What is curvature in differential geometry?

Answer 1

There are two most important types of curvature: extrinsic curvature and intrinsic curvature.

The extrinsic curvature of curves in two- and three-space was the first type of curvature to be studied historically, culminating in the Frenet formulas, which describe a space curve entirely in terms of its "curvature," torsion, and the initial starting point and direction.

There is also a curvature of surfaces in three-space. The main curvatures that emerged from this scrutiny are the mean curvature, Gaussian curvature, and the shape operator.

I advice to read the following article:

http://mathworld.wolfram.com/Curvature.html

Moreover, I advise add-on for Mathematica CAS, which do calculations in differential geometry.

http://digi-area.com/Mathematica/atlas

There is a tutorial about the invariants including curvature which calculates for curves and surfaces.

http://digi-area.com/Mathematica/atlas/ref/Invariants.php