Math 562: January 2013 Qualifying Exam
PUID Number:
Work four out of ve of the following problems. The time limit is two hours. Please
explicitly indicate which four problems you want graded as otherwise this decision will
be made for you with no guarante
MA562 Qualifying Exam
1/2/07
Print your name:
1. The hyperbolic space Hn is the unit disc cfw_(x1 , . . . , xn ) |
gH =
F (D2 )
1
2
Px
i
2
2
i
2
i xi
< 1 with the hyperbolic metric
gE , where gE is the Euclidean metric. Consider the parametrized surface S
Qualifying Exam MA 56200 Jan 2010
Name:
Each problem is worth 6 points.
(1) (a) Prove that
M := cfw_(x, y, z ) R3 | x2 + y 2 + z 2 1
is a smooth manifold. Determine its dimension.
(b) Does M have boundary? If yes, determine M .
(2) Let M be a manifold wit
Qualifying Examination
January 2006
Math 562 Professor Donnelly
1. If M is a compact manifold, show that every continuous map M S p can be
uniformly approximated by a smooth map.
2. If dim(M ) < p, show that every continuous map M S p is homotopic to a
co
Math 562: August 2013 Qualifying Exam (McReynolds)
PUID Number:
Work four out of ve of the following problems. The time limit is two hours. Please
explicitly indicate which four problems you want graded as otherwise this decision will
be made for you with
562 Qualifying Exam2001 Spring
1(a). Let X1 = z x + x y y z , X2 = y x z y + x z be vector elds in R3 . Do they span the
tangent space of a two dimensional surface at (1, 1, 1)? Explain.
(b). Given a dimension one distribution of vector elds on a dierenti
QUALIFYING EXAMINATION
JANUARY 2004
MATH 562 - Professor Donnelly
Each question is worth ten points.
1. Prove that a d-dimensional manifold X , for which there exists an immersion
f : X Rd+1 , is orientable if and only if there is a smooth nowhere vanishi
QUALIFYING EXAMINATION
JANUARY 2005
MATH 562 - Prof. Catlin
1. Let X1 , . . . , Xn and Y1 , . . . , Yn be linearly independent vector elds dened in
neighborhoods about p M and q N , respectively, where M and N are manifolds of dimension n. Let 1 , . . . ,