Levi-Civita connection: Difference between revisions

Latest revision as of 17:59, 6 January 2012

This lives as an element of: the space of all linear connections, which in turn sits inside the space of all $\mathbb {R}$ -bilinear maps $\Gamma (TM)\times \Gamma (TM)\to \Gamma (TM)$

Definition

Given data

A Riemannian manifold $(M,g)$ (here, $M$ is a differential manifold and $g$ is the additional structure of a Riemannian metric on it).

More generally, we can also look at a pseudo-Riemannian manifold, or a manifold with a pseudo-Riemannian metric: a smoothly varying nondegenerate (not necessarily positive definite) symmetric bilinear form $g$ in each tangent space.

Definition part

A Levi-Civita connection on $(M,g)$ is a linear connection $\nabla$ on $M$ satisfying the following two conditions:

The connection is metric, viz $Xg(Y,Z)=g(\nabla _{X}Y,Z)+g(Y,\nabla _{X}Z)$
The connection is torsion-free, viz $\nabla _{X}Y-\nabla _{Y}X=[X,Y]$ .

Definition by formula

The formula for the Levi-Civita connection requires us to use the fact that $g$ is nondegenerate. So, instead of directly specifying $\nabla _{X}Y$ for vector fields $X$ and $Y$ , the formula specifies $g(\nabla _{X}Y,Z)$ for $X,Y,Z$ vector fields, as follows:

$g(\nabla _{X}Y,Z)={\frac {Xg(Y,Z)+Yg(Z,X)-Zg(X,Y)+g(Y,[Z,X])+g(Z,[X,Y])-g(X,[Y,Z])}{2}}$ .

Additional use

The term Levi-Civita connection is sometimes also used for the connection induced on any tensor product involving the tangent and cotangent bundle, using the rule for tensor product of connections and the dual connection.

Facts

The Levi-Civita connection is unique

Further information: Levi-Civita connection exists and is unique

The proof combines the metric condition, the torsion-free condition, and the nondegeneracy of $g$ .

Note that the nondegeneracy of $g$ is very important, otherwise knowing the value of the inner product for any three vectors may not necessarily help us in computing the value of $\nabla _{X}Y$

Christoffel symbols

Further information: Christoffel symbols of a connection

The Levi-Civita connection on a manifold $M$ is a map $\nabla :\Gamma (TM)\times \Gamma (TM)\to \Gamma (TM)$ . This means that at any point $p\in M$ , it gives a map $T_{p}(M)\times \Gamma (TM)\to \Gamma (TM)$ , which roughly differentiates one tangent vector along another.

Let $\partial _{1},\partial _{2},\ldots ,\partial _{n}$ form a basis for the tangent space $TM$ . Then, the Christoffel symbol $\Gamma _{ij}^{k}$ is the component along $e_{k}$ of the vector $\nabla _{\partial _{i}}\partial _{j}$ .

The Christoffel symbols thus give an explicit description of the Levi-Civita connection. Namely, the Levi-Civita connection can be expressed using the Christoffel symbols.

@@ Line 5: / Line 5: @@
 ===Given data===
-A [[Riemmanian manifold]] <math>(M,g)</math> (here, <math>M</math> is a [[differential manifold]] and <math>g</math> is the additional structure of a [[Riemannian metric]] on it).
+A [[Riemannian manifold]] <math>(M,g)</math> (here, <math>M</math> is a [[differential manifold]] and <math>g</math> is the additional structure of a [[defining ingredient::Riemannian metric]] on it).
+More generally, we can also look at a [[pseudo-Riemannian manifold]], or a manifold with a [[defining ingredient::pseudo-Riemannian metric]]: a smoothly varying nondegenerate (not necessarily positive definite) symmetric bilinear form <math>g</math> in each tangent space.
-More generally, we can also look at a [[pseudo-Riemannian manifold]], or a manifold with a smoothly varying nondegenerate (not necessarily positive definite) symmetric positive definite bilinear form in each tangent space.
 ===Definition part===
-A '''Levi-Civita connection''' on <math>(M,g)</math> is a [[linear connection]] <math>\nabla</math> on <math>M</math> satisfying the following two conditions:
+A '''Levi-Civita connection''' on <math>(M,g)</math> is a [[defining ingredient::linear connection]] <math>\nabla</math> on <math>M</math> satisfying the following two conditions:
-* <math>X g (Y,Z) = g (\nabla_X Y, Z) + g(Y, \nabla_X Z)</math>
+* The connection is [[defining ingredient::metric connection|metric]], viz <math>X g (Y,Z) = g (\nabla_X Y, Z) + g(Y, \nabla_X Z)</math>
-* The connection is [[torsion-free linear connection|torsion-free]], viz <math>\nabla_X Y - \nabla_Y X = [X,Y]</math>
+* The connection is [[defining ingredient::torsion-free linear connection|torsion-free]], viz <math>\nabla_X Y - \nabla_Y X = [X,Y]</math>.
+===Definition by formula===
-An equivalent way of seeing this is that the Levi-Civita connection is the unique connection such that the isomorphism with
+The formula for the Levi-Civita connection requires us to use the fact that <math>g</math> is nondegenerate. So, instead of directly specifying <math>\nabla_XY</math> for vector fields <math>X</math> and <math>Y</math>, the formula specifies <math>g(\nabla_XY,Z)</math> for <math>X,Y,Z</math> vector fields, as follows:
-==Facts==
-===The Levi-Civita connection is unique===
+<math>g(\nabla_XY,Z) = \frac{Xg(Y,Z) + Yg(Z,X) - Zg(X,Y) + g(Y,[Z,X]) + g(Z,[X,Y]) - g(X,[Y,Z])}{2}</math>.
-The proof of the uniqueness of the Levi-Civita connection is as follows.
+===Additional use===
-* Take three vector fields <math>X, Y, Z</math>. Now, consider the three equations obtained by cycling <math>X, Y, Z</math> in the first condition. Solving this system of linear equations, we can express <math>g(\nabla_XY,Z)</math> in terms of <math>X,Y,Z</math>.
+The term '''Levi-Civita connection''' is sometimes also used for the connection induced on any tensor product involving the tangent and cotangent bundle, using the rule for [[tensor product of connections]] and the [[dual connection]].
-Explicitly:
+==Facts==
-<math>g(\nabla_XY,Z) + g(Y,\nabla_XZ) = Xg(Y,Z)</math>
+===The Levi-Civita connection is unique===
-<math>g(\nabla_YX,Z) + g(X,\nabla_YZ) = Yg(Z,X)</math>
+{{further|[[Levi-Civita connection exists and is unique]]}}
-<math>g(\nabla_ZX,Y) + g(X,\nabla_ZY) = Zg(X,Y)</math>
+The proof combines the metric condition, the torsion-free condition, and the nondegeneracy of <math>g</math>.
-Now let's choose to focus only on the clockwise cyclic expressions, that is, the three expressions <math>p = g(\nabla_XY,Z), q = g(\nabla_YZ,X), q = g(\nabla_ZX,Y)</math>.
+Note that the nondegeneracy of <math>g</math> is very important, otherwise knowing the value of the inner product for any three vectors may not necessarily help us in computing the value of <math>\nabla_XY</math>
-Writing everything in terms of these three (we here make use of the torsion tensor vanishing):
+===Christoffel symbols===
-<math>p + q = Xg(Y,Z) - g(Y,[X,Z])</math>
+{{further|[[Christoffel symbols of a connection]]}}
-and similarly for the other three variables.
-We thus get expressions for <math>g(\nabla_XY,Z)</math> in terms of <math>g</math> and the Lie bracket.
-* Now use the fact that <math>g</math> is nondegenerate to conclude that knowledge of the function <math>g(\nabla_XY,Z)</math> for ''all'' <math>Z</math> helps us fix <math>\nabla_XY</math> uniquely.
-* This means that we have a unique definition for <math>\nabla</math>.
-Note that the nondegeneracy of <math>g</math> is very important, otherwise knowing the value of the inner product for any three vectors may not necessarily help us in computing the value of <math>nabla_XY</math>
-To show that the Levi-Civita connection exists, it suffices to check that the map sending <math>Z</math> to what we propose for <math>g(\nabla_X Y, Z)</math> is actually a linear map.
-===Christoffel symbols===
-The Levi-Civita connection on a manifold <math>M</math> is a map <math>\nabla: \Gamma(TM) \times \Gamma(TM) \to \Gamma(TM)</math>. This means that at any point <math>p \in M</math>, it gives a map <math>T_p(M) \times T_p(M) \to T_p(M)</math>, which roughly ''differentiates'' one tangent vector along another.
+The Levi-Civita connection on a manifold <math>M</math> is a map <math>\nabla: \Gamma(TM) \times \Gamma(TM) \to \Gamma(TM)</math>. This means that at any point <math>p \in M</math>, it gives a map <math>T_p(M) \times \Gamma(TM) \to \Gamma(TM)</math>, which roughly ''differentiates'' one tangent vector along another.
-Let <math>\partial_1, \partial_2, \ldots, \partial_n</math> form a basis for the tangent space <math>TM</math>. Then, the Christoffel symbol <math>\Gamma_ij^k</math> is the component along <math>e_k</math> of the vector <math>\nabla_{\partial_i}\partial_j</math>.
+Let <math>\partial_1, \partial_2, \ldots, \partial_n</math> form a basis for the tangent space <math>TM</math>. Then, the Christoffel symbol <math>\Gamma_{ij}^k</math> is the component along <math>e_k</math> of the vector <math>\nabla_{\partial_i}\partial_j</math>.
 The Christoffel symbols thus give an ''explicit description'' of the Levi-Civita connection. Namely, the Levi-Civita connection can be expressed using the Christoffel symbols.