Metric connection

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This lives as an element of: the space of all connections, which in turn sits inside the space of all \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:

X g (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.