ON GLOBAL RIEMANN-CARTAN GEOMETRY

We know that
S
b
∈C
∞ Λ
2 M ⊗TM
( M ) ⊕ Ω ( M ) ⊕ ( M )
∗
M ⊗ T M ≅ Ω
2
3
Λ 2 Ω
1
. In turn, the following pointwise O(q)-irreducible decomposition holds
. Here, q = g(x) for an arbitrary point x ∈ M
projections on the components of this decomposition are defined by the following relations:
(1)
S
b
( )
b
b
b
( X , Y,
Z ) = −1 S ( X,
Y,
Z ) + S ( Y,
Z,
X ) + S ( Z,
X,
Y )
3 ;
( X,
Y,
Z ) g( Y,
Z ) θ ( X ) g( X,
Z ) θ( Y )
(2) S b
−
(3)
S
b
= ;
( )
b
(1) b
(2) b
( X , Y,
Z ) S ( X,
Y,
Z ) − S ( X,
Y,
Z ) − S ( Z,
X,
Y )
= ,
b
where S ( X,
Y,
Z ) = g( S( X,
Y ),
Z ) and := ( n −1) −1 trace S
θ .
. In this case, the orthogonal
[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 Ω
α or
the tensor field
b
S is a section of corresponding tensor bundle α
( Ě )
Ω or ( Ě ) ⊕ Ω ( Ě )
Ωα
⊕ Ω
β for α , β = 1 , 2,
3 and α < β if
Ω .
α
β
[2] Capozziello S., Lambiase G., Stornaiolo C. Geometric classification of the torsion tensor in space-time. Annalen
Phys., 10 (2001), 713-727.
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