1
2
4
axis of symmetry
3
It is clear that the full symmetry group ofTisS4, the symmetric group on the four vertices, and that the rotation group ofTisA4. Apart from the identity, the rotations either have an axis passing through a vertex and the midpoint of an opposite face, the angle of rotation being±2π/3, or have an axis passing though the midpoints of opposite edges, the angle of rotation beingπ. There are 8 rotations of the first type and 3 rotations of the second type, consistent withA4having order 12.
We now consider the symmetries of T which are not rotations. For each edge of T, there is a plane of symmetry passing through it, and hence a pure reflection.
There are therefore 6 such pure reflections. This leaves us searching for 6 more elements. These are in fact rotated reflections, where the axis for the rotation is a line passing though the midpoints of opposite edges, but the angle of rotation is on this occasion±π/2. Note that neither the rotation about this axis through an angle±π/2, nor the pure reflection in the orthogonal plane, represents a symmetry ofT, but the composite does.
1.4 Curves and their lengths
Crucial to the study of all the geometries in this course will be the curves lying on them. We consider first the case of a general metric space(X,d), and then we consider the specific case of curves inRn.
Definition 1.8 A curve (or path) in a metric space (X,d) is a continuous function :[a,b] → X, for some real closed interval[a,b]; by an obvious linear reparametrization, we may assume if we wish that:[0, 1] →X. A metric space is calledpath connectedif any two points ofX may be joined by a continuous path.
This is closely related to the concept of a metric (or topological) space X being connected; that is, when there is no decomposition of X into the union of two disjoint non-empty open subsets. Equivalently, this is saying that there is no continous function fromX onto the two element set{0, 1}. If there is such a functionf, then
X = f−1(0)∪f−1(1)andX is not connected (it isdisconnected); conversely, if X = U0∪U1, withU0,U1 disjoint non-empty open subsets, then we can define a continuous functionf fromX onto{0, 1}, by stipulating that it takes value 0 on U0and 1 onU1. From the definitions, it is easily checked that both connectedness and path connectedness are topological properties, in that they are invariant under homeomorphisms.
IfX is path connected, then it is connected. If not, there would be a surjective continuous functionf : X → {0, 1}; we can then choose pointsP,Q at whichf takes the value 0, 1 respectively, and letbe a path joiningPtoQ. Thenf ◦ : [a,b] → {0, 1}is a surjective continuous function, contradicting the Intermediate Value theorem. All the metric spaces we wish to consider in this course will however have the further property of beinglocally path connected, that is each point ofX has a path connected open neighbourhood; for such spaces, it is easy to see conversely that connectedness implies path connectedness (Exercise1.7), and so the two concepts coincide (although this is not true in general). In particular, the two concepts coincide for open subsets ofRn.
Definition 1.9 For a curve:[a,b] →X on a metric space(X,d), we consider dissections
D:a=t0<t1<ã ã ã<tN =b of[a,b], withNarbitrary. We setPi =(ti)andsD:=
d(Pi,Pi+1).
Thelength lofis defined to be l=sup
D sD,
if this is finite. For curves inRn, this is illustrated below.
P = P0 P1
P2
PN ⫺1
Q = PN
IfDis arefinementofD(i.e. with extra dissection points), the triangle inequality impliessD ≤sD. Moreover, given dissectionsD1andD2, we can find a common refinement D1∪D2, by taking the union of the dissection points. Therefore, we may also define the length asl=limmesh(D)→0sD, where by definition mesh(D)= maxi(ti−ti−1). Note thatl is the smallest number such thatl ≥ sDfor allD. By taking the dissection just consisting ofaandb, we see thatl≥d((a),(b)). In the
1.4 CURVES AND THEIR LENGTHS 13 Euclidean case, any curve joining the two end-points which achieves this minimum length is a straight line segment (Exercise1.8).
There do exist curves:[a,b] →R2(where[a,b]is a finite closed real interval) which fail to have finite length (see for instance Exercise1.9), but by Proposition1.10 below this is not the case for sufficiently nice curves. IfX denotes a path connected open subset of Rn, it is the case that any two points may be connected by a curve of finite length. This property however fails for example for R2 with the British Rail metric: this space is certainly path connected, but it is easily checked that any non-constant curve has infinite length.
A metric space(X,d)is called alength spaceif for any two pointsP,QofX, d(P,Q)=inf{length() : a curve joiningPtoQ},
and the metric is sometimes called anintrinsicmetric. In fact, if we start from a metric space(X,d0)satisfying the property that any two points may be joined by a curve of finite length, then we can define a metricd onX via the above recipe,defining d(P,Q)to be the infimum of lengths of curves joining the two points; it is easy to see that this is a metric, and(X,d)is then a length space by Exercise1.17.
Example IfX denotes a path connected open subset of R2, and d0 denotes the Euclidean metric, we obtain an induced intrinsic metric d, where d(P,Q)is the infimum of the lengths of curves inX joiningPtoQ. Easy examples show that, in general, this is not the Euclidean metric.
P
X Q
Moreover, the distanced(P,Q)will not in general be achievable as the length of a curve joiningPtoQ. If for instanceX =R2\ {(0, 0)}, then the intrinsic metricdis just the Euclidean metricd0, but forP=(−1, 0)andQ=(1, 0), there is no curve of lengthd(P,Q)=2 joiningPtoQ.
The geometries we study in this course will have underlying metric spaces which are length spaces. Moreover, for most of the important geometries, the space will have the property that the distance between any two points is achieved as the length of some curve joining them; a length space with this property is called ageodesic space. This curve of minimum length is often called ageodesic, although the definition we give in Chapter7will be slightly different (albeit closely related). It might be observed that the London Underground metric (as defined on the appropriate quotient ofR2) determines a geodesic space, in that between any two points there will be a (possibly non-unique) route of minimum length.
Having talked in the abstract about curves on metric spaces, let us now consider the important case of curves inR3. In the geometries described in this course, we shall usually wish to impose a stronger condition on a curvethan just that of continuity;
from Chapter4onwards, the property of beingpiecewise continuously differentiable will nearly always be the minimum we assume. Given such a curve inR3, by definition it may be subdivided into a finite number of continuously differentiable parts; to find the length of the curve, we need only find the lengths of these parts. We reduce therefore to the case whenis continuously differentiable.
Proposition 1.10 If:[a,b] →R3is continuously differentiable, then
length= b
a
(t)dt,
where the integrand is the Euclidean norm of the vector(t)∈R3.
Proof We write(t) = (f1(t),f2(t),f3(t)). Thus givens = t ∈ [a,b], the Mean Value theorem implies
(t)−(s)
t−s =(f1(ξ1),f2(ξ2),f3(ξ3))
for some ξi ∈ (s,t). Since the fi are continuous on [a,b], they are uniformly continuous in the sense of Lemma1.13: so for anyε > 0, there existsδ > 0 such that, for 1≤i≤3,
|t−s|< δ =⇒ fi(ξi)−fi(ξ)< ε
3 for allξ ∈(s,t).
Therefore, if|t−s|< δ, then
(t)−(s)−(t−s)(ξ)< ε(t−s) for allξ ∈(s,t).
Now take a dissection
D:a=t0<t1<ã ã ã<tN =b
of [a,b], with mesh(D) < δ. The Euclidean distance d((ti−1),(ti)) equals (ti)−(ti−1). The triangle inequality implies
(ti−ti−1)(ti−1) −ε(b−a) <sD
<
(ti−ti−1)(ti−1) +ε(b−a).