Page 21 - Curvature and Homology

1.1. DIFFERENTIABLE MANIFOLDS 3
there is a finite or countable open covering {U,} and, for each a a homeo-
morphism u, : U, -+ Rn of U, onto an open subset in Rn;
(ii) For any two open sets U, and Up with non-empty intersection
the map usu;;l : ua(Ua n Us) -+ Rn is of class k (that is, it possesses
continuous derivatives of order k) with non-vanishing Jacobian.
The functions defining u, are called local coordinates in U,. Clearly,
one may also speak of structures of class c;o (that is, structures of class k
for every positive integer k) and analytic structures (that is, every map
uBu;' is expressible as a convergent power series in the n variables). The
local coordinates constitute an essential tool in the study of M. However,
the geometrical properties should be independent of the choice of local
coordinates.
The space M with the property (i) will be called a topological mani-
fold. We shall generally assume that the spaces considered are connected
although many of the results are independent of this hypothesis.
Examples: 1. The Euclidean space En is perhaps the simplest example
of a topological manifold with a differentiable structure. The identity
map I in En together with the unit covering (Rn, I) is its natural differen-
tiable structure: (U,, u,) = (Rn, I).
2. The (n - 1)-dimensional sphere in En defined by the equation

It can be covered by 2n coordinate neighborhoods defined by xi > 0
and xi < 0 (i = 1, ..., n).
3. The general linear group: Let V be a vector space over R (the real
numbers) of dimension n and let (el, ..., en) be a basis of V. The group
of all linear automorphisms a of V may be expressed as the group of all
non-singular matrices (a:) ;

called the general linear group and denoted by GL(n, R). We shall also
denote it by GL(V) when dealing with more than one vector space.
(The Einstein summation convention where repeated indices implies
addition has been employed in formula (1.1.2) and, in the sequel we
shall adhere to this notation.) The multiplication law is

GL(n, R) may be considered as an open set [and hence as an open