A Curvature & Connection Without Metric

[pervect]

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 :(.

 

[Orodruin]

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
$$
0 = \nabla_\mu (E^i \cdot E_j) = E^i \cdot \Gamma_{\mu j}^k E_k + (\nabla_\mu E^i)\cdot E_j = \Gamma_{\mu j}^i + (\nabla_\mu E^i)\cdot E_j
$$
leading to
$$
(\nabla_\mu E^i)\cdot E_j = -\Gamma_{\mu j}^i.
$$
In other words, the connection coefficients on the dual bundle is the same as those on the vector bundle with opposite sign.

As for the curvature tensor, it is uniquely defined by
$$
R(X,Y) A = \nabla_X \nabla_Y A - \nabla_Y \nabla_X A - \nabla_{[X,Y]} A,
$$
where $X$ and $Y$ are in the tangent space and $A$ in the fiber.