# 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…