ON GLOBAL RIEMANN-CARTAN GEOMETRY
ON GLOBAL RIEMANN-CARTAN GEOMETRY ON GLOBAL RIEMANN-CARTAN GEOMETRY
The E. Cartan idea was reflected in the well-known books in differential geometry of the first half of the last century. Now there are hundreds works published in the frameworks of ECT and MAG, and moreover, the published results are of applied physical character. [11] Schouten J.A., Struik D.J. Einführung in die neuere methoden der differntialgeometrie, I. Noordhoff: Groningen- Batavia, 1935. [12] Schouten J.A., Struik D.J. Einführung in die neuere methoden der differntialgeometrie, II. Noordhoff: Groningen- Batavia, 1938. [13] Eisenhart L.P. Non Riemannian geometry. New York: Amer. Math. Soc. Coll. Publ., 1927. [14] Eisenhart L.P. Continuous groups of transformations. Prinseton: Prinseton Univ. Press, 1933. [15] Puetzfeld D. Prospects of non-Riemannian cosmology. Proceeding of the of 22 nd Texas Symposium on Relativistic Astrophysics at Stanford University (Dec. 13-17, 2004). California: Stanford Univ. Press, 2004, 1-5. [16] Hehl F.W., Heyde P., Kerlick G.D., Nester J.M. General Relativity with spin and torsion: Foundations and prospects. Rev. Mod. Phys, 48: 3 (1976), 393-416. 4
Classification of known kinds of metrically-affine spaces (manifolds) is presented in the following diagram. Metric-affine Metric-affine manifolds manifolds Q = n -1 tr Q g Weyl-Cartan Weyl-Cartan = 0 manifolds manifolds Q = 0 Weyl Weyl manifolds manifolds Q = 0 = 0 Q = 0 Riemannian Riemannian manifolds manifolds = 0 Flat Flat manifolds manifolds Riemannian Riemannian – – Cartan Cartan manifolds manifolds manifolds manifolds = 0 = 0 = 0 Weitzenböck Weitzenböck manifolds manifolds 5
- Page 1 and 2: ON GLOBAL RIEMANN-CARTAN GEOMETRY P
- Page 3: T. Kibble and D. Sciama have found
- 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 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
The E. Cartan idea was reflected in the well-known books in differential geometry of the first half of the last century.<br />
Now there are hundreds works published in the frameworks of ECT and MAG, and moreover, the published results<br />
are of applied physical character.<br />
[11] Schouten J.A., Struik D.J. Einführung in die neuere methoden der differntialgeometrie, I. Noordhoff: Groningen-<br />
Batavia, 1935.<br />
[12] Schouten J.A., Struik D.J. Einführung in die neuere methoden der differntialgeometrie, II. Noordhoff: Groningen-<br />
Batavia, 1938.<br />
[13] Eisenhart L.P. Non Riemannian geometry. New York: Amer. Math. Soc. Coll. Publ., 1927.<br />
[14] Eisenhart L.P. Continuous groups of transformations. Prinseton: Prinseton Univ. Press, 1933.<br />
[15] Puetzfeld D. Prospects of non-Riemannian cosmology. Proceeding of the of 22 nd Texas Symposium on<br />
Relativistic Astrophysics at Stanford University (Dec. 13-17, 2004). California: Stanford Univ. Press, 2004, 1-5.<br />
[16] Hehl F.W., Heyde P., Kerlick G.D., Nester J.M. General Relativity with spin and torsion: Foundations and<br />
prospects. Rev. Mod. Phys, 48: 3 (1976), 393-416.<br />
4
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