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


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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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).

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  • Riemannian manifolds
  • Sasaki metric
  • sphere bundles

Mathematics Subject Classification

  • 53C25
  • 53D17
  • 53C07