Geometry of Manifolds-Final MSc. Exam-1386-10-15
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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
and
its manifold topology and metric topology are equalent.
4.
4.
Show that if 
is 
and 
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
as a
module.
6.
6.
Show that iffis a 
a closed regular submfd N of M then f is restriction of a
on M.