70
4 Examples of Classical Mechanical Projections
m
ϕ
0 be a complementary subspace in g N to k
ϕ
0 .
Since the mapping P P
ϕ
+ restricted to
O
m
ϕ+ is smooth and of constant rank m ϕ , it is a
subimmersion (compare [51, 5.10.6.]), hence there is a manifold Z
m
ϕ+ of dimension
m ϕ and a submersion s
ϕ
+ :
O
m
ϕ+ → Z
m
ϕ+ as well as an immersion i
ϕ
+ : Z
m
ϕ+ → P(H
+
N )
such, that
P P
ϕ
+ = i
ϕ
+ ◦ s
ϕ
+ on
O
m
ϕ+ .
(4.3.26a)
This means, that the image P P
ϕ
+ (
O
m
ϕ+ ) ⊂ P(H
+
N ) can be considered as an
immersed submanifold (with possible selfintersections) of P(H
+
N ):
P P
ϕ
+ (
O
m
ϕ+ ) = i
ϕ
+ (Z
m
ϕ+ ).
(4.3.26b)
A basis of the tangent space to Z
m
ϕ+ is generated in the point ν := s
ϕ
+ (ϕ) by curves
t → s
ϕ
+ (exp(−it X ξ )ϕ with ξ ∈ m
ϕ
0 . The image by T ν i
ϕ
+ of this tangent space in
T ϕ
(+) P(H
+
N ) is generated by vectors which, in the chart ϕ (+) (see 2.1.5, 2.1.8), have
the form
T ϕ
(+) ϕ
(+) (v ξ ) := −i (I − P ϕ
(+) )P + X ξ ϕ, ξ ∈ m
ϕ
0 .
(4.3.27)
The values of the symplectic form on P(H N ) on these vectors are:
ϕ
(+) (v η , v ξ ) = −2ϕ
(+)
−2 Im(P + X η ϕ, (I − P ϕ
(+) )P + X ξ ϕ).
(4.3.28)
The pull-back of by i
ϕ
+ makes Z
m
ϕ+ a manifold endowed with a canonical twoform. It is known, that the factorization of the subimmersion P P
ϕ
+ (together with
the choice of the manifold Z
m
ϕ+ ) can be chosen in a canonical way, see [51, 5.10.7].
We assume here, that the mapping s
ϕ
+ is onto (i.e. surjective), what is possible,
because any submersion is an open mapping. The form i
ϕ∗
+ on Z
m
ϕ+ is closed.
The subset of Z
m
ϕ+ on which the form i
ϕ∗
+ has its maximal rank is an open set,
hence a submanifold Z ϕ+ of Z
m
ϕ+ . Denote by
◦
+ the restriction of i
ϕ∗
+ onto Z ϕ+ .
Since d
◦
+ = 0, the characteristic bundle of
◦
+ (consisting of vector fields on Z ϕ+
annihilating the form
◦
+ ) is an integrable subbundle of T Z ϕ+ , see e.g. [1, 5.1.2],
determining a natural foliation of Z ϕ+ ; any leaf of this foliation is an immersed
connected submanifold of Z ϕ+ . Let M
+
ϕ be the factor space obtained from Z ϕ+ by
its decomposition into the leaves of this foliation and let p
+
M : Z ϕ+ → M
+
ϕ be the
natural projection. If the equivalence relation on Z ϕ+ given by classes identical with
leaves [ p
+
M ]
−1
(x) (x ∈ M
+
ϕ ) is regular (see [51, 5.9.5]), then there is unique manifold
structure on M
+
ϕ such that p
+
M is a submersion. In this case there is, on the malnifold
M
+
ϕ , a unique symplectic form
M
+ satisfying
4 Examples of Classical Mechanical Projections
m
ϕ
0 be a complementary subspace in g N to k
ϕ
0 .
Since the mapping P P
ϕ
+ restricted to
O
m
ϕ+ is smooth and of constant rank m ϕ , it is a
subimmersion (compare [51, 5.10.6.]), hence there is a manifold Z
m
ϕ+ of dimension
m ϕ and a submersion s
ϕ
+ :
O
m
ϕ+ → Z
m
ϕ+ as well as an immersion i
ϕ
+ : Z
m
ϕ+ → P(H
+
N )
such, that
P P
ϕ
+ = i
ϕ
+ ◦ s
ϕ
+ on
O
m
ϕ+ .
(4.3.26a)
This means, that the image P P
ϕ
+ (
O
m
ϕ+ ) ⊂ P(H
+
N ) can be considered as an
immersed submanifold (with possible selfintersections) of P(H
+
N ):
P P
ϕ
+ (
O
m
ϕ+ ) = i
ϕ
+ (Z
m
ϕ+ ).
(4.3.26b)
A basis of the tangent space to Z
m
ϕ+ is generated in the point ν := s
ϕ
+ (ϕ) by curves
t → s
ϕ
+ (exp(−it X ξ )ϕ with ξ ∈ m
ϕ
0 . The image by T ν i
ϕ
+ of this tangent space in
T ϕ
(+) P(H
+
N ) is generated by vectors which, in the chart ϕ (+) (see 2.1.5, 2.1.8), have
the form
T ϕ
(+) ϕ
(+) (v ξ ) := −i (I − P ϕ
(+) )P + X ξ ϕ, ξ ∈ m
ϕ
0 .
(4.3.27)
The values of the symplectic form on P(H N ) on these vectors are:
ϕ
(+) (v η , v ξ ) = −2ϕ
(+)
−2 Im(P + X η ϕ, (I − P ϕ
(+) )P + X ξ ϕ).
(4.3.28)
The pull-back of by i
ϕ
+ makes Z
m
ϕ+ a manifold endowed with a canonical twoform. It is known, that the factorization of the subimmersion P P
ϕ
+ (together with
the choice of the manifold Z
m
ϕ+ ) can be chosen in a canonical way, see [51, 5.10.7].
We assume here, that the mapping s
ϕ
+ is onto (i.e. surjective), what is possible,
because any submersion is an open mapping. The form i
ϕ∗
+ on Z
m
ϕ+ is closed.
The subset of Z
m
ϕ+ on which the form i
ϕ∗
+ has its maximal rank is an open set,
hence a submanifold Z ϕ+ of Z
m
ϕ+ . Denote by
◦
+ the restriction of i
ϕ∗
+ onto Z ϕ+ .
Since d
◦
+ = 0, the characteristic bundle of
◦
+ (consisting of vector fields on Z ϕ+
annihilating the form
◦
+ ) is an integrable subbundle of T Z ϕ+ , see e.g. [1, 5.1.2],
determining a natural foliation of Z ϕ+ ; any leaf of this foliation is an immersed
connected submanifold of Z ϕ+ . Let M
+
ϕ be the factor space obtained from Z ϕ+ by
its decomposition into the leaves of this foliation and let p
+
M : Z ϕ+ → M
+
ϕ be the
natural projection. If the equivalence relation on Z ϕ+ given by classes identical with
leaves [ p
+
M ]
−1
(x) (x ∈ M
+
ϕ ) is regular (see [51, 5.9.5]), then there is unique manifold
structure on M
+
ϕ such that p
+
M is a submersion. In this case there is, on the malnifold
M
+
ϕ , a unique symplectic form
M
+ satisfying