Curvature and connection without a metric

• A
Staff Emeritus
In the absence of a metric, we can not raise and lower indices at will.

There are two sorts of Christoffel symbols, Christoffels of the first kind, ##\Gamma^a{}_{bc}## in component notation, and Christoffel symbols of the second kind, ##\Gamma_{abc}##. What's the relationship between the two kinds of Christoffel symbols? Is perhaps one of them a connection betwen vectors, and the other a connection between covectors?

Similarly, is ##R^a{}_{bcd}## "the" curvature tensor?

I suppose it'd be better to express this in terms of geometry rather than components, but I'm struggling a bit to do that.

This is all very basic, but I'm just not used to thinking about differential geometry without a metric :(.

Staff Emeritus
Homework Helper
Gold Member
Typically the nomenclature ”Christoffel symbols” is generally reserved for the connection coefficients of the Levi-Civita connection. As such, they don't really hold meaning outside of a Riemannian or pseudo-Riemannian manifold.

The more general concept of connection coefficients ##\Gamma_{\mu i}^j## relate to the connection on a vector bundle where I have written the fiber indices with Latin letters. Choosing a basis ##E_i## for the fiber, the connection coefficients are defined by
$$\nabla_\mu E_i = \Gamma_{\mu i}^j E_j.$$
For a the tangent bundle, the indices are the same as the indices of the base manifold itself and you would write ##\Gamma_{\mu\nu}^\lambda## etc. (So if you don't want to think about general vector bundle, just replace ##i## and ##j## by Greek letters.)

The corresponding connection on the dual bundle is found by using a basis ##E^i## with the property ##E^i\cdot E_j = \delta^i_j##. From this follows that
$$0 = \nabla_\mu (E^i \cdot E_j) = E^i \cdot \Gamma_{\mu j}^k E_k + (\nabla_\mu E^i)\cdot E_j = \Gamma_{\mu j}^i + (\nabla_\mu E^i)\cdot E_j$$
$$(\nabla_\mu E^i)\cdot E_j = -\Gamma_{\mu j}^i.$$
$$R(X,Y) A = \nabla_X \nabla_Y A - \nabla_Y \nabla_X A - \nabla_{[X,Y]} A,$$