Composite bundles
Y
→
Σ
→
X
play a prominent role in gauge theory with symmetry breaking, e.g., gauge gravitation theory, non-autonomous mechanics where
X
=
R
is the time axis, e.g., mechanics with time-dependent parameters, and so on. There are the important relations between connections on fiber bundles
Y
→
X
,
Y
→
Σ
and
Σ
→
X
.
In differential geometry by a composite bundle is meant the composition
π
:
Y
→
Σ
→
X
(
1
)
of fiber bundles
π
Y
Σ
:
Y
→
Σ
,
π
Σ
X
:
Σ
→
X
.
It is provided with bundle coordinates
(
x
λ
,
σ
m
,
y
i
)
, where
(
x
λ
,
σ
m
)
are bundle coordinates on a fiber bundle
Σ
→
X
, i.e., transition functions of coordinates
σ
m
are independent of coordinates
y
i
.
The following fact provides the above mentioned physical applications of composite bundles. Given the composite bundle (1), let
h
be a global section of a fiber bundle
Σ
→
X
, if any. Then the pullback bundle
Y
h
=
h
∗
Y
over
X
is a subbundle of a fiber bundle
Y
→
X
.
For instance, let
P
→
X
be a principal bundle with a structure Lie group
G
which is reducible to its closed subgroup
H
. There is a composite bundle
P
→
P
/
H
→
X
where
P
→
P
/
H
is a principal bundle with a structure group
H
and
P
/
H
→
X
is a fiber bundle associated with
P
→
X
. Given a global section
h
of
P
/
H
→
X
, the pullback bundle
h
∗
P
is a reduced principal subbundle of
P
with a structure group
H
. In gauge theory, sections of
P
/
H
→
X
are treated as classical Higgs fields.
Given the composite bundle
Y
→
Σ
→
X
(1), let us consider the jet manifolds
J
1
Σ
,
J
Σ
1
Y
, and
J
1
Y
of the fiber bundles
Σ
→
X
,
Y
→
Σ
, and
Y
→
X
, respectively. They are provided with the adapted coordinates
(
x
λ
,
σ
m
,
σ
λ
m
)
,
(
x
λ
,
σ
m
,
y
i
,
y
^
λ
i
,
y
m
i
)
,
, and
(
x
λ
,
σ
m
,
y
i
,
σ
λ
m
,
y
λ
i
)
.
There is the canonical map
J
1
Σ
×
Σ
J
Σ
1
Y
→
Y
J
1
Y
,
y
λ
i
=
y
m
i
σ
λ
m
+
y
^
λ
i
.
This canonical map defines the relations between connections on fiber bundles
Y
→
X
,
Y
→
Σ
and
Σ
→
X
. These connections are given by the corresponding tangent-valued connection forms
γ
=
d
x
λ
⊗
(
∂
λ
+
γ
λ
m
∂
m
+
γ
λ
i
∂
i
)
,
A
Σ
=
d
x
λ
⊗
(
∂
λ
+
A
λ
i
∂
i
)
+
d
σ
m
⊗
(
∂
m
+
A
m
i
∂
i
)
,
Γ
=
d
x
λ
⊗
(
∂
λ
+
Γ
λ
m
∂
m
)
.
A connection
A
Σ
on a fiber bundle
Y
→
Σ
and a connection
Γ
on a fiber bundle
Σ
→
X
define a connection
γ
=
d
x
λ
⊗
(
∂
λ
+
Γ
λ
m
∂
m
+
(
A
λ
i
+
A
m
i
Γ
λ
m
)
∂
i
)
on a composite bundle
Y
→
X
. It is called the composite connection. This is a unique connection such that the horizontal lift
γ
τ
onto
Y
of a vector field
τ
on
X
by means of the composite connection
γ
coincides with the composition
A
Σ
(
Γ
τ
)
of horizontal lifts of
τ
onto
Σ
by means of a connection
Γ
and then onto
Y
by means of a connection
A
Σ
.
Given the composite bundle
Y
(1), there is the following exact sequence of vector bundles over
Y
:
0
→
V
Σ
Y
→
V
Y
→
Y
×
Σ
V
Σ
→
0
,
(
2
)
where
V
Σ
Y
and
V
Σ
∗
Y
are the vertical tangent bundle and the vertical cotangent bundle of
Y
→
Σ
. Every connection
A
Σ
on a fiber bundle
Y
→
Σ
yields the splitting
A
Σ
:
T
Y
⊃
V
Y
∋
y
˙
i
∂
i
+
σ
˙
m
∂
m
→
(
y
˙
i
−
A
m
i
σ
˙
m
)
∂
i
of the exact sequence (2). Using this splitting, one can construct a first order differential operator
D
~
:
J
1
Y
→
T
∗
X
⊗
Y
V
Σ
Y
,
D
~
=
d
x
λ
⊗
(
y
λ
i
−
A
λ
i
−
A
m
i
σ
λ
m
)
∂
i
,
on a composite bundle
Y
→
X
. It is called the vertical covariant differential. It possesses the following important property.
Let
h
be a section of a fiber bundle
Σ
→
X
, and let
h
∗
Y
⊂
Y
be the pullback bundle over
X
. Every connection
A
Σ
induces the pullback connection
A
h
=
d
x
λ
⊗
[
∂
λ
+
(
(
A
m
i
∘
h
)
∂
λ
h
m
+
(
A
∘
h
)
λ
i
)
∂
i
]
on
h
∗
Y
. Then the restriction of a vertical covariant differential
D
~
to
J
1
h
∗
Y
⊂
J
1
Y
coincides with the familiar covariant differential
D
A
h
on
h
∗
Y
relative to the pullback connection
A
h
.
