CLASSIFICATION OF LOCALLY EUCLIDEAN SPACES

The classification of Riemann surfaces has largely reached its completion. The purpose of the present paper is to lay the foundation for a new intriguing field in the classification theory: Riemannian spaces with Euclidean metrics. The paper is self-contained, both for the Riemann surface expert and...

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Bibliographic Details
Main Author: Sario,Leo
Format: Report
Language:English
Subjects:
ALGEBRAIC TOPOLOGY
BOUNDARY VALUE PROBLEMS
CLASSIFICATION
COMPLEX VARIABLES
FUNCTIONAL ANALYSIS
OPERATORS(MATHEMATICS)
POTENTIAL THEORY
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Summary:The classification of Riemann surfaces has largely reached its completion. The purpose of the present paper is to lay the foundation for a new intriguing field in the classification theory: Riemannian spaces with Euclidean metrics. The paper is self-contained, both for the Riemann surface expert and the reader whose main interest is with higher dimensions. The significance of locally Euclidean spaces lies, first of all, in that their function-theoretic nature differs for dimensions n2 and n=2. The existence or nonexistence of Green's functions and positive or bounded harmonic functions in Rn, punctured Rn, and in the punctured flat torus offer simple examples. A striking phenomenon is that, despite such differences, the basic inclusion relations remain valid. Moreover, capacities and null-classes can be defined for components of point sets in Rn. Pub. in Nagoya Mathematical Journal v25 p87-111 Mar 1965 (Copies available only to DDC users).