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.

令\（（M，\ langle \;，\; \ rangle _ {TM}）\）为黎曼流形。众所周知，TM上的Sasaki度量标准非常严格，但是当限制为\（T ^ {（r）} M = \ {u \ in TM，| u | = r \} \）时，它具有很好的属性。在本文中，我们考虑一般情况，我们用向量束\（E \ longrightarrow M \）替换TM，该向量束具有欧几里得积\（\ langle \;，\; \ rangle _E \）和连接\（\ nabla ^ E \）保留\（\ langle \;，\; \ rangle _E \）。我们在E上定义Sasaki度量，并考虑其对\（E ^ {{r}} = \ {a \ in E，\ langle a，a \ rangle _E = r ^ 2 \} \）的限制h。我们研究\（（E ^ {（r）}，h）\）的黎曼几何，归纳了最初在\（T ^ {（r）} M \）上首先获得的许多结果，并建立了新结果。我们将在一般情况下获得的结果应用于作者在Boucetta和Essoufi J Geom Phys 140：161–177，2019中引入的欧几里得Atiyah向量束的类别。最后，我们证明了任何单模三维李群G都带有左不变黎曼度量，因此\（（T ^ {（（1）} G，h）\）具有正标量曲率。