Chapter 1

Classical

Differential Geometry

In this chapter we discuss some classical results of the differential geome-

try of nets (parametrized surfaces and coordinate systems) in

RN

, mainly

concentrated around the topics of transformations of nets and of their per-

mutability properties. This classical area was very popular in the differential

geometry of the 19th and of the first quarter of the 20th century, and is well

documented in the fundamental treatises by Bianchi, Darboux, Eisenhart

and others. Our presentation mainly follows these classical treatments, of

course with modifications which reflect our present points of view. We do

not trace back the exact origin of the concrete classical results: often enough

this turns out to be a complicated task in the history of mathematics, which

still waits for its competent investigation.

For the classes of nets described by essentially two-dimensional systems

(special classes of surfaces such as surfaces with a constant negative Gaussian

curvature or isothermic surfaces), the permutability theorems, mainly due

to Bianchi, are dealing with a quadruple of surfaces (depicted as vertices of

a so-called Bianchi quadrilateral). Given three surfaces of such a quadruple,

the fourth one is uniquely defined; see Theorems 1.27 and 1.31.

For the classes of nets described by essentially three-dimensional systems

(conjugate nets; Moutard nets; asymptotic line parametrized surfaces; or-

thogonal nets, including curvature line parametrized surfaces), the situation

is somewhat different. The corresponding permutability theorems (Theo-

rems 1.3, 1.10, 1.15, and 1.20) consist of two parts. The first part of each

theorem presents the traditional view and deals with Bianchi quadrilater-

als. In our opinion, this is not the proper setting in the three-dimensional

1

http://dx.doi.org/10.1090/gsm/098/01