Riemannian Geometry | Elementary Differential Geometry by Barrett O'Neil | 1-Forms Exercise

Differential geometry explores how curves, surfaces, and higher-dimensional spaces bend and behave by applying ideas from calculus and linear algebra. Instead of treating shapes as fixed objects, it focuses on their local properties—tangent directions, curvature, and metrics—which describe how distance and direction change across a space. These tools allow us to analyze both simple surfaces and highly curved geometric structures. Its power lies in linking small-scale behavior to large-scale stru