Differential Geometry Sunday

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 f_* in the following context:

Consider two manifolds M and M' and a mapping f:M\to M' of the prior into the latter. Then for a point p\in M, the differential of f at p is a linear mapping f_*:T_p(M)\to T_{f(p)}(M') which is defined as follows: Given a vector X\in T_p(M), choose a path x(t) with p=x(t_0). Then f_*(X) is the vector tangent to the curve f(x(t)) at f(p)=f(x(t_0)).

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…

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