Narrow Results

Species definitions appear to have proliferated in the past 15 years; all aim to characterise much the same biological entities, but differ in how these entities are best viewed. The Phylogenetic Species Concept is easiest to apply in practice. In the fossil record, species are easiest to diagnose where there are gaps, but applying empirically determined standard parameters of variation gives additional clues where gaps are absent. The Composite Species Concept is the most logical where evolving lineages can be identified.

We develop variational formulae for the total m-th mean curvatures of codimension-one distributions on a compact Riemannian manifold Mⁿ⁺¹. Based on the method of our recent work Rovenski–Walczak (2007) we show that the total generalized mean curvatures over M(c) of constant curvature c do not depend on the choice of k orthonormal vector fields, that for k = 1 was proved by Brito–Langevin–Rosenberg (1981). We calculate the variations of the total generalized mean curvatures for k orthonormal vector fields on a compact Riemannian manifold M.