A nontrivial smooth steady incompressible Euler flow in three dimensions with compact support is constructed. Another uncommon property of this solution is the dependence between the Bernoulli function and the pressure.

We study the double exponential map which is a composition of a special form of two exponential maps on a manifold with connection. We relate this map and the composition of covariant derivations as well as the composition of pseudodifferential operators on these manifolds.

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We obtain a composition formula for the higher covariant derivatives on a vector bundle over a manifold. The formula generalizes the classical Leibniz product rule for the derivative. We also obtain, as a corollary, a generalization of the author's theorem about the double exponential map to the case of multiple maps.