In this chapter, we are interested in measuring distances, angles and areas on manifolds.
For an embedded surface S in R3, the tool that is used to perform such measurements is the dot product
u · v = u1v1 + u2v2 + u3v3,
where u = (u1, u2, u3) and v = (v1, v2, v3) are vectors in R 3. The com-
ponents are expressed with respect to the standard basis of R3. The length of a vector u is given by ‖u‖ = √u · u.