Lecture 3 27
Before doing so, we would like to fix a linguistic ambiguity; for
example, what should the word ‘sphere’ mean? How will we indicate
whether we are treating a particular surface as a geometric object,
or as a topological one (that is, one which may be deformed without
changing the nature of the surface)? Our convention will be as follows:
an indefinite article in front of the name, as in ‘a sphere’, ‘a torus’, or
‘a projective plane’, will mean that we consider the object in the topo-
logical sense, up to a homeomorphism. The use of an adjective or the
definite article will generally signify a smaller class of objects, as in ‘a
sphere given by an equation’. Then ‘a round sphere’ would mean any
sphere which has ‘spherical geometry’, that is, which is isometric to
the actual sphere in Euclidean space. Similarly, ‘a flat torus’ signifies
any torus with locally Euclidean geometry, while ‘the flat torus’ or
‘the torus’ will indicate the unit square with opposite sides identified,
endowed with the appropriate geometry inherited from
we will call this object ‘the standard flat torus’. ‘The elliptic plane’
indicates the factor space of the unit sphere in which antipodal points
are identified, with geometry inherited from the sphere, and so on for
various other examples which will arise.
Exercise 1.13. Write parametric representations for a projective
plane in each of the following:
(1) R3 (with self-intersections).
e. Regularity conditions for parametrically defined surfaces.
A parametrisation of a surface in R3 is given by a region U ⊂ R2
with coordinates (t, s) ∈ U and a set of three maps f1, f2, f3; the
surface is then the image of F = (f1, f2, f3), the set of all points
(x, y, z) = (f1(t, s), f2(t, s), f3(t, s)).
As with parametric representations of curves, we need a regular-
ity condition to ensure that our surface is in fact smooth, without
cusps or singularities. We once again require that the functions fi be
continuously differentiable, but now it is insuﬃcient to simply require
that the matrix of derivatives Df be non-zero. Rather, we require