Peks̨en, Ömer (1989) Homogeneous exact fillings. University of Southampton, Doctoral Thesis.
Abstract
A subset X of the Grassmann manifold Γk(n) of k-planes in n-space Rn is said to be an exact filling of Rn by k-planes if and only if for any S bset X, the union of all the elements of S is Rn if and only if S= X. Thus the union of all k-planes in X is Rn and every such k-plane in X is required to achieve this result. The first part of the thesis concerns basic definitions and theorems about transformation groups, and gives an account of what has been achieved to date on various aspects of exact filling theory, including descriptions of several examples of exact fillings, and of a class of exact fillings obtained by considering maximal tori of compact connected Lie groups and their Lie algebras. The main contribution of the thesis to the theory of exact fillings is the concept of `homogeneous' exact filling, which encompasses most of the examples now known. Some unsolved problems are discussed.
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