The Geometry of the Sasaki Metric on the Sphere Bundles of Euclidean Atiyah Vector Bundles

Abstract

Let \((M,\langle \;,\;\rangle _{TM})\) be a Riemannian manifold. It is well known that the Sasaki metric on TM is very rigid, but it has nice properties when restricted to \(T^{(r)}M=\{u\in TM,|u|=r \}\). In this paper, we consider a general situation where we replace TM by a vector bundle \(E\longrightarrow M\) endowed with a Euclidean product \(\langle \;,\;\rangle _E\) and a connection \(\nabla ^E\) which preserves \(\langle \;,\;\rangle _E\). We define the Sasaki metric on E and we consider its restriction h to \(E^{(r)}=\{a\in E,\langle a,a\rangle _E=r^2 \}\). We study the Riemannian geometry of \((E^{(r)},h)\) generalizing many results first obtained on \(T^{(r)}M\) and establishing new ones. We apply the results obtained in this general setting to the class of Euclidean Atiyah vector bundles introduced by the authors in Boucetta and Essoufi J Geom Phys 140:161–177, 2019). Finally, we prove that any unimodular three dimensional Lie group G carries a left invariant Riemannian metric, such that \((T^{(1)}G,h)\) has a positive scalar curvature.

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References

  1. 1.

    Besse, A.: Einstein Manifolds. Springer, Berlin (1987)

    Google Scholar 

  2. 2.

    Boeckx, E., Vanhecke, L.: Unit tangent sphere bundle with constant scalar curvature. Czechoslov. Math. J. 51(126), 523–544 (2001)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Borisenko, A.A., Yampolsky, A.L.: On the Sasaki metric of the tangent and the normal bundles. Sov. Math. Dokl. 35, 479–482 (1987)

    MATH  Google Scholar 

  4. 4.

    Boucetta, M., Essoufi, H.: The geometry of generalized Cheeger-Gromoll metrics on the total space of transitive Euclidean Lie algebroids. J. Geom. Phys. 140, 161–177 (2019)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Jensen, G.R.: Homogeneous Einstein spaces of dimension four. J. Differ. Geom. 3, 309–349 (1969)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Kowalski, O.: Curvature of the induced Riemannian metric of the tangent bundle of Riemannian manifold. J. Reine Angew. Math. 250, 124–129 (1971)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Kowalski, O., Sekizawa, M.: On tangent sphere bundles with small or large constant radius. Ann. Glob. Anal. Geom. 18, 207–219 (2000)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Knapp, A.: Lie Groups Beyond an Introduction. Progress in Mathematics, vol. 140. Springer, USA (1996)

  9. 9.

    Milnor, J.: Curvatures of left invariant metrics on Lie groups. Adv. Math. 21, 293–329 (1976)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Musso, E., Tricerri, F.: Riemannian metrics on tangent bundles. Ann. Math. Pura Appl. (4)(150), 1–20 (1988)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Nagy, P.T.: Geodesics on the tangent sphere bundle of a Riemannian manifold. Geom. Dedic. 7, 233–243 (1978)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Nash, J.C.: Positive Ricci curvature on fibre bundles. J. Differ. Geom. 14(2), 241–254 (1979)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

We thank the referee for having read the paper carefully, for valid suggestions and useful corrections which improved considerably the paper.

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Correspondence to Mohamed Boucetta.

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Boucetta, M., Essoufi, H. The Geometry of the Sasaki Metric on the Sphere Bundles of Euclidean Atiyah Vector Bundles. Mediterr. J. Math. 17, 178 (2020). https://doi.org/10.1007/s00009-020-01614-3

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Keywords

  • Riemannian manifolds
  • Sasaki metric
  • sphere bundles

Mathematics Subject Classification

  • 53C25
  • 53D17
  • 53C07