ON GLOBAL RIEMANN-CARTAN GEOMETRY

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[2] Bochner S., Yano K. Tensor-fields in non-symmetric connections. The Annals of Mathematics, 2 nd Ser. 56(1952), No. 3, 504-519. [3] Yano K., Bochner S. Curvature and Betti number. Princeton: Princeton University Press, 1953. [4] Goldberg S.I. On pseudo-harmonic and pseudo-Killing vector in metric manifolds with torsion. The Annals of Mathematics, 2 nd Ser. 64 (1956), No. 2, 364-373. [5] Kubo Y. Vector fields in a metric manifold with torsion and boundary. Kodai Math. Sem. Rep. 24 (1972), 383-395. [6] Rani N., Prakash N. Non-existence of pseudo-harmonic and pseudo-Killing vector and tensor fields in compact orientable generalized Riemannian space (metric manifold with torsion) with boundary. Proc. Natl. Inst. Sci. India. 32 (1966), No. 1, 23-33. 18
9. Scalar and complete scalar curvature of Riemann-Cartan manifolds It is well known that the curvature tensor R of the linear non-symmetric connection ∇ of a Riemann-Cartan manifold ( ,g,∇) 2 2 M is a section of the tensor bundle Λ M ⊗ Λ M . Therefore the scalar curvature of the Riemann- Cartan manifold ( ,g,∇) a Riemannian manifold (M, g). n M we can define by the formula s ∑ R( e ,e ,e ,e ) _ = as an analogy to the scalar curvature s of The dependence between the scalar curvatures s and s is described in the following formula i = 1 ( 1) b 2 ( ) ( 2 ) b 2 ( 3 ) b 2 b = s − S −2 n −2 S + 2 S − div ( trace S ). (9.1) s 4 In particular, for the Weitzenböck connection ∇ we have the identity s = 0. Then the formula (9.1) can be rewritten in the following form ( 1) b 2 ( ) ( 2 ) b 2 ( 3 ) b 2 b S + 2 n −2 S −2 S 4div ( traceS ) s = + . i j i j [1] Yano K., Bochner S. Curvature and Betti number. Princeton: Princeton University Press, 1953. [2] Stepanov S.E., Gordeeva I.A. Pseudo-Killing and pseudo harmonic vector field on a Riemann-Cartan manifold. Mathematical Notes, 87: 2 (2010), 238-247. 19
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 and 14: 6. Examples of Riemann-Cartan manif
Page 15 and 16: The classification of almost Hermit
Page 17: 8. Green theorem for a Riemann-Cart
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

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