Page 20 - Curvature and Homology

2 I. RIEMANNIAN MANIFOLDS

defined over it is continuous or not. A discussion of the properties of
the classical surfaces in differential geometry requires more than
continuity, however, for the functions considered. By a regular closed
surface S in'P is meant an ordered pair {So, X} consisting of a topological
space So and a differentiable map X of So into I!?. As a topological space,
So is to be a separable, Hausdorff space with the further propetties:
(i) So is compact (that is X(So) is closed and bounded);
(ii) So is connected (a topological space is said to be connected if it
cannot be expressed as the union of two non-empty disjoint open
subsets) ;
(iii) Each point of So has an open neighborhood homeomorphic
with EZ: The map X : P -+ (x (P), y(P), z(P)), P E So where x(P), y(P)
and z(P) are differentiable functions is to have rank 2 at each point
P E SO, that is the matrix

of partial derivatives must be of rank 2 where u, u are local parameters
at P. Let U and V be any two open neighborhoods of P homeomorphic
with E2 and with non-empty intersection. Then, their local parameters
or coordinates (cf. definition given below of a differentiable structure)
must be related by differentiable functions with non-vanishing Jacobian.
It follows that the rank of X is invariant with respect to a change of
coordinates.
That a certain amount of differentiability is necessary is clear from
several points of view. In the first place, the condition on the rank of X
implies the existence of a tangent plane at each point of the surface.
Moreover, only those local parameters are "allowed" which are related
by differentiable functions.
A regular closed surface is but a special case of a more general concept
which we proceed to define.
Roughly speaking, a differentiable manifold is a topological space in
which the concept of derivative has a meaning. Locally, the space is to
behave like Euclidean space. But first, a topological space M is said to be
separable if it contains a countable basis for its topology. It is called a
Hausdorff space if to any two points of M there are disjoint open sets each
containing ixactly one of the points.
A separable Hausdorff space M of dimension n is said to have a
dtj'kmtiable structure of class k > 0 if it has the following properties:
(i) Each point of M has an open neighborhood homeomorphic with
an open subset in Rrr the (number) space of n real variables, that is,