§ 4. Differential Manifolds
235
13 – Submanifolds and Subimmersions
The most immediately obvious manifolds are certain subsets X of a Cartesian
space E which, for simplicity’s sake, we will often suppose to be R
n . There
are several equivalent and equally important methods to equip them with a
natural differentiable structure;
56 these methods were conceived in the 17th
−1
1
Fig. 13.7.
56 It should not be thought that all possible methods for defining a manifold structure on a given set, however familiar they may be, lead to the same result. First,
a method to construct two different, though isomorphic, differentiable structures
on a manifold X involves choosing a non-differentiable homeomorphism σ from
X onto X and to declare that the differentiable functions for the second structure are those obtained by composing σ with a differentiable function for the first
one. This procedure, which can, to start with, be applied in R, being in everyone’s reach, consider equivalent two such manifold structures. The real question
is whether there are others. Some experts in algebraic topology (M. Kervaire
and J. Milnor, Annals of Math., 77, 1963) have calculated the number νn, which
happens to be finite, of non-equivalent C
∞ structures that can be defined on the
unit sphere of R
n :
n : ≤ 6
7
8 9 10
11
12 13 14
15
16 17
νn : 1
28 2 9
6
992
1
3
2
16256
2
16
.
Others have explicitly described some of these bizarre structures that were unknown. C
∞ structures essentially different from those of everyone else can also
be defined on R
4 (but not on R
n with n ≤ 3). Finally, there are topological
manifolds, i.e of class C
0 , on which no C
1 structure can be defined. Naive ideas
are sometimes incorrect.
235
13 – Submanifolds and Subimmersions
The most immediately obvious manifolds are certain subsets X of a Cartesian
space E which, for simplicity’s sake, we will often suppose to be R
n . There
are several equivalent and equally important methods to equip them with a
natural differentiable structure;
56 these methods were conceived in the 17th
−1
1
Fig. 13.7.
56 It should not be thought that all possible methods for defining a manifold structure on a given set, however familiar they may be, lead to the same result. First,
a method to construct two different, though isomorphic, differentiable structures
on a manifold X involves choosing a non-differentiable homeomorphism σ from
X onto X and to declare that the differentiable functions for the second structure are those obtained by composing σ with a differentiable function for the first
one. This procedure, which can, to start with, be applied in R, being in everyone’s reach, consider equivalent two such manifold structures. The real question
is whether there are others. Some experts in algebraic topology (M. Kervaire
and J. Milnor, Annals of Math., 77, 1963) have calculated the number νn, which
happens to be finite, of non-equivalent C
∞ structures that can be defined on the
unit sphere of R
n :
n : ≤ 6
7
8 9 10
11
12 13 14
15
16 17
νn : 1
28 2 9
6
992
1
3
2
16256
2
16
.
Others have explicitly described some of these bizarre structures that were unknown. C
∞ structures essentially different from those of everyone else can also
be defined on R
4 (but not on R
n with n ≤ 3). Finally, there are topological
manifolds, i.e of class C
0 , on which no C
1 structure can be defined. Naive ideas
are sometimes incorrect.