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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 :(.