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10 XVI DIFFERENTIAL MANIFOLDS

that obtained by transporting the structure on X by means of w. Two differ-
ential manifolds X, Y are said to be diffeomorphic if there exists a diffeo-
morphism of X onto Y.

Remarks

(16.2.7) Consider the real line R, endowed with its canonical structure of
differential manifold (16.2.2), and let u be the real-valued function such that
u(t) = t for t ^ 0, and u(t) = 2t for t ^ 0. It is clear that u is a homeomor-
phism of R onto itself (4.2.2), and we may therefore endow R with the struc-
ture of differential manifold defined by the single chart u(c), where c =
(R, 1_R, 1) is the single chart defining the canonical structure. Since u is not
differentiate at the point t — 0, the charts c and u(c) are not compatible.
If Xj and X₂ are the differential manifolds defined on the underlying space
R by c and u(c), respectively, then u is a diffeomorphism of Xj onto X₂,
and we have therefore defined on R two distinct (but isomorphic) structures
of differential manifold. In other words, the identity mapping 1_R is not a
difFeomorphism of Xj onto X₂.

It can be shown that, for certain values of n ^ 7, there exist on the topo-
logical space S_n several nonisomorphic structures of differential manifold,
having the same underlying topology.

It can also be shown that the only connected differential manifolds of
dimension 1 are (up to diffeomorphism) R and S_x (Problem 6).

(16.2.8) Let X be a differential manifold, X' a set, «:X-»X' a bijection
of X onto X'. We can begin by transporting the topology of X to X' by means
of w, by defining the open sets in X' to be the images under u of the open sets
in X. Since u then becomes a homeomorphism of X onto X' we can transport
to X' (again by means of u) the structure of differential manifold on X, as
explained in (16,2.6),

PROBLEMS

1. Show that the space T" = R"/Z" (12.11) is endowed with a structure of real-analytic
manifold for which there exists an atlas of n + 1 charts whose images are translations in
R" of the open cube P, where I « ]0,1[ (cf. (16.10.6)).

2. (a) Let K be a compact subset of R" and B a closed ball whose interior contains K.
Show that for each e > 0 there exists a homeomorphism / of R" onto itself, such that
f(x) = x for all x $ B and such that the diameter of/(K) is ^'e. (We may assume that
0 is the center of B; take/to be of the form/(jc) = x(p(\\x\\\ where cp is a suitably chosen
real-valued function.)



	10 XVI DIFFERENTIAL MANIFOLDS that obtained by transporting the structure on X by means of w. Two differ- ential manifolds X, Y are said to be diffeomorphic if there exists a diffeo- morphism of X onto Y. Remarks (16.2.7) Consider the real line R, endowed with its canonical structure of differential manifold (16.2.2), and let u be the real-valued function such that u(t) = t for t ^ 0, and u(t) = 2t for t ^ 0. It is clear that u is a homeomor- phism of R onto itself (4.2.2), and we may therefore endow R with the struc- ture of differential manifold defined by the single chart u(c), where c = (R, 1_R, 1) is the single chart defining the canonical structure. Since u is not differentiate at the point t — 0, the charts c and u(c) are not compatible. If Xj and X₂ are the differential manifolds defined on the underlying space R by c and u(c), respectively, then u is a diffeomorphism of Xj onto X₂, and we have therefore defined on R two distinct (but isomorphic) structures of differential manifold. In other words, the identity mapping 1_R is not a difFeomorphism of Xj onto X₂. It can be shown that, for certain values of n ^ 7, there exist on the topo- logical space S_n several nonisomorphic structures of differential manifold, having the same underlying topology. It can also be shown that the only connected differential manifolds of dimension 1 are (up to diffeomorphism) R and S_x (Problem 6). (16.2.8) Let X be a differential manifold, X' a set, «:X-»X' a bijection of X onto X'. We can begin by transporting the topology of X to X' by means of w, by defining the open sets in X' to be the images under u of the open sets in X. Since u then becomes a homeomorphism of X onto X' we can transport to X' (again by means of u) the structure of differential manifold on X, as explained in (16,2.6),

	PROBLEMS 1. Show that the space T" = R"/Z" (12.11) is endowed with a structure of real-analytic manifold for which there exists an atlas of n + 1 charts whose images are translations in R" of the open cube P, where I « ]0,1[ (cf. (16.10.6)). 2. (a) Let K be a compact subset of R" and B a closed ball whose interior contains K. Show that for each e > 0 there exists a homeomorphism / of R" onto itself, such that f(x) = x for all x $ B and such that the diameter of/(K) is ^'e. (We may assume that 0 is the center of B; take/to be of the form/(jc) = x(p(\\x\\\ where cp is a suitably chosen real-valued function.)