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

Show that if is and

then an F-related vector field Y on M , if it exists, is uniquely determined iff

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

Show that iffis a

a closed regular submfd N of M then f is restriction of a