ON GLOBAL RIEMANN-CARTAN GEOMETRY

We know that
5. Vanhecke-Tricerri classification of Riemann-Cartan manifolds
Ň
b
∈ŇĚ
⊗C
∞
2
Λ M
. In turn, the following pointwise O(q)-irreducible decomposition holds
T*M ⊗ Λ 2 M ≅ Ψ 1 (M) ⊕ Ψ 2 (M) ⊕ Ψ 3 (M). In this case, the orthogonal projections on the components of this
decomposition are defined by the following relations:
(1)
Т
Т
b
( )
1 b
b
b
( X ,Y ,Z ) 3 Т ( X ,Y ,Z ) + Т ( Y ,Z, X ) + Т ( Z, X ,Y )
= − ;
( X ,Y ,Z ) = g( X ,Z ) ω( Y ) − g( X ,Y ) ω( Z )
(2) b
;
(3)
Т
b
( )
b
(1) b
(2) b
( X ,Y ,Z ) Т ( X ,Y ,Z ) − Т ( Y ,Z, X ) − Т ( Z, X ,Y )
= ,
b
−1
where Т ( X,
Y,
Z ) = g( T ( X,
Y ),
Z ) and : = ( n −1) trace Т
ω .
[1] Bourguignon J.P. Formules de Weitzenbök en dimension 4. Géométrie Riemannienne en dimension 4: Seminaire
Arthur Besse 1978/79. Paris: Cedic-Fernand Nathan, 1981.
We say that a Riemann-Cartan manifold ( M ,g,∇)
belongs to the class Ψ α or
b
tensor field Т is a section of corresponding tensor bundle Ψ α ( М ) or α ( М ) ⊕Ψβ
( М )
Ψ ⊕Ψ
for , β = 1 , 2,
3
α
Ψ .
β
α and α < β if the
[2] Tricerri F., Vanhecke L. Homogeneous structures. Progress in mathematics (Differential geometry), 32 (1983),
234-246.
The spaces
12
∗
Λ 2 М ⊗Т
М and Т М ⊗Λ М
∗
2
, as well as their irreducible components, are isomorphic. Therefore these
two classifications are equivalent. Moreover, corresponding classes of Riemann-Cartan manifolds from these two
classifications coincide.