We explicitly calculate the Riemannian curvature of D-dimensional metrics recently discussed by Chen, Lü and Pope. We find that they can be concisely written by using a single function. The Einstein condition which corresponds to the Kerr-NUT-de Sitter metric is clarified for all dimensions. It is shown that the metrics are of type D.

In this paper, we prove Hardy-Leray and Rellich-Leray inequalities for curl-free vector fields with sharp constants. This complements the former work by Costin-Maz'ya [2] on the sharp Hardy-Leray inequality for axisymmetric divergence-free vector fields.

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This is a complement of the former works by Costin-Maz'ya [1] and Hamamoto-Takahashi [5] on sharp Hardy-Leray inequality for solenoidal (i.e., divergence-free) fields in R 3 with some axisymmetry conditions. Here we derive the same best constant, with no assumption of any symmetry.

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