Page 22 - Curvature and Homology

4 I. RIEMANNIAN MANIFOLDS

submanifold (cf. $IS)] of E~~. this structure (as an analytic
With
manifold), GL(n, R) is a Lie group (cf. $3.6).
Let f be a real-valued continuous function defined in an open subset
S of M. Let P be a point of S and Ua a coordinate neighborhood
containing P. Then, in S n Ua, f can be expressed as a function of the
local coordinates ul, ..., un in Ua. (If xl, .. ., xn are the n coordinate func-
tions on Rn, then ui(P) = xi(ua(P)), i = 1, ..., n and we may write
ut = xi u,). The function f is said to be diflerentiable at P if f(ul, ..., un)
possesses all first partial derivatives at P. The partial derivative of f
with respect to ui at P is defined as

This property is evidently independent of the choice of Ua. The function f
is called diflerentiable in S, if it is differentiable at every point of S.
Moreover, f is of the form g ua if the domain is restricted to S n Ua
where g is a continuous function in ua(S n U,) c Rn. Two differentiable
structures are said to be equivalent if they give rise to the same family
of differentiable functions over open subsets of M. This is an equivalence
relation. The family of functions of class k determines the differentiable
structures in the equivalence class.
A topological manifold M together with an equivalence class of
differentiable structures on M is called a dzgerentiable manqold. It has
recently been shown that not every topological manifold can be given
a differentiable structure [44]. On the other hand, a topological manifold
may carry differentiable structures belonging to distinct equivalence
classes. Indeed, the 7-dimensional sphere possesses several inequivalent
differentiable structures [60].
A differentiable mapping f of an open subset S of Rn into Rn is called
sense-preserving if the Jacobian of the map is positive in S. If, for any
pair of coordinate neighborhoods with non-empty intersection, the
mapping usu;l is sense-preserving, the differentiable structure is said to
be oriented and, in this case, the differentiable manifold is called orientable.
Thus, if fs,(x) denotes the Jacobian of the map uauil at xi(ua(P)),
i = 1, ..., n, then fYB(x) fSa(x) = fYa(x), P E Ua n Us n U,,.
The 2-sphere in E3 is an orientable manifold whereas the real
projective plane (the set of lines through the origin in E3) is not
(cf. I.B. 2).
Let M be a differentiable manifold of class k and S an open subset of
M. By restricting the functions (of class k) on M to S, the differentiable
structure so obtained on S is called an induced structure of class k.
In particular, on every open subset of El there is an induced structure