer|title=Levi-Civita connection|id=p/l058230}}
* [http://mathworld.wolfram.com/Levi-CivitaConnection.html MathWorld: Levi-Civita Connection]
* [http://planetmath.org/encyclopedia/LeviCivitaConnection.html 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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[[Category:Riemannian geometry]]
[[Category:Connection (mathematics)]]

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Algebraic variety