Blaga P. Lectures on the differential geometry of - tiera.ru

4.3. The equivalence of local parameterizations 119
Figure 4.1: The torus
4.3 The equivalence of local parameterizations
Definition. Let S be a surface, (U, r) – a local parameterization and W = r(U). Then the
map r −1 : W → U is a bijection, associating to each point of W a pair of real numbers
(u, v) ∈ U. This correspondence is called a curvilinear coordinate system on S or a
chart on S .
Remark. We ought to mention here that in many books the fundamental objects used
to describe a surface are not the local parameterizations, but the charts. Of course,
the two approaches are completely equivalent, but they came from different directions.
The description of surfaces by using local parameterizations has, probably, the origin in
mathematical analysis, where the surfaces are thought off either as images of functions
or as graphs of functions, in any case, the central objects is the function. The “charts’s
approach” has the origin in chartography. In fact, assigning a local chart to a surface
simply means to apply that surface on a portion of a plane, in other words, what is meant
is the construction of a “map” of the surface and, actually, in the advanced differential
geometry, a collection of charts that cover the entire surface is called an atlas, exactly as
happens in chartography.
Theorem 4.3.1. (of the parameters’ change) Let (U, r) and (U1, r1) be two local parameterizations
of a surface S and r(U) = r(U1). Then there is a diffeomorphism λ : U → U1
such that r = r1 ◦ λ. The diffeomorphism λ is called a parameters’ change.
Before proving the theorem, some remarks are in order. If a change of parameters
λ does exist, then, from the relation r = r1 ◦ λ, we should have, of course, λ = r−1 1 ◦ r.
The real difficulty is to show that λ and its inverse are smooth. Although r is smooth,
r−1 is not1 , because its domain, r1(U1), is not an open subset of the Euclidean space R3 .
1 not in the classical sense, at least