1386/9/26 Midterm Geometry of manifolds Dr. H.Ghane

Geometry of manifolds

Midterm MSc. Exam

Department of MathematicsFerdowsiUniversity of Mashhad

1386/9/26

Dr. H.Ghane’

1. Let be a regular submanifold of M. show that ( inclusion map ) is an imbedding.

2. If H is a regular submanifold and subgroup of a Lie group G , then H is closed as asubset of G.

3. Show that real projective space is a .

4. Let M and N be two with dimension m and n respectively with . let be a 1-1 immersion . show that F is an open mapping .

5. Show that is an n-dimentional submanifold of M with dimension n if and only if it is an open submanifold ofM.

6. Let G be a Lie grup and denote the connected component of G containing the identity element e. show that is the only connected open subgroup of G.

7. Show that acts transitively on and determine the isotropy subgroup of .