SavedAdd your journal

Journals / International Electronic Journal of Geometry / 2026 / Volume: 13 Issue: 1

Infinitesimal Projective Transformations on the Tangent Bundle of a Riemannian Manifold with a Class of Lift Metrics

Mosayeb Zohrehvand0000-0002-3876-4060
Pages
—
Publication Date
2020-01-30
🏛️Bibliographic metadata and full text access for this article are provided via DergiPark.
DergiPark Record ↗
✨

AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

We characterized this type of transformations and then ‎we prove that if $ (TM,‎\tilde{g}‎) $ admits a non-affine infinitesimal projective transformation‎, ‎then $ M $ and $ TM‎‎ $ are locally flat.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:Let ‎$‎ (M,g) $ be a Riemannian manifold and ‎$‎ TM $ be its tangent bundle. ‎In the present paper‎, ‎we study infinitesimal projective transformations on $ TM $ with respect to the Levi-Civita connection of a class of (pseudo-)Riemannian metrics ‎$ ‎‎\tilde{g}‎ $ which is a generalization of the three classical lifts of the metric ‎$‎g$.
📊
Key Findings & Data:Let ‎$‎ (M,g) $ be a Riemannian manifold and ‎$‎ TM $ be its tangent bundle. ‎In the present paper‎, ‎we study infinitesimal projective transformations on $ TM $ with respect to the Levi-Civita connection of a class of (pseudo-)Riemannian metrics ‎$ ‎‎\tilde{g}‎ $ which is a generalization of the three classical lifts of the metric ‎$‎g$.
💡
Conclusion & Impact:We characterized this type of transformations and then ‎we prove that if $ (TM,‎\tilde{g}‎) $ admits a non-affine infinitesimal projective transformation‎, ‎then $ M $ and $ TM‎‎ $ are locally flat.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Lift metrics · infinitesimal projective transformations · Riemannian manifold

Abstract

Let ‎$‎ (M,g) $ be a Riemannian manifold and ‎$‎ TM $ be its tangent bundle. ‎In the present paper‎, ‎we study infinitesimal projective transformations on $ TM $ with respect to the Levi-Civita connection of a class of (pseudo-)Riemannian metrics ‎$ ‎‎\tilde{g}‎ $ which is a generalization of the three classical lifts of the metric ‎$‎g$. We characterized this type of transformations and then ‎we prove that if $ (TM,‎\tilde{g}‎) $ admits a non-affine infinitesimal projective transformation‎, ‎then $ M $ and $ TM‎‎ $ are locally flat.

🔖

Cite This Article

Zohrehvand, M. (2020). Infinitesimal Projective Transformations on the Tangent Bundle of a Riemannian Manifold with a Class of Lift Metrics. International Electronic Journal of Geometry, Volume: 13 Issue: 1. https://doi.org/10.36890/iejg.566914

📄 Full Text PDF Viewer

📄

PDF görüntüleyici hazırlanıyor...

References (18)

  1. Abbassi, M. T. K., Sarih, M.: On natural metrics on tangent bundles of Riemannian manifolds. Arch. Math. (Brno). 41 (1), 71-92 (2005).
  2. Abbassi, M. T. K., Sarih, M.: On Riemannian $g$-natural metrics of the form $ag^s+bg^h+cg^v$ on the tangent bundle of a Riemannian manifold $(M,g)$. Mediterr. J. Math. 2 (1), 19-43 (2005).
  3. Dan’shin, A.: Infinitesimal projective transformations in the tangent bundle of general space of path. Izv. Vyssh. Uchebn. Zaved. Mat. 41 (9), 8-12 (1997).
  4. Gezer, A., Bilen, L.: Projective vector fields on the tangent bundle with a class of Riemannian metrics. C. R. Acad. Bulgare Sci. 71 (5), 587-596 (2018).
  5. Hasegawa, I., Yamauchi, K.: Infinitesimal projective transformations on contact Riemannian manifolds. J. Hokkaido Univ. Educ. Nat. Sci. 51 (1), 1-7 (2000).
  6. Hasegawa, I., Yamauchi, K.: Infinitesimal projective transformations on tangent bundles with the horizontal lift connection. J. Hokkaido Univ. Educ. Nat. Sci. 52 (1), 1-5 (2001).
  7. Hasegawa, I., Yamauchi, K.: Infinitesimal projective transformations on tangent bundles with lift connections. Sci. Math. Jpn. 7 (13), 489-503 (2002).
  8. Hasegawa, I., Yamauchi, K.: Infinitesimal projective transformations on tangent bundles. In: M. Anastasiei, P.L. Antonelli (editors). Finsler and Lagrange Geometries. New York, USA: Springer Science+Business Media, Springer Netherlands. 91-98 (2003).
  9. Kobayashi, S.: A theorem on the affine transformation group of a Riemannian manifold. Nagoya Math. J. 9, 39-41 (1955).
  10. Nagano, T.: The projective transformation on a space with parallel Ricci tensor. Kodai Math. J. 11 (3), 131-138 (1959).
  11. Okumura, M.: On infinitesimal conformal and projective transformation of normal contact spaces. Tohoku Math. J. 14 (4), 389-412 (1962).
  12. Yamauchi, K.: On Riemannian manifolds admitting infinitesimal projective transformations. Hokkaido Math. J. 16 (2), 115-125 (1987).
  13. Yamauchi, K.: On infinitesimal projective transformations of the tangent bundles with the complete lift metric over Riemannian manifolds. Ann. Rep. Asahikawa Med. Coll. 19, 49-55 (1998).
  14. Yamauchi, K.: On infinitesimal projective transformations of tangent bundle with the metric II+III. Ann. Rep. Asahikawa Med. Coll. 20, 67-72 (1999).
  15. Yamauchi, K.: On infinitesimal projective transformations of tangent bundles over Riemannian manifolds. Mathematica Japonica 49 (3), 433-440 (1999).
  16. Yano, K.: The Theory of Lie Derivatives and Its Applications. Franklin Classics Trade Press (2018).
  17. Yano, K., Ishihara, S.: Tangent and Cotangent Bundles: differential geometry. Marcel Dekker Inc., New York (1973).
  18. Yano, K., Kobayashi, S.: Prolongation of tensor fields and connections to tangent bundles I, II, III. J. Math. Soc. Japan 18 (2), 194-210 (1966), 18 (3), 236-246 (1966), 19 (4), 486-488 (1967).