and consider the tangent space at p,
Prove that 
is the linear subspace of 
consisting of all vectors 
such that 
.
,
be open subsets,
and 
a smooth map.
-
Suppose there is a smooth map
of rank 1 such that
on U.
Show that the rank r of the Jacobian matrix Jf
is everywhere <m.
-
Suppose that f does have constant rank r<m
on U. Show that, on some open subset
,
there is a smooth map 
of rank 1 such that 
on U.
where 
denotes the transpose of A, and I is the
identity 
matrix.
Consider the map 
,
defined by 
. Prove:
- Show that
is a compact subspace of 
.
-
is smooth.
- Relative to the standard identification
,
the differential 
is given by:
- The map
has constant rank 
.
- Using the above, conclude that the orthogonal group
is a smooth,
compact submanifold of
dimension 
.
- Show that the vector subspace
is the space of skew-symmetric matrices.
alexsuciu@neu.edu