In symplectic geometry, the symplectic frame bundle of a given symplectic manifold
(
M
,
ω
)
is the canonical principal
S
p
(
n
,
R
)
-subbundle
π
R
:
R
→
M
of the tangent frame bundle
F
M
consisting of linear frames which are symplectic with respect to
ω
. In other words, an element of the symplectic frame bundle is a linear frame
u
∈
F
p
(
M
)
at point
p
∈
M
,
i.e. an ordered basis
(
e
1
,
…
,
e
n
,
f
1
,
…
,
f
n
)
of tangent vectors at
p
of the tangent vector space
T
p
(
M
)
, satisfying
ω
p
(
e
j
,
e
k
)
=
ω
p
(
f
j
,
f
k
)
=
0
and
ω
p
(
e
j
,
f
k
)
=
δ
j
k
for
j
,
k
=
1
,
…
,
n
. For
p
∈
M
, each fiber
R
p
of the principal
S
p
(
n
,
R
)
-bundle
π
R
:
R
→
M
is the set of all symplectic bases of
T
p
(
M
)
.
The symplectic frame bundle
π
R
:
R
→
M
, a subbundle of the tangent frame bundle
F
M
, is an example of reductive G-structure on the manifold
M
.
