ON GLOBAL RIEMANN-CARTAN GEOMETRY

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A homogeneous Riemannian manifold (M, g) is the connected Riemannian manifold (M, g) whose isometry group is transitive. By the Tricerri and Vanhecke theorem, a complete connected Riemannian manifold (M, g) is 2 homogeneous iff a tensor field T ∈С ∞ ТМ ⊗Λ М such that = 0 ∇R and = 0 ∇T for the connection = ∇ + T ∇ . In this case, ∇g = 0 and, therefore, a homogeneous Riemannian manifold is an example of the Riemannian-Cartan manifold( M, g,∇) . [4] Tricerri F., Vanhecke L. Homogeneous structures on Riemannian manifolds. London Math. Soc.: Lecture Note Series., Vol. 83. Cambridge University Press, London, 1983. An almost Hermitian manifold is defined as the triple (M, g, J), where the pair (M, g) is a Riemannian 2m-dimensional manifold with almost complex structure J [5] Kobayshi S., Nomizu K. Foundations of Riemannian geometry, Vol. 2. Interscience publishers, New York-London, 1969. ∗ ∈ ТМ ⊗Т М compatible with the metric g, i.e. J 2 −Id TM = and g ( J, J) = g . In this case, ∇g = 0 for the connection∇ =∇+∇J , and, therefore, an almost Hermitian manifold (M, g, J), together with the connection , ∇ =∇+∇J , is an example of the Riemannian-Cartan manifold (M, g,∇). 14
The classification of almost Hermitian manifolds is well known, it is based on the pointwise U(m)-irreducible decomposition of the tensor ∇ Ω, where Ω (X, Y) = g(X, JY). Almost semi-Kählerian manifolds are isolated by the condition trace ∇ J = 0 and are an example of Riemann- Cartan manifolds of class Ω 1 ⊕ Ω 2 . Almost Kählerian manifolds are isolated by the condition dΩ = 0 and are an example of Riemann-Cartan manifolds of class Ω 2 ⊕ Ω 3 . Nearly Kählerian manifolds are isolated by the condition dΩ = 3 ∇Ω and are an example of Riemann-Cartan manifolds of class Ω 1 . [6] Gray A., Hervella L. The sixteen class of almost Hermitean manifolds. Ann. Math. Pura Appl., 123 (1980), 35-58. 15
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 and 10: We know that S b ∈C ∞ Λ 2 M
Page 11 and 12: 4. The class Ω 1 ⊕ Ω 2 of Cappo
Page 13: 6. Examples of Riemann-Cartan manif
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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