ON GLOBAL RIEMANN-CARTAN GEOMETRY
ON GLOBAL RIEMANN-CARTAN GEOMETRY ON GLOBAL RIEMANN-CARTAN GEOMETRY
2. Riemann-Cartan manifolds A Riemann-Cartan manifold is a triple ( M ,g,∇) , where ( M ,g ) is a Riemannian n-dimensional ( n ≥ 2) manifold with linear connection ∇ having nonzero torsion S such that ∇g = 0 . [1] Trautman A. The Einstein-Cartan theory. Encyclopedia of Mathematical Physics: Edited by Fracoise J.-P., Naber G.L., Tsou S.T. – Oxford: Elsevier, 2 (2006), 189-195. The deformation tensor T defined by the identity following properties (i) T is uniquely defined; (ii) S ( X, Y ) ( T ( X, Y ) −T ( Y, X )) ; (iii) = 2 1 g ( T ( X ,Y ),Z ) + g ( T ( X ,Z ), Y ) = 0 2 T ∈C ∞ TM ⊗Λ M since ∇ = 0 ⇔ g ; (iV) g ( T ( Y ,Z ), X ) g ( S( X ,Y ),Z ) + g ( S( X ,Z ),Y ) + g( S( Y ,Z ), X ) (V) trace T = 2 trace S . = ; Т := ∇ −∇ where ∇ is the Levi-Civita connection on (M, g) has the [2] Yano K., Bochner S. Curvature and Betti number. Princeton: Princeton University Press, 1953. 8 3. Cappozziello-Lambiase-Stornaiolo classification of Riemann-Cartan manifolds
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. 9
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2. Riemann-Cartan manifolds<br />
A Riemann-Cartan manifold is a triple ( M ,g,∇)<br />
, where ( M ,g ) is a Riemannian n-dimensional ( n ≥ 2)<br />
manifold with<br />
linear connection ∇ having nonzero torsion S such that ∇g = 0 .<br />
[1] Trautman A. The Einstein-Cartan theory. Encyclopedia of Mathematical Physics: Edited by Fracoise J.-P., Naber<br />
G.L., Tsou S.T. – Oxford: Elsevier, 2 (2006), 189-195.<br />
The deformation tensor T defined by the identity<br />
following properties<br />
(i)<br />
T is uniquely defined;<br />
(ii) S ( X, Y ) ( T ( X,<br />
Y ) −T<br />
( Y,<br />
X ))<br />
;<br />
(iii)<br />
= 2<br />
1<br />
g ( T ( X ,Y ),Z<br />
) + g ( T ( X ,Z ),<br />
Y ) = 0<br />
2<br />
T ∈C<br />
∞ TM ⊗Λ M since ∇ = 0 ⇔ g ;<br />
(iV) g ( T ( Y ,Z ),<br />
X ) g ( S( X ,Y ),Z<br />
) + g ( S( X ,Z ),Y<br />
) + g( S( Y ,Z ),<br />
X )<br />
(V) trace T = 2 trace S .<br />
= ;<br />
Т := ∇ −∇<br />
where ∇ is the Levi-Civita connection on (M, g) has the<br />
[2] Yano K., Bochner S. Curvature and Betti number. Princeton: Princeton University Press, 1953.<br />
8<br />
3. Cappozziello-Lambiase-Stornaiolo classification of Riemann-Cartan manifolds
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