AbstractIn this paper we prove a result which can be regarded as a sub-Riemannian version of de Rham decomposition theorem. More precisely, suppose that (M, H, g) is a contact and oriented sub-Riemannian manifold such that the Reeb vector field $$\xi $$
ξ
is an infinitesimal isometry. Under such assumptions there exists a unique metric and torsion-free connection on H. Suppose that there exists a point $$q\in M$$
q
∈
M
such that the holonomy group $$\Psi (q)$$
Ψ
(
q
)
acts reducibly on H(q) yielding a decomposition $$H(q) = H_1(q)\oplus \cdots \oplus H_m(q)$$
H
(
q
)
=
H
1
(
q
)
⊕
⋯
⊕
H
m
(
q
)
into $$\Psi (q)$$
Ψ
(
q
)
-irreducible factors. Using parallel transport we obtain the decomposition $$H = H_1\oplus \cdots \oplus H_m$$
H
=
H
1
⊕
⋯
⊕
H
m
of H into sub-distributions $$H_i$$
H
i
. Unlike the Riemannian case, the distributions $$H_i$$
H
i
are not integrable, however they induce integrable distributions $$\Delta _i$$
Δ
i
on $$M/\xi $$
M
/
ξ
, which is locally a smooth manifold. As a result, every point in M has a neighborhood U such that $$T(U/\xi )=\Delta _1\oplus \cdots \oplus \Delta _m$$
T
(
U
/
ξ
)
=
Δ
1
⊕
⋯
⊕
Δ
m
, and the latter decomposition of $$T(U/\xi )$$
T
(
U
/
ξ
)
induces the decomposition of $$U/\xi $$
U
/
ξ
into the product of Riemannian manifolds. One can restate this as follows: every contact sub-Riemannian manifold whose holonomy group acts reducibly has, at least locally, the structure of a fiber bundle over a product of Riemannian manifolds. We also give a version of the theorem for indefinite metrics.
On Existence Theorems to Symmetric Functional Set-Valued Differential Equations