ABSTRACT
The term curvature simply refers to the amount of bending of a
curve or a surface. In simple terms, curvature at a point indicates
how quickly a curve or a surface bends away from its local tangent
line or tangent plane, respectively. While tangent planes are useful
for studying first-order behavior of surfaces, curvatures deal with
second-order behavior. However, a rigorous study of curvatures
can be quite involved, and in general, there are different notions
of curvatures depending on the context. The interested reader is
referred to the comprehensive survey of curvature in the classical
differential geometry text by Do Carmo [dC76]. In this chapter,
we will explain the basic ideas, develop some intuitions, and study
the necessary formulations that allow us to better understand the
notion of curvature and help us to compute various types of cur-
vatures.
