Metric connection: Difference between revisions

Latest revision as of 19:48, 18 May 2008

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

Template:Connection property

Definition

Given data

A differential manifold $M$ .

A metric bundle $E$ over $M$ (viz, a vector bundle with a smoothly varying metric structure $g$ on each fibre of $E$ over $M$ ).

Definition part

A metric connection on $(M,g)$ is a connection $\nabla$ on the vector bundle $E$ over $M$ satisfying the following condition:

$Xg(Y,Z)=g(\nabla _{X}Y,Z)+g(Y,\nabla _{X}Z)$

Here $X$ is a vector field (viz a section of $TM$ ) and $Y$ and $Z$ are sections of $E$ .

In other words, it is a connection such that the dual connection on the dual bundle to $E$ is the same as the connection obtained by the natural isomorphism between $E$ and its dual (induced by the metric).

We are in particular interested in metric linear connections, which are metric connections over the tangent bundle for a Riemannian manifold (viz, the tangent bundle is endowed with a Riemannian metric). Of these, a very special one is the Levi-Civita connection, which is the only torsion-free metric linear connection.

@@ Line 7: / Line 7: @@
 ===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 [[differential manifold]] <math>M</math>.
+A [[metric bundle]] <math>E</math> over <math>M</math> (viz, a [[vector bundle]] with a smoothly varying metric structure <math>g</math> on each fibre of <math>E</math> over <math>M</math>).
-A vector bundle <math>E</math> over <math>M</math>.
-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 '''metric connection''' on <math>(M,g)</math> is a [[linear connection]] <math>\nabla</math> on <math>M</math> satisfying the following condition:
+A '''metric connection''' on <math>(M,g)</math> is a [[connection]] <math>\nabla</math> on the vector bundle <math>E</math> over <math>M</math> satisfying the following condition:
 <math>X g (Y,Z) = g (\nabla_X Y, Z) + g(Y, \nabla_X Z)</math>
-In other words, it is a connection such that the dual connection on the cotangent bundle is the same as the connection obtained by the natural isomorphism between the tangent and cotangent bundle (induced by the metric).
+Here <math>X</math> is a vector field (viz a section of <math>TM</math>) and <math>Y</math> and <math>Z</math> are sections of <math>E</math>.
+In other words, it is a connection such that the dual connection on the dual bundle to <math>E</math> is the same as the connection obtained by the natural isomorphism between <math>E</math> and its dual (induced by the metric).
+We are in particular interested in [[metric linear connection]]s, which are metric connections over the tangent bundle for a [[Riemannian manifold]] (viz, the tangent bundle is endowed with a [[Riemannian metric]]). Of these, a very special one is the [[Levi-Civita connection]], which is the only [[torsion-free linear connection|torsion-free]] metric linear connection.