be a vector bundle.
-
Show there is a map of vector bundles (over M)
which is surjective on each fiber, if and only if there are sections 
of E over M such that, for each 
, 
span the fiber 
.
-
Show there is a map of vector bundles (over M)
which is injective on each fiber, if and only if there are sections 
of E over M such that, for each 
, 
are linearly independent in the fiber 
.
-
Show that E is isomorphic to the trivial rank n bundle
if and only if there exist sections 
of E over M such that, for each 
, 
form a basis for the fiber 
.
-
Let
, 
. For which 
in 
is there a neighborhood U of 
such that 
is a coordinate system?
-
For which real numbers c is the locus
a submanifold of 
?
be the real projective space, with the standard differentiable structure. Let 
be given by
Show that 
is a well-defined, smooth embedding. Compute 
.
-
Prove that the group
of orthogonal 
matrices with determinant 1 is an embedded submanifold of the 9-dimensional Euclidean space of all 
matrices.
-
Prove that
with the differentiable structure described above is diffeomorphic to the real projective space 
with the standard differentiable structure.
by explicitly defining coordinate charts and calculating transition functions.
Let M be a differentiable manifold and 
a diffeomorphism. Consider the direct product 
with the identification of pairs of points 
and 
, for all 
. Let 
be the quotient space (called the mapping torus of f, or, the suspension construction on f.)
-
Show that
possesses a natural structure of differentiable manifold.
-
Show that
is a locally trivial fiber bundle with base 
and fiber M.
-
Apply the suspension construction to the following three cases:
-
.
-
.
-
.
Identify the resulting mapping tori. In which of the three cases is the bundle
trivial?
-
alexsuciu@neu.edu