242
IX – Multivariate Differential and Integral Calculus
(iii) One-Parameter Subgroups of a Torus are the classical examples of
immersions that are not necessarily open.
59 To see this, take X = R and
for Y , take the“ torus ” T
2 , where T is the unit circle of C. It is a compact
submanifold of C
2 and, at the same time, is a (multiplicative) group which
plays the same role for periodic functions with two variable as T in Chapter
VII and leads to the same theory; the reader will invent it without difficulty
and will even be able to generalize it to n variables. . . As always, setting
e(t) = exp(2πit) ,
the map
f (t) = (e(at), e(bt)) ,
(13.7)
where a, b ∈ R are given and both non-zero, is both a homomorphism of
the additive group R to the additive group T
2 and an immersion since its
derivative
f
(t) = (2πiae(at), 2πibe(bt))
is never zero. To deduce that the “ one-parameter subgroup ” Z = f (R) is
a 1-dimensional submanifold of T
2 , we need to make sure that f is an open
map from R onto its image equipped with the topology of T
2 . As will be
shown, that is the case if and only if the ratio a/b is rational. Otherwise, f (R)
is everywhere dense in T
2 , is neither locally compact nor locally connected
with respect to the topology of T
2 and so is not a submanifold of T
2 .
Let us first investigate the case when a/b is rational. Multiplying a and b
by a convenient real number, a and b can be supposed to be coprime integers.
Then there are integers u and v such that au + bv = 1 (Bezout’s theorem,
who before the Revolution was the author of a famous Mathematics Course
used in artillery schools that were predecessors of the ´
Ecole Polytechnique).
Then map (7) has period 1. Moreover, relation f (t) = (1, 1), where this is
the identity element of the group T
2 , requires at ∈ Z and bt ∈ Z, so that
t = t(au + bv) ∈ Z. As f is a homomorphism from the additive group R to
the multiplicative group T
2 , it follows that
f (t) = f (t
) ⇐⇒ t − t
∈ Z .
(13.8)
Hence f (R) = f (I), where I = [0, 1], so thatf (R) is compact and in particular
closed in T
2 . But (8) shows that f is obtained by composing the map R −→
R/Z with the map ϕ from R/Z to T
2 , which is obviously continuous and
injective. As the space R/Z is compact, a generalization of theorem 12 of
Chap. III shows that ϕ is a homeomorphism from R/Z onto its image f (R):
59 Since the results of this section are not used in the rest of this Chapter, it can
be skipped in the immediate.
IX – Multivariate Differential and Integral Calculus
(iii) One-Parameter Subgroups of a Torus are the classical examples of
immersions that are not necessarily open.
59 To see this, take X = R and
for Y , take the“ torus ” T
2 , where T is the unit circle of C. It is a compact
submanifold of C
2 and, at the same time, is a (multiplicative) group which
plays the same role for periodic functions with two variable as T in Chapter
VII and leads to the same theory; the reader will invent it without difficulty
and will even be able to generalize it to n variables. . . As always, setting
e(t) = exp(2πit) ,
the map
f (t) = (e(at), e(bt)) ,
(13.7)
where a, b ∈ R are given and both non-zero, is both a homomorphism of
the additive group R to the additive group T
2 and an immersion since its
derivative
f
(t) = (2πiae(at), 2πibe(bt))
is never zero. To deduce that the “ one-parameter subgroup ” Z = f (R) is
a 1-dimensional submanifold of T
2 , we need to make sure that f is an open
map from R onto its image equipped with the topology of T
2 . As will be
shown, that is the case if and only if the ratio a/b is rational. Otherwise, f (R)
is everywhere dense in T
2 , is neither locally compact nor locally connected
with respect to the topology of T
2 and so is not a submanifold of T
2 .
Let us first investigate the case when a/b is rational. Multiplying a and b
by a convenient real number, a and b can be supposed to be coprime integers.
Then there are integers u and v such that au + bv = 1 (Bezout’s theorem,
who before the Revolution was the author of a famous Mathematics Course
used in artillery schools that were predecessors of the ´
Ecole Polytechnique).
Then map (7) has period 1. Moreover, relation f (t) = (1, 1), where this is
the identity element of the group T
2 , requires at ∈ Z and bt ∈ Z, so that
t = t(au + bv) ∈ Z. As f is a homomorphism from the additive group R to
the multiplicative group T
2 , it follows that
f (t) = f (t
) ⇐⇒ t − t
∈ Z .
(13.8)
Hence f (R) = f (I), where I = [0, 1], so thatf (R) is compact and in particular
closed in T
2 . But (8) shows that f is obtained by composing the map R −→
R/Z with the map ϕ from R/Z to T
2 , which is obviously continuous and
injective. As the space R/Z is compact, a generalization of theorem 12 of
Chap. III shows that ϕ is a homeomorphism from R/Z onto its image f (R):
59 Since the results of this section are not used in the rest of this Chapter, it can
be skipped in the immediate.