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Submanifolds. Let n be a positive integer.

Summary: Submanifolds.
Let n be a positive integer.
Definition. We say f is an n-diffeomorphism if
(1) f is function whose domain and range of f are open subsets of Rn
(2) f is smooth;
(3) f is univalent and, for each x dmn f, f(x) carries Rn
isomorphically onto itself.
Whenever U and V are open subsets of Rn
we let
be the set of ordered triples (U, F, V ) such that F is an n-diffeomorphism with domain U and range V .
Proposition. We have
(1) is an n-diffeormorphism;
(2) if F is an n-diffeormorphism and W is an open subset of Rn
then F|W is an n-diffeormorphism;
(3) if U is a family of open subsets of Rn
, F : U Rn
, F is univalent and F|U is a n-diffeormorphism
for each U U then F is an n-diffeormorphism;


Source: Allard, William K. - Department of Mathematics, Duke University


Collections: Mathematics