A vector-bundle version of a theorem of V. Doležal
A well known theorem of \textit{V. Doležal} [Časopis Mat. 89, 466--468 (1964; Zbl 0187.37103)] states that given a square matrix \(A(t)\) that is a \(C^k\) function of the scalar parameter \(t\in\mathbb{R}\) 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 \(C^k\) functions of \(t\). The paper explores some generalizations of Doležal's theorem in which the parameter space \(\mathbb{R}\) is replaced by a topological space when \(k= 0\) or a differentiable manifold when \(k\geq 1\). The author shows that for a broad class of parameter spaces one can associate two vector bundles to a constant-rank, \(C^k\) parameterized matrix function and that Dolžal'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.
scientific article; zbMATH DE number 2119127
scientific article