ON GLOBAL RIEMANN-CARTAN GEOMETRY

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All these classes of Riemann-Cartan manifolds are presented in the following diagram. Riemannian Riemannian - - - - Cartan Cartan manifolds manifolds Ω 1 ⊕ Ω 1 ⊕ 2 2 Ω 1 Ω 2 Ω 1 Ω 1 ⊕ Ω 1 ⊕ 3 Ω 2 3 Ω 2 ⊕ Ω 2 ⊕ 3 Ω 3 3 3 Riemannian Riemannian manifolds manifolds 10
4. The class Ω 1 ⊕ Ω 2 of Cappozziello-Lambiase-Stornaiolo classification Lemma 4.1. A Riemann-Cartan manifold ( ,∇) satisfies an algebraic equation of the form b of Riemann-Cartan manifolds M,g belongs to the class Ω1 ⊕ Ω 2 if and only if its torsion tensor b ( X, Y, Z ) + S ( X, Z, Y ) = g( X, Z ) B( Y ) + g( X, Y ) B( Z ) + g( Y Z ) A( X ) S , for some smooth A, B ∈ С ∞ Т ∗ М and arbitrary vector fields X, Y, Z ∈С ∞ ТМ Lemma 4.2. The class Ω 2 of Riemann-Cartan manifolds ( M,g ,∇) consists of semisymmetric Riemannian-Cartan manifolds. Lemma 4.3. A Riemann-Cartan manifold ( M,g ,∇) belongs to the class Ω 1 if and only if its torsion tensor satisfies the property S b ∈C ∞ Λ 3 M . In particular, this class includes spaces of semisimple groups. [1] Eisenhart L.P. Continuous groups of transformations. Princeton: Princeton Univ. Press, 1933. [2] Yano K., Bochner S. Curvature and Betti number. Princeton: Princeton University Press, 1953. [3] Fabri L. On a completely antisymmetric Cartan torsion tensor. Annalen de la Foundation de Broglie 32: 2-3 (2007), 215-228.. 11
Page 1 and 2: ON GLOBAL RIEMANN-CARTAN GEOMETRY P
Page 3 and 4: T. Kibble and D. Sciama have found
Page 5 and 6: Classification of known kinds of me
Page 7 and 8: The development of geometry of metr
Page 9: We know that S b ∈C ∞ Λ 2 M
Page 13 and 14: 6. Examples of Riemann-Cartan manif
Page 15 and 16: The classification of almost Hermit
Page 17 and 18: 8. Green theorem for a Riemann-Cart
Page 19 and 20: 9. Scalar and complete scalar curva
Page 21 and 22: For a compact Riemann-Cartan manifo
Page 23 and 24: The differential equation ( ∇ ξ,
Page 25 and 26: 11. Vanishing theorems for pseudo-K
Page 27 and 28: 12. Pseudo-harmonic vector fields A

ON GLOBAL RIEMANN-CARTAN GEOMETRY

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