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Journals / International Electronic Journal of Geometry / 2020 / Cilt: 13 - Sayı: 1

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

Mosayeb ZOHREHVAND
Pages
50–60
Publication Date
2022-10-18
DOI
—
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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 ‎$‎g$.
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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 ‎$‎g$.
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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.

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Cite This Article

ZOHREHVAND, M. (2022). Infinitesimal Projective Transformations on the Tangent Bundle of a Riemannian Manifold with a Class of Lift Metrics. International Electronic Journal of Geometry, Cilt: 13 - Sayı: 1, 50–60. https://ajindex.com/articles/539612

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