I was able to actually stick with my plan a little earlier and spend the day parsing through some stuff in Kobayashi and Nomizu. That’s not a bad way to spend a Sunday.
Somewhere in the middle of that, I ended up stumbling upon something I’d always been somewhat privy to and I did so almost by accident. In the text, I ran across the definition in the following context:
Consider two manifolds and and a mapping of the prior into the latter. Then for a point , the differential of at is a linear mapping which is defined as follows: Given a vector , choose a path with . Then is the vector tangent to the curve at .
The notation reminded me of something I saw during my very first foray into Differential Geometry, namely the Hodge star/dual operator. It was a notion that was so novel when I first saw it that I contemplated preparing a seminar talk at BGSU for my peers, none of whom were geometers of any kind; now that I’ve rediscovered it, I’m having similar ideas for my non-geometer peers here at FSU. But I’m getting ahead of myself…