The object of our study is a Lie algebroid $A$ or a Cartan-Lie algebroid$(A,abla)$ (a Lie algebroid with a compatible connection) over a basemanifold $M$ equipped with appropriately compatible geometrical structures. Themain focus is on a Riemannian base $(M,g)$, but we also consider symplectic andgeneralized Riemannian structures. For the Riemannian case, we show that the compatibility implies that thefoliation induced by the Lie algebroid is a Riemannian foliation; thus, inparticular, if the leaf space $Q = M/!sim$ is smooth, $Q$ permits a metricsuch that the quotient projection is a Riemannian submersion. For otherstructures on $M$ with a smooth leaf space $Q$, the reduced geometrical type on$Q$ can be different: for example, $(M,omega)$ symplectic provides in generalonly a symplectic realization of a Poisson manifold $(Q,mathcal{P})$. Building upon a result of del Hoyo and Fernandes, we prove that any Liealgebroid integrating to a proper Lie groupoid admits a compatible Riemannianbase. We also show that, given an arbitrary connection on an anchored bundle,there is a unique Cartan connection on the corresponding free Lie algebroid.
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