A vector-bundle version of a theoremof V. Doležal
A vector-bundle version of a theoremof V. Doležal
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V. Doležal 定理的向量丛版本
DOI：
10.1016/j.laa.2004.05.014
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发表时间：
2004
影响因子：
1.1
通讯作者：
K. Grasse
中科院分区：
文献类型：
--
作者：
K. Grasse
Doležal的一个著名定理指出，给定一个方阵A(T)，它是标量参数t∈R的Ck函数，且其秩随参数变化不变，可以通过线性无关的也是t的Ck函数的向量来跨越A(T)的值域和核。与Doležal定理的发展同时(且独立于)Doležal定理的发展，Sibuya单独地并与谢家华联合证明了关于Rnor Cn中参数的参数矩阵分解的相关结果，由此Doležal定理可以很容易地扩展到依赖于有限维欧氏空间的矩形子集中的参数的矩阵。本文进一步推广了Doležal定理，即当k=0时，用拓扑空间R代替参数空间R，当k⩾1时，用可微流形代替参数空间R。我们证明了对于一类广泛的参数空间，可以把两个向量丛与一个常秩常数参数矩阵函数联系起来，当且仅当这两个向量丛都是平凡的时，Doležal定理将继续成立。特别地，这一结果将Dolež等人的定理推广到参数空间是可压缩的(但当k=0时可能是无限维的)的情况，并包含了Dolež等人以前已知的结果。
A well known theorem of Doležal states that given a square matrix A(t) that is a Ckfunction of the scalar parameter t∈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 Ckfunctions 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 Rnor 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 k⩾1. We show that for a broad class of parameter spaces one can associate two vector bundles to a constant-rank, Ckparameterized 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.