Post on 12-Jul-2020
Introduction to Algebraic andGeometric Topology
Week 3
Domingo Toledo
University of Utah
Fall 2016
Recall Maps f : (X , d) ! (Y , d 0)
I Continuous
I Uniformly continuous.
I Lipschitz
I bi-Lipschitz
I Isometry
Recall equivalences f : (X , d) ! (Y , d 0)
I Homeomorphism
I bi-Lipschitz equivalence
I Isometry
QuestionsI Is there an isometry f : (R2, d(2)) ! (R2, d(1))?
I Is there an isometry f : (R2, d(1)) ! (R2, d(1))?
Tool: Equality set for triangle inequality
I Given x , z 2 (X , d), let
E
d
(x , z) = {y 2 X |d(x , z) = d(x , y) + d(y , z)}
I If f : (X , d) ! (Y , d 0) is an isometry, then f takesequality sets to equality sets.:
E
d
0(f (x), f (z)) = f (Ed
(x , z))
I In particular, equality sets must be homeomorphic.
ExamplesI (R2, d(2)) and (R2, d(1)).
I (R2, d(1)) and (R2, d(1)).
Higher Dimensions
I Are (Rn, d(1)) and (Rn, d(1)) isometric for n � 3?
I Look at unit spheres for n = 3:
In > 3?
I Duality?
I What’s special about n = 2?
Groups of IsometriesI (X , d) metric space. The collection of all isometries is
a group.
I Composition
I Inverse
Isometries of Rn
If : (Rn, d(2)) ! (Rn, d(2)) Euclidean isometry.
I Example: translation f (x) = x + b for fixed b 2 Rn
I Example: rotation f (x) = Ax , A orthogonal matrix,det(A) = 1.
I More general: f (x) = Ax , A orthogonal matrix.
I Composition f (x) = Ax + b, A orthogonal matrix,b 2 Rn,
Orthogonal Matrices
ITheorem: These are all the Euclidean isometries.
IMain Point Euclidean isometries are Affine linear
Back to Euclidean Isometries
I Orthogonal matrices
IA : Rn ! Rn linear.
Ix ! Ax is a Euclidean isometry
() A is orthogonal.
IA : Rn ! Rn linear.
Ib 2 Rn.
I Then x ! Ax + b is called an Affine Linear
transformation.
I Example: if A is orthogonal,
x ! Ax + b
is an isometry of Rn.
ITheorem These are all the isometries of Rn.
I Main point of proof: All isometries are affine-linear.
If Euclidean isometry
I Reduce to f (0) = 0,
I Use the equality sets E(x , z).
I Prove f (rx) = r f (x) for all r 2 R
I Prove f (x + y) = f (x) + f (y) for all x , y 2 Rn.
Structure of the Euclidean group E(n)I Linear Part
I Subgroup of index two:
I Subgroup of translations
I Non-normal subgroups
Fixed Points, Classification
Isometries of the sphere