Aug 21, 20071
Math 4441
Dierential Geometry
Fall 2007, Georgia Tech
Lecture Notes 0
Basics of Euclidean Geometry
By R we shall always mean the set of real numbers. The set of all n-tuples of
real numbers Rn := cfw_(p1 , . . . , pn ) | pi R is called the E
Math 497C Curves and Surfaces Fall 2004, PSU
Oct 7, 20041
Lecture Notes 9
2.2 Denition of Gaussian Curvature
Let M R3 be a regular embedded surface, as we dened in the previous lecture, and let p M . By the tangent space of M at p, denoted by Tp M , we me
Math 598 Geometry and Topology II Spring 2005, PSU
Mar 20, 20051
Lecture Notes 10
3.4 Proof of Sards Theorem
This section will be typeset later. In the meantime the reader is refered to Milnors book on Topology from Dierentiable View Point. Exercise 1. Sh
Math 497C Curves and Surfaces Fall 2004, PSU
Mar 3, 20041
Lecture Notes 10
2.3 Meaning of Gaussian Curvature
In the previous lecture we gave a formal definition for Gaussian curvature K in terms of the differential of the gauss map, and also derived expli
Oct 2, 20061
Math 6455
Dierential Geometry I
Fall 2006, Georgia Tech
Lecture Notes 11
Orientability
Any ordered basis (b1 , . . . , bn ) of Rn may be viewed as a matrix B GL(n) whose ith
column is bi . Thus the set of ordered basis of Rn are in one-to-one
Math 497C Curves and Surfaces Fall 2004, PSU
Mar 3, 20041
Lecture Notes 11
2.4 Intrinsic Metric and Isometries of Surfaces
distM (p, q) := infcfw_Length[] | : [0, 1] M, (0) = p, (1) = q. Exercise 1. Show that (M, distM ) is a metric space. Lemma 2. Show t
Oct 10, 20061
Math 6455
Dierential Geometry I
Fall 2006, Georgia Tech
Lecture Notes 12
Riemannian Metrics
0.1
Denition
If M is a smooth manifold then by a Riemannian metric g on M we mean a smooth
assignment of an innerproduct to each tangent space of M .
Math 497C Curves and Surfaces Fall 2004, PSU
Nov 11, 20041
Lecture Notes 12
2.6 Gauss's formulas, and Christoffel Symbols
Let X : U R3 be a proper regular patch for a surface M , and set Xi := Di X. Then cfw_X1 , X2 , N may be regarded as a moving bases
Oct 10, 20061
Math 6455
Dierential Geometry I
Fall 2006, Georgia Tech
Lecture Notes 13
Integration on Manifolds, Volume, and Partitions of Unity
Suppose that we have an orientable Riemannian manifold (M, g ) and a function
f : M R. How can we dene the int
Math 497C Curves and Surfaces Fall 2004, PSU
Nov 11, 20041
Lecture Notes 13
2.9 The Covariant Derivative, Lie Bracket, and Riemann Curvature Tensor of Rn
Let A Rn , p A, and W be a tangent vector of A at p, i.e., suppose there exists a curve : (- , ) A wi
Nov 1, 20061
Math 6455
Dierential Geometry I
Fall 2006, Georgia Tech
Lecture Notes 14
Connections
Suppose that we have a vector eld X on a Riemannian manifold M . How can we
measure how much X is changing at a point p M in the direction Yp Tp M ?
The main
Math 497C Curves and Surfaces Fall 2004, PSU
Nov 11, 20041
Lecture Notes 14
2.11 The Induced Lie Bracket on Surfaces; The SelfAdjointness of the Shape Operator Revisited
[V, W ]M := V W W V, which is again a tangent vector eld on M . Note that since, as w
Nov 14, 20061
Math 6455
Dierential Geometry I
Fall 2006, Georgia Tech
Lecture Notes 15
Riemannian Geodesics
Here we show that every Riemannian manifold admits a unique connection, called
the Riemanninan or Levi-Civita connection, which satises two propert
Nov 30, 20041
Math 497C
Curves and Surfaces
Fall 2004, PSU
Lecture Notes 15
2.13
The Geodesic Curvature
Let : I M be a unit speed curve lying on a surface M R3 . Then the
absolute geodesic curvature of is dened as
|g | := ( )
= , n() n() ,
where n is a lo
Nov 22, 20061
Math 6455
Dierential Geometry I
Fall 2006, Georgia Tech
Lecture Notes 16
Exponential Map
0.1
ODEs revisited; Local ows of vector elds
Recall that earlier we proved:
Theorem 0.1. Let U Rn be an open set and F : U Rn be C 1 , then for every
p0
Math 598 Geometry and Topology II Spring 2005, PSU
Mar 2, 20051
Lecture Notes 9
3
3.1
Some Topics from Dierential Topology
Regular points and values; Fundamental Theorem of Algebra
Let f : M N be a smooth map. We say that p M is a regular point of f provi
Math 497C Curves and Surfaces Fall 2004, PSU
Oct 8, 20041
Lecture Notes 8
2
2.1
Surfaces
Denition of a regular embedded surface
n Br (p) := cfw_ x Rn | dist(x, p) < r .
An n-dimensional open ball of radius r centered at p is dened by
We say a subset U Rn
Math 598 Geometry and Topology II Spring 2005, PSU
Feb 22, 20051
Lecture Notes 8
2.10 Measure of C 1 maps
If X is a topological space, we say that A X is dense is X provided that A = X, where A denotes the closure of A. In other words, A is dense in X if
Jan 11, 20051
Math 528
Geometry and Topology II
Fall 2005, PSU
Lecture Notes 1
1
Top ological Manifolds
The basic ob jects of study in this class are manifolds. Roughly speaking, these are
ob jects which locally resemble a Euclidean space. In this section
Aug 28, 20071
Math 4441
Dierential Geometry
Fall 2007, Georgia Tech
Lecture Notes 1
1
Curves
1.1
Denition and Examples
A (parametrized) curve (in Euclidean space) is a mapping : I Rn , where
I is an interval in the real line. We also use the notation
t (t
Math 528 Geometry and Topology II Fall 2005, USC
Jan 11, 20051
Lecture Notes 2
1.4 Definition of Manifolds
By a basis for a topological space (X, T ), we mean a subset B of T such that for any U T and any x U there exists a V B such that x V and V U . Exe
Math 497C Curves and Surfaces Fall 2004, PSU
Sep 9, 20041
Lecture Notes 2
1.5 Isometries of the Euclidean Space
Let M1 and M2 be a pair of metric space and d1 and d2 be their respective metrics. We say that a mapping f : M1 M2 is an isometry provided that
Math 598 Geometry and Topology II Spring 2005, PSU
Jan 21, 20051
Lecture Notes 3
1.8 Immersions and Embeddings
Let X and Y be topological spaces and f : X Y be a continuous map. We say that f is an immersion provided that it is locally one-to-one, and f i
Math 497C Curves and Surfaces Fall 2004, PSU
Sep 17, 20041
Lecture Notes 3
1.8 The general definition of curvature; Fox-Milnor's Theorem
Let : [a, b] Rn be a curve and P = cfw_t0 , . . . , tn be a partition of [a, b], then (the approximation of) the tota
Math 598 Geometry and Topology II Spring 2005, PSU
Jan 21, 20051
Lecture Notes 4
2
2.1
Differentiable manifolds
Differential structures and maps
Recall that a mapping f : Rn Rm is differentiable of class C r provided that all of its partial derivatives ex
Math 497C Curves and Surfaces Fall 2004, PSU
Sep 17, 20041
Lecture Notes 4
1.9 Curves of Constant Curvature
Here we show that the only curves in the plane with constant curvature are lines and circles. The case of lines occurs precisely when the curvature
Math 598 Geometry and Topology II Spring 2005, PSU
Feb 2, 20051
Lecture Notes 5
2.2 Denition of Tangent Space
If M is a smooth n-dimensional manifold, then to each point p of M we may associate an n-dimensional vector space Tp M which is dened as follows.
Sep 20, 20071
Math 4441
Dierential Geometry
Fall 2007, Georgia Tech
Lecture Notes 5
1.13
Osculating Circle and Radius of Curvature
Recall that in a previous section we dened the osculating circle of a planar
curve : I R2 at a point a of nonvanishing curva
Feb 11, 20051
Math 598
Geometry and Topology II
Spring 2005, PSU
Lecture Notes 6
2.5
The inverse function theorem
Recall that if f : M N is a dieomorphism, then dfp is nonsingular at all
p M (by the chain rule and the observation that f f 1 is the identit
Sep 14, 20071
Math 4441
Dierential Geometry
Fall 2007, Georgia Tech
Lecture Notes 6
1.15
The four vertex theorem for convex curves
A vertex of a planar curve : I R2 is a point where the signed curvature
of has a local max or min.
Exercise 1. Show that an