I don’t understand modern abstract differential geometry, but the elementary theory of curves and surfaces in $\mathbb{R}^3$, as expounded by Darboux and others, is very useful in engineering and manufacturing. A few examples:
 1. The curvature properties of a surface determine how light reflects from it. These reflections are what determine the aesthetic qualities of car bodies and consumer products.
 2. The Gaussian curvature of a surface determines how easily it can be made by deforming flat stock material (as in stamping of sheet metal). In particular, if the Gaussian curvature is zero, the surface is developable, and manufacturing is easy.
 3. The curvature of a curve (and derivatives of curvature) determines the forces that will be experienced by anything moving along that curve. This is important in the design of mechanisms, and in the layout of roads and rail tracks.
 4. When curved surfaces like aircraft wings are made by layup of carbon fiber tape, the tape will want to follow geodesic curves on the surface. It’s important to know whether these curves are nearly parallel or wildly divergent.
 5. When you’re making a surface by 5-axis CNC milling, the end of the cutting tool is typically a toroidal shape. By tilting the axis of the tool, you can arrange for the toroidal shape to closely match the surface you’re trying to produce. The right amount of tilt depends on the curvature of the surface.