Table Of ContentA course in low-dimensional geometry
Mark Steinberger
Date: January 2, 2016. Copyright Mark Steinberger. All rights reserved.
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2 MARK STEINBERGER
Contents
Preface 7
Terminology 7
Functions 7
Relations 8
Acknowledgements 10
1. Some linear algebra 11
1.1. Vector spaces and linear maps 11
1.2. Spans and linear independence 13
1.3. Matrices 16
1.4. Block matrices and their multiplication 19
1.5. Dimension 20
1.6. Rank 23
1.7. Direct sums 24
1.8. Base change 26
1.9. Exercises 32
2. Basic Euclidean geometry 34
2.1. Lines in Rn 34
2.2. Lines in the plane 38
2.3. Inner products and distance 39
2.4. Euclidean isometries are affine 44
2.5. Affine functions and linearity 47
2.6. Affine automorphisms of Rn 49
2.7. Similarities 50
2.8. Convex and affine hulls; affine subspaces and maps 51
2.8.1. Convex and affine hulls 52
2.8.2. Joins 56
2.8.3. Affine maps 58
2.8.4. Affine and convex hulls of infinite sets 62
2.8.5. Convex subsets of lines 63
2.9. Affine independence, interiors and faces 64
2.9.1. Affine independence 64
2.9.2. Interiors 68
2.9.3. Faces 74
2.9.4. Examples 80
2.10. Exercises 82
3. Groups 83
3.1. Definition and examples 83
A COURSE IN LOW-DIMENSIONAL GEOMETRY 3
3.2. Orders of elements 86
3.3. Conjugation and normality 87
3.4. Homomorphsims 89
3.5. A matrix model for isometries and affine maps 94
3.6. G-sets 96
3.7. Direct products 98
4. Linear isometries 99
4.1. Orthonormal bases and orthogonal matrices 99
4.2. Gramm–Schmidt 105
4.3. Orthogonal complements 106
4.4. Applications to rank 109
4.5. Invariant subspaces for linear isometries 110
5. Isometries of R2 112
5.1. Reflections 112
5.2. Trigonometric functions 121
5.3. Linear isometries of R2: calculation of O(2) 128
5.4. Angles in R2 and Rn; the cosine law; orientation in R2 132
5.4.1. Angles in R2 132
5.4.2. Angles in Rn 133
5.4.3. Orientation in R2 135
5.5. Calculus of isometries of R2 137
5.5.1. Glide reflections 138
5.5.2. Calculating composites of isometries 140
5.5.3. Calculus of reflections 145
5.6. Classical results from Euclidean geometry 149
5.7. Exercises 149
6. Groups of symmetries: planar figures 151
6.1. Symmetry in Rn; congruence and similarity 151
6.1.1. The group of symmetries of X ⊂ Rn 151
6.1.2. The subgroups T(X) and O(X) of S(X) 152
6.1.3. Congruence and similarity 152
6.2. Symmetries of polytopes 154
6.2.1. Generalities 154
6.2.2. Centroids 157
6.2.3. Symmetries of the n-cube 158
6.2.4. Symmetries of the regular n-gon in R2 159
6.3. Geometry meets number theory: the golden mean 159
6.4. Symmetries of points and lines in R2 161
4 MARK STEINBERGER
6.5. Dihedral groups 162
6.6. Index 2 subgroups 165
6.7. Left cosets; orbit counting; the first Noether theorem 168
6.8. Leonardo’s theorem 172
6.9. Orbits and isotropy in the plane 174
6.10. Frieze groups 176
6.11. Fundamental regions and orbit spaces 184
6.12. Translation lattices in Rn 188
6.13. Orientation-preserving wallpaper groups 199
6.13.1. Groups admitting 4-centers 201
6.13.2. Groups admitting 6-centers 204
6.13.3. Groups admitting 3-centers but not 6-centers 207
6.13.4. The remaining cases 210
6.14. General wallpaper groups 213
6.14.1. Wallpaper groups with O(W) = W 218
4
6.14.2. Wallpaper groups with O(W) = W 224
6
6.14.3. Wallpaper groups with O(W) = W 227
3
6.14.4. Wallpaper groups with O(W) = W 232
2
6.14.5. Wallpaper groups with O(W) = W 239
1
6.15. Exercises 246
7. Linear isometries of R3 262
7.1. Linear orientations of Rn 262
7.2. Rotations 264
7.3. Cross products 269
7.4. Reflections 271
7.5. Rotation-reflections 274
7.6. Symmetries of the Platonic solids 276
7.6.1. The cube and the regular tetrahedron 276
7.6.2. The regular tetrahedron 277
7.6.3. Calculation of O(C) 278
7.6.4. The dodecahedron 279
7.6.5. Duality 290
7.6.6. The octahedron 291
7.6.7. Dual of the tetrahedron 292
7.6.8. The icosahedron 293
7.7. Exercises 296
8. Spheres and other manifolds 297
8.1. Some advanced calculus 297
A COURSE IN LOW-DIMENSIONAL GEOMETRY 5
8.2. Orientation properties of nonlinear mappings in Rn 302
8.3. Topological manifolds; Sn−1 303
8.4. Smooth manifolds 306
8.5. Products of manifolds 311
8.6. Oriented atlases 312
8.7. Exercises 313
9. Spherical geometry 315
9.1. Arc length and distance in Sn; isometries of Sn 315
9.2. Lines and angles in S2 326
9.3. Spherical triangles 329
9.4. Isometries of S2 330
9.5. Perpendicular bisectors 332
9.6. Exercises 333
10. Tangent bundles 335
10.1. The local model 335
10.2. The tangent bundle of a smooth manifold 336
10.3. Tangent bundles of products 341
10.4. Immersions and embeddings; submersions 342
10.5. Orientation of manifolds 345
10.6. Vector fields 346
11. Riemannian manifolds 347
11.1. Riemannian metrics 347
11.2. Arc length, distance and angles 350
11.3. Geodesics 355
11.3.1. Geodesics in the local model 355
11.3.2. Geodesics in general Riemannian manifolds 359
11.4. The exponential map 361
12. Hyperbolic geometry 367
12.1. Boundary of H and compactification of C. 368
12.2. M¨obius transformations 371
12.3. Isometric properties of M¨obius transformations 377
12.4. Hyperbolic lines and geodesics 379
12.5. Incidence relations and transitivity properties 390
12.6. Hyperbolic line segments 392
12.7. Parallels and perpendiculars 394
12.8. Reflections 397
12.9. Generalized M¨obius transformations 401
12.10. Calculus of isometries 405
6 MARK STEINBERGER
12.11. Exercises 406
Appendix A. Spaces with identifications 408
A.1. Metric topology 408
A.2. Subspace topology 412
A.3. Quotient topology 413
A.4. Group actions and orbit spaces 418
A.5. Basis for a topology 421
A.6. Properly discontinuous actions 424
A.6.1. Product topology 425
A.6.2. Disjoint unions 429
A.7. Topology of the orbit space 429
Appendix B. Compactness 434
B.1. Heine–Borel 434
B.2. Maps out of compact spaces 438
B.3. Cones and convex bodies 439
B.3.1. Cones 439
B.3.2. Convex bodies 440
References 444
A COURSE IN LOW-DIMENSIONAL GEOMETRY 7
Preface
This book has grown out of two courses the author has taught at the Uni-
versityatAlbany. Thefirstcourseinvestigatestherigidmotions(isometries)
of the Euclidean plane and develops rosette, frieze and wallpaper groups. In
particular, we study discrete subgroups of the group of isometries of the
plane (i.e., two-dimensional crystallographic groups).
The second course develops Euclidean, spherical and hyperbolic geome-
try in dimension two from an analytic point of view, providing realizations
of Euclidean and non-Euclidean geometry from an analytic, rather than
axiomatic, point of view, and develops tools like the cosine law and the
Gauss–Bonnettheoremthatprovideadeeperinsightintothegeometrythan
is provided by the axioms alone. This has particular value for prospective
high school teachers, as the cosine law is actually used in the high school
curriculum, andaunifieddevelopmentofboththatandEuclideangeometry
puts things in a modern perspective that might be very useful to the high
school students themselves.
In particular, we show how isometries provide clean and direct proofs of
the basic theorems in Euclidean geometry and their analogues on the sphere
and in hyperbolic space.
Terminology. We shall be primarily interested in subpsaces of Euclidean
n-space, Rn, defined as the space of all n-tuples of real numbers:
Rn = {(x ,...,x ) : x ,...,x ∈ R}.
1 n 1 n
Thus, formally, Rn is the cartesian product of n copes of the real line. More
generally, if X ,...,X are sets, their cartesian product is given by
1 n
X ×···×X = {(x ,...,x ) : x ∈ X for i = 1,...,n}.
1 n 1 n i i
If X is a finite set, we write |X| for the number of elements in X. An
elementary counting argument gives:
Lemma 0.0.1. Let X ,...,X be finite sets. Then X ×···×X is finite
1 n 1 n
and the number of elements in it is the product of the numbers of elements
in the individual sets:
(0.0.1) |X ×···×X | = |X |···|X |.
1 n 1 n
Given subset X,Y ⊂ Z, we write X (cid:114)Y for the set-theoretic difference:
X (cid:114)Y = {x ∈ X : x (cid:54)∈ Y}
= X (cid:114)(X ∩Y).
Functions. A function
f : X → Y
is sometimes called a map, a mapping, or a transformation. We call X its
domain and Y its codomain.
8 MARK STEINBERGER
We say that f is one-to-one or injective if
f(x) = f(y) ⇒ x = y.
An injective function is called an injection.
The image or range of f is
imf = f(X) = {y ∈ Y : y = f(x) for some x ∈ X}.
For Z ⊂ X, we write f(Z) = {f(z) : z ∈ Z} for the image of f| : Z → Y,
Z
the restriction of f to Z.
We say that f : X → Y is onto or surjective if its range is all of Y,
i.e., if the range is equal to the codomain. A surjective function is called a
surjection.
If f is both injective and surjective, it is said to be bijective. A bijective
function is called a bijection or one-to-one correspondence.
If f : X → Y is bijective, its inverse function
f−1 : Y → X
is defined by setting f−1(y) to be the unique x ∈ X with f(x) = y: x exists
because f is surjective and is unique because f is injective. The inverse
function f−1 is easily seen to be bijective.
Relationships between functions are often expressed via a diagram. We
say
h (cid:47)(cid:47)
X (cid:63)(cid:63)Z
f (cid:32)(cid:32) g
Y
commutes if h = g◦f. A more general diagram commutes if any two ways
of getting from one node to another give the same function. Thus,
f (cid:47)(cid:47)
X Y
g h
(cid:15)(cid:15) (cid:15)(cid:15)
k (cid:47)(cid:47)
Z W
commutes if k◦g = h◦f.
Relations. Formally, a relation R on a set X is a subset R ⊂ X ×X. It
is customary to write xRy if the ordered pair (x,y) is in R (and usually a
symbolic operator such as ∼ is used rather than a letter like R. Relations
express a relationship between x and y. One generally does not talk about
the subset R at all, but gives a criterion for xRy to hold.
Examples 0.0.2.
(1) The usual ordering ≤ on R is a relation. It is an example of an order
relation to be discussed below.
A COURSE IN LOW-DIMENSIONAL GEOMETRY 9
(2) Let X be a set and let P(X) be the set of all subsets of X. (P(X) is
called the power set of X.) Then the standard inclusion of subsets
S ⊂ T gives a relation on P(X). It is also an order relation.
(3) Among the integers, there is a relation, let’s call it ≡, defined by
setting a ≡ b if they have the same parity (i.e., either both are odd
or both are even). Note that we can test for parity by taking powers
of (−1): n is even if (−1)n = 1 and is odd if (−1)n = (−1). Thus,
a ≡ b if and only if (−1)a = (−1)b. This relation is generally called
congruence mod 2, and is written a ≡ b mod 2. It is an exmple of
what’s known as an equivalence relation to be discussed below.
(4) Let f : X → Y be a function. Define the relation ∼ on X by setting
x ∼ x if f(x ) = f(x ). We call this the relation on X induced by
1 2 1 2
f. This will also be seen to be an equivalence relation.
With these examples in mind, we give some potential properties of rela-
tions.
Definition 0.0.3. Let R be a relation on X
• R is reflexive if xRx for all x ∈ X.
• R is symmetric if xRy ⇒ yRx.
• R is antisymmetric if (xRy and yRx) ⇒ x = y.
• R is transitive if (xRy and yRz) ⇒ xRz.
We say R is an order relation or partial ordering if it is reflexive, anti-
symmetric and transitive. We say it is a total ordering if in addition each
pair of elements in X is comparable, i.e., for x,y ∈ X, either xRy or yRx.
We say R is an equivalence relation if it is reflexive, symmetric and tran-
sitive.
Remark 0.0.4.
• The relation ≤ on R is a total ordering.
• Therelation⊂onP(X)isapartialordering,butnotatotalordering
if X has more than one element: if x (cid:54)= y ∈ X, then neither {x}
nor {y} is contained in the other, so these subsets are not related in
P(X).
• For a set X, let P (X) ⊂ P(X) be the finite subsets of X. Then
f
there is an equivalence relation on P (X) given by setting S ∼ T if
f
S and T have the same number of elements.
Equivalence relations are about grouping similar things together, while
order relations are about comparing different things. In the former case the
notion of equivalence classes is valuable.
Definition 0.0.5. If ∼ is an equivalence relation on X we define the equiv-
alence class of x ∈ X by
[x] = {y ∈ X : x ∼ y} ⊂ X.
10 MARK STEINBERGER
The set of all equivalence classes, X/∼, is
X/∼ = {[x] : x ∈ X}
and the canonical map
π : X → X/∼
is defined by π(x) = [x] for all x ∈ X.
Example 0.0.6. Let f : X → Y be a function and let ∼ be the relation on
X induced by f (Example 0.0.2(4)). Then the equivalence classes are the
subsets f−1(y) with y in the image of f.
The key result about equivalence relations is the following.
Lemma 0.0.7. Let ∼ be an equivalence relation on X. Then every element
of X belongs to exactly one equivalence class. A given element x lies in [x],
and if [x]∩[y] (cid:54)= ∅, then [x] = [y]. Finally, if π : X → X/∼ is the canonical
map, then π is onto and
(0.0.2) [x] = π−1([x]).
Of course, here the [x] on the right is a point in the set X/∼ (which consists
of subsets of X), while the [x] on the left is a subset of X.
Proof. Ifz ∈ [x]∩[y],thenx ∼ z ∼ y bysymmetry,sox ∼ y bytransitivity.
Transitivity then shows that [y] ⊂ [x]. By symmetry, y ∼ x, so [x] ⊂ [y],
and both must coincide with [z] by the argument just given. In particular,
y ∈ [x] ⇔ [y] = [x],
and (0.0.2) follows. (cid:3)
Acknowledgements. Manymathematicianshavebeengenerouswiththeir
thoughtsandinsights. IwishtothankNoamElkies, CharlesFrohman, Greg
Kuperberg, John Randall, Peter Shalen and Deane Yang.
I have also learned from my students, both past and present. In addition
to contributing ideas and asking interesting and useful questions, they have
caught mistakes and typos for which I am grateful. Special thanks go to
Gabe Holmes and Aaron Wolff.
