CHAPTER 1
BACKGROUND
The prerequisites for this course are a background in general topology, advanced calculus
(analysis), and linear algebra.
The background in general topology is a knowledge of the notions of a topology as given
in a one semester or qua
CHAPTER 6
IMMERSIONS AND EMBEDDINGS
In this chapter we turn to inclusion maps of one manifold to another. If f : N M is
an inclusion, then the image should also be a manifold. In chapter 3, we saw one situation
where a subset of f (N ) M inherited the str
CHAPTER 5
TANGENT VECTORS
In Rn tangent vectors can be viewed from two perspectives
(1) they capture the innitesimal movement along a path, the direction, and
(2) they operate on functions by directional derivatives.
The rst viewpoint is more familiar as
CHAPTER 4
PARTITIONS OF UNITY
AND SMOOTH FUNCTIONS
In this section, we construct a technical device for extending some local constructions
to global constructions. It is called a partition of unity. We also use the opportunity to
discuss C functions. We b
CHAPTER 2
MANIFOLDS
In this chapter, we address the basic notions: What is a manifold and what is a map
between manifolds. Several examples are given.
An n dimensional manifold is a topological space that appears to be Rn near a point,
i.e., locally like
CHAPTER 3
SUBMANIFOLDS
One basic notion that was not addressed in Chapter 1 is the concept of containment:
When one manifold is a subset on a second manifold. This notion is subtle in a manner
the reader may have experienced in general topology. We give a
CHAPTER 7
VECTOR BUNDLES
We next begin addressing the question: how do we assemble the tangent spaces at various
points of a manifold into a coherent whole? In order to guide the decision, consider the
case of U Rn an open subset. We reect on two aspects.
CHAPTER 9
MULTILINEAR ALGEBRA
In this chapter we study multilinear algebra, functions of several variables that are linear
in each variable separately. Multilinear algebra is a generalization of linear algebra since a
linear function is also multilinear i
The Tangent Bundle
Jimmie Lawson
Department of Mathematics
Louisiana State University
Spring, 2006
1
The Tangent Bundle on Rn
The tangent bundle gives a manifold structure to the set of tangent vectors on
Rn or on any open subset U . Unlike the practice i
Calculating Submanifold Charts
Jimmie Lawson
Department of Mathematics
Louisiana State University
Spring, 2006
1
Introduction
The full-rank submanifold theorem states
Theorem 1.1 The set Q = f 1 (q ) is a submanifold of M of dimension
m n for f : M N , wh
Basic Dierentiable Calculus Review
Jimmie Lawson
Department of Mathematics
Louisiana State University
Spring, 2003
1
Introduction
Basic facts about the multivariable dierentiable calculus are needed as background for dierentiable geometry and its applicat
HOMEWORK SOLUTIONS
Scattered Homework Solutions for Math 7550, Dierential Geometry, Spring 2006. If students have solutions written in some form of TeX that they would like to submit to me for problems not posted, Ill check them and, if correct, post them
Math 7550, Spring 2006
QUIZ 2
Name:
Do all three problems
1. (4 points) Suppose that A = cfw_(Ui , i ) : i I is an atlas for M . What condition must
a chart (U, ) satisfy to be in the dierentiable structure generated by this atlas. What
must one prove ab