CHARACTERISTIC CLASSES
BY
JOHN W. MILNOR
AND
JAMES D. STASHEFF
PRINCETON UNIVERSITY PRESS
AND
UNIVERSITY OF TOKYO PRESS
PRINCETON, NEW JERSEY
1974
Copyright 1974 by Princeton University Press
ALL RIGHTS RESERVED
Published in Japan exclusively by
Universit
Chapter 1
Introduction
1.1 Some history
In the words of S.S. Chern, the fundamental objects of study in differential geometry are manifolds. 1 Roughly, an n-dimensional manifold is a mathematical object
that locally looks like Rn . The theory of manifolds
Version 2.1, May 2009
Allen Hatcher
c
Copyright 2003
by Allen Hatcher
Paper or electronic copies for noncommercial use may be made freely without explicit permission from the author.
All other rights reserved.
Table of Contents
Introduction
. . . . . . .
Solving
Differential Equations
on Manifolds
Ernst Hairer
Universite de Gen`eve
Section de mathematiques
2-4 rue du Li`evre, CP 64
CH-1211 Gen`eve 4
June 2011
Acknowledgement. These notes have been distributed during the lecture Equations
differentielles s
MATH 208, Fall 2012
Manifolds I
Final
1. Consider two vector fields v(x) = v0 + A(x) and w(x) = w0 + B(x)
on Rn , where v0 and w0 are constant vectors, and A and B are linear
maps Rn Rn .
a. Find the flow t of v.
b. Find the braket [v, w].
2. Let = g r D2
MATH 2210: SUPPLEMENTARY NOTES ON WEEK 2
Abstract. In the following we review some important concepts in Sec 2.1 and 2.2.
Keywords: (1) Stereographic projection; (2) graphs of functions of multivariables; (3)
continuity and uniform continuous.
Without spe
MATH 208, Fall 2012
Manifolds I, Midterm
1. Let M be a smooth manifold. Prove that for any continuous function
f : M R and any > 0, there exists a smooth function h such that
supxM |f (x) h(x)| < . (In other words, every continuous function
can be C 0 -ap
Prof. A. Cattaneo
Institut f
ur Mathematik
Universit
at Z
urich
Fr
uhjahrsemester 2015
Differentiable Manifolds
Solutions Exercise Sheet 5
Due date: April 20
Exercise 1 (1-forms). (a) Let f : U R, (x, y) 7 arctan(y/x) be a function on U = R2 cfw_x =
+ x y
KOC
UNIVERSITY, FALL 2011, MATH 554 MANIFOLDS, MIDTERM 1
OCTOBER 27, INSTRUCTOR: BURAK OZBAGCI, 180 Minutes
Solutions by Fatih C
elik
PROBLEM 1 (20 points): Let N = (0, 0, 1) be the north pole in the sphere S2 R3 , and let S = (0, 0, 1) denote the south
Hopf Fibration and Cliord Translation
of the 3-sphere
Most rotations of the 3-dimensional sphere S3 are
quite dierent from what we might expect from familiarity with 2-sphere rotations. To begin with, most
of them have no xed points, and in fact, certain
Manifolds I
Midterm Makeup Problem 2
Richard Klevan
November 21, 2016
1. Show that df (p) = 0 implies
Proof Note that dxi dxj =
2f
i
j
xi xj |p dx dx
xi xj
y y |p dy dy
=
2f
|pdy i dy j
y y
(using summation convention) and that
2f
f
=
i
j
j
x x
xi x
y
MATH 549, SPRING 2013, MIDTERM
(1) Let O(n) be the set of n n orthogonal matrices, viewed as a subset
of all n n real matrices.
(a) Prove that O(n) is a submanifold and describe its tangent space
at each point (Hint : use the regular value theorem applied
HW 3.
3.1 A). Look up and state the definition of a manifold being parallelizable.
3.1 B) Show that S 1 is parallelizable.
3.1 C) Show that O(n) is paralleliable. Hint: Use the diffeos O(n) O(n) given
by LA (g) = Ag and the previous HW where you computed
Selected Solutions to
Loring W. Tus
An Introduction to Manifolds (2nd ed.)
Prepared by Richard G. Ligo
Chapter 1
Problem 1.1: Let g : R R be defined by
Z t
Z t
3
g(t) =
f (s)dt =
s1/3 dt = t4/3 .
4
0
0
Rx
Show that the function h(x) = 0 g(t)dt is C 2 but
HW Assignment 1.
1.1 A. Prove that the locus xy = 0 in the plane, is NOT a topological manifold.
B. For what values of c is the locus xy = c a smooth manifold?
1.2. Set-up: Consider the space of all lines in the plane. Since a line can be given
in slope-i
:
Copyright: Mehdi Nadjakhah, Ahmad Reza Forough.
e-mail : m_nadjakhah@iust.ac.ir, a_forough@iust.ac.ir
Web : http:/webpages.iust.ac.ir/m_nadjakhah
Last edition of this book : http:/webpages.iust.ac.ir/m_nadjakhah/NDEB.htm
pc.
pc.
. . . .
Manifolds I
Midterm
Richard Klevan
November 16, 2016
1. Proof The straightening lemma implies that there are coordinates xj such that X =
Putting Y = f j x
j we have that
x1 .
2
(f j j ) f j j 1
1
x
x
x x
2
j
f
2
j
= fj j 1 +
f
x x
x1 xj
xj x1
f j
=
x1 x
MA455 Manifolds Solutions 1 May 2008
1. (i) Given real numbers a < b, find a diffeomorphism (a, b) R.
Solution: For example first map (a, b) to (0, /2) and then map (0, /2) diffeomorphically to R using the function
tan.
(ii) Find a diffeomorphism (0, ) R.
MATH 209, MANIFOLDS II, WINTER 2015
Final
due Tuesday 3/17
Throughout the exam all manifolds, maps, and homotopies are assumed to be smooth.
1. Prove that the complex projective space CP n is orientable.
2. Let D be a closed two-dimensional disk.
(a) Show
MATH 209, MANIFOLDS II, WINTER 2015
Homework Assignment III: Linear Algebra
Throughout this assignment, we use the notation and conventions used in
class rather than in the textbook. In particular, V is always assumed to
be a real vector space of dimensio
MATH 209, MANIFOLDS II, WINTER 2015
Homework Assignment I: One-forms and integration
1. Let f : R R be given by y = x3 . Calculate f (/x) and f dy. Is
f (/x) smooth?
2. Let f be a smooth function on a manifold M . Show that df = f dy,
where y is the natur
MATH 209, MANIFOLDS II, WINTER 2017
Midterm
due Thursday 2/23
1. Let X and Y be smooth vector fields on a manifold M and (M ). Prove that LX (iY ) =
i[X,Y ] + iY (LX ).
2. Let be a non-vanishing 1-form on a manifold M . Prove that the following two condit
MATH 209, MANIFOLDS II, WINTER 2015
Homework Assignment V: Orientations and integration
1. Let F : C C be a holomorphic function. Show that F is necessarily orientation
preserving at its regular points, i.e., F dx dy = f dx dy with f 0.
2. Let N be a hype
MATH 209, MANIFOLDS II, WINTER 2015
Midterm
due 2/24
1. Let X and Y be smooth vector fields on a manifold M and (M ). Prove that LX (iY ) =
i[X,Y ] + iY (LX ).
2. Let be a non-vanishing 1-form on a manifold M . Prove that the following two conditions are
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