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The completeness assumption is automatically fulfilled for a surface which is embedded as a closed subset of Euclidean space.
en.wikipedia.org In general, the Euclidean algorithm is convenient in such applications, but not essential; for example, the theorems can often be proven by other arguments.
en.wikipedia.org The pair wise distance between sequences is calculated using Euclidean distance measure.
en.wikipedia.org Strictly speaking it is the ring of integers that is Euclidean since fields are trivially Euclidean domains, but the terminology is standard.
en.wikipedia.org In addition, there are properties of surfaces which depend on an embedding of the surface into Euclidean space.
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