And then came the curvature tensor. Not Riemann's original, messy component form, but the clean, coordinate-free definition: For vector fields ( X, Y, Z ),
A year later, Nina defended her PhD. Her thesis was on "A Coordinate-Free Approach to Perturbative Gravity," and the first sentence of the introduction read: "We will not start with physics. We will start with geometry." Her committee, including her grumpy advisor, passed her unanimously. frederic schuller lecture notes pdf
"These lecture notes were transcribed by students," it read. "Errors are their own. Clarity is mine. If you find a mistake, prove it. If you find a better way, write your own notes. The cathedral of knowledge is never complete. You are the next stonemason." And then came the curvature tensor