Geometry of Manifolds-Final MSc. Exam-1386-10-15
Geometry of Manifolds
Final MSc. Exam
Department ofMathematics Ferdowsi University ofMashhad
1386-10-15
Dr H.ghane
1. Let M be a
and . if
is a chart at P with coordinate function
, then show that
Is a basis for .
2.
Let
. Show that for each
is an open set in M and
is a diffeomorphism of
onto
with inverse .
3.
Let M be a connected Riemannian manifold and
. let
be a chart at P with and
. suppose and and
denote the maximum and minimum value of mapping
. if we have the following inequality
Show that M is a metric space with metric and
its manifold topology and metric topology are equalent.
4.
4.
Show that if is and
then an F-related vector field Y on M , if it exists, is uniquely determined iff
is dense in M.
5.
5.
Show that
is infinite dimensional over
but locally finitely generated over
, i.e. each
has a neighborhoodV on which there is a finite set of vector fields wich generated as a
module.
6.
6.
Show that iffis a
a closed regular submfd N of M then f is restriction of a
on M.