A vector-bundle version of a theoremof V. Doležal Original Research Article

A well known theorem of Doležal states that given a square matrix A(t) that is a Ck function of the scalar parameter tset membership, variantR and whose rank is constant as the parameter varies, one can span the range and kernel of A(t) by linearly independent vectors that are also Ck functions of t. Contemporaneously with (and independently of) the development of Doležal’s theorem, Sibuya, individually and in joint work with Hsieh, proved related results on parametric matrix decompositions for parameters in Rn or Cn, from which Doležal’s theorem can be easily extended to matrices depending on parameters in a rectangular subset of a finite-dimensional Euclidean space. In this paper we explore further generalizations of Doležal’s theorem in which the parameter space R is replaced by a topological space when k = 0 or a differentiable manifold when kgreater-or-equal, slanted1. We show that for a broad class of parameter spaces one can associate two vector bundles to a constant-rank, Ck parameterized matrix function and that Doležal’s theorem will continue to hold if and only if both of these vector bundles are trivial. In particular, this result generalizes Doležal’s theorem to the case where the parameter space is contractible (but possibly infinite-dimensional when k = 0) and subsumes the previously known results of Doležal et al.