 
Summary: Differential geometry upstairs
William K. Allard
Mathematics Department
Duke University
January 29, 2010
Department of Mathematics, Duke University
Preliminaries.
Let M be a smooth ndimensional manifold and let F(M) be the
algebra of smooth real valued functions on M.
Let A be the set of (x, U) such that x : U Rn is a chart. For
each p M let Ap = {(x, U) A : p U}.
Whenever (x, U) A, V is an open subset of M, W is a finite
dimensional vector space and f : V W is smooth we let
f
x
: U V Hom(Rn
, W )
be such that
f
x
