In differential geometry, the Kosmann lift, named after Yvette Kosmann-Schwarzbach, of a vector field
X
on a Riemannian manifold
(
M
,
g
)
is the canonical projection
X
K
on the orthonormal frame bundle of its natural lift
X
^
defined on the bundle of linear frames.
Generalisations exist for any given reductive G-structure.
In general, given a subbundle
Q
⊂
E
of a fiber bundle
π
E
:
E
→
M
over
M
and a vector field
Z
on
E
, its restriction
Z
|
Q
to
Q
is a vector field "along"
Q
not on (i.e., tangent to)
Q
. If one denotes by
i
Q
:
Q
↪
E
the canonical embedding, then
Z
|
Q
is a section of the pullback bundle
i
Q
∗
(
T
E
)
→
Q
, where
i
Q
∗
(
T
E
)
=
{
(
q
,
v
)
∈
Q
×
T
E
∣
i
(
q
)
=
τ
E
(
v
)
}
⊂
Q
×
T
E
,
and
τ
E
:
T
E
→
E
is the tangent bundle of the fiber bundle
E
. Let us assume that we are given a Kosmann decomposition of the pullback bundle
i
Q
∗
(
T
E
)
→
Q
, such that
i
Q
∗
(
T
E
)
=
T
Q
⊕
M
(
Q
)
,
i.e., at each
q
∈
Q
one has
T
q
E
=
T
q
Q
⊕
M
u
,
where
M
u
is a vector subspace of
T
q
E
and we assume
M
(
Q
)
→
Q
to be a vector bundle over
Q
, called the transversal bundle of the Kosmann decomposition. It follows that the restriction
Z
|
Q
to
Q
splits into a tangent vector field
Z
K
on
Q
and a transverse vector field
Z
G
,
being a section of the vector bundle
M
(
Q
)
→
Q
.
Let
F
S
O
(
M
)
→
M
be the oriented orthonormal frame bundle of an oriented
n
-dimensional Riemannian manifold
M
with given metric
g
. This is a principal
S
O
(
n
)
-subbundle of
F
M
, the tangent frame bundle of linear frames over
M
with structure group
G
L
(
n
,
R
)
. By definition, one may say that we are given with a classical reductive
S
O
(
n
)
-structure. The special orthogonal group
S
O
(
n
)
is a reductive Lie subgroup of
G
L
(
n
,
R
)
. In fact, there exists a direct sum decomposition
g
l
(
n
)
=
s
o
(
n
)
⊕
m
, where
g
l
(
n
)
is the Lie algebra of
G
L
(
n
,
R
)
,
s
o
(
n
)
is the Lie algebra of
S
O
(
n
)
, and
m
is the
A
d
S
O
-invariant vector subspace of symmetric matrices, i.e.
A
d
a
m
⊂
m
for all
a
∈
S
O
(
n
)
.
Let
i
F
S
O
(
M
)
:
F
S
O
(
M
)
↪
F
M
be the canonical embedding.
One then can prove that there exists a canonical Kosmann decomposition of the pullback bundle
i
F
S
O
(
M
)
∗
(
T
F
M
)
→
F
S
O
(
M
)
such that
i
F
S
O
(
M
)
∗
(
T
F
M
)
=
T
F
S
O
(
M
)
⊕
M
(
F
S
O
(
M
)
)
,
i.e., at each
u
∈
F
S
O
(
M
)
one has
T
u
F
M
=
T
u
F
S
O
(
M
)
⊕
M
u
,
M
u
being the fiber over
u
of the subbundle
M
(
F
S
O
(
M
)
)
→
F
S
O
(
M
)
of
i
F
S
O
(
M
)
∗
(
V
F
M
)
→
F
S
O
(
M
)
. Here,
V
F
M
is the vertical subbundle of
T
F
M
and at each
u
∈
F
S
O
(
M
)
the fiber
M
u
is isomorphic to the vector space of symmetric matrices
m
.
From the above canonical and equivariant decomposition, it follows that the restriction
Z
|
F
S
O
(
M
)
of an
G
L
(
n
,
R
)
-invariant vector field
Z
on
F
M
to
F
S
O
(
M
)
splits into a
S
O
(
n
)
-invariant vector field
Z
K
on
F
S
O
(
M
)
, called the Kosmann vector field associated with
Z
, and a transverse vector field
Z
G
.
In particular, for a generic vector field
X
on the base manifold
(
M
,
g
)
, it follows that the restriction
X
^
|
F
S
O
(
M
)
to
F
S
O
(
M
)
→
M
of its natural lift
X
^
onto
F
M
→
M
splits into a
S
O
(
n
)
-invariant vector field
X
K
on
F
S
O
(
M
)
, called the Kosmann lift of
X
, and a transverse vector field
X
G
.
