A generalization of a locally most powerful unbiased (LMPU) test, for the single parameter case, to the k-parameter case is proposed. In particular, we define a locally most mean power unbiased (LMMPU) test based on the mean curvature of the power hypersurface. Compared with the type C tests (Neyman and Pearson 1938) and the type D tests, LMMPU tests possess better theoretical properties and enjoy ease of construction of critical regions in practical situations. LMMPU tests are obtained for the important practical case (Perng and Littel 1976) of a two-parameter univariate normal population, for which Isaacson (1951, p. 233) was unable to find a type D test, and for the case of means of a multivariate normal population with independent coordinates.