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GEOMETRY Contents 1. Euclidean Geometry 2 1.1. Metric Spaces 2 1.2 GEOMETRY JOUNI PARKKONEN Contents 1. Euclidean geometry 2 1.1. Metric spaces 2 1.2. Euclidean space 2 1.3. Isometries 4 2. The sphere 7 2.1. More on cosine and sine laws 10 2.2. Isometries 11 2.3. Classification of isometries 12 3. Map projections 14 3.1. The latitude-longitude map 14 3.2. Stereographic projection 14 3.3. Inversion 14 3.4. Mercator’s projection 16 3.5. Some Riemannian geometry. 18 3.6. Cylindrical projection 18 4. Triangles in the sphere 19 5. Minkowski space 21 5.1. Bilinear forms and Minkowski space 21 5.2. The orthogonal group 22 6. Hyperbolic space 24 6.1. Isometries 25 7. Models of hyperbolic space 30 7.1. Klein’s model 30 7.2. Poincaré’s ball model 30 7.3. The upper halfspace model 31 8. Some geometry and techniques 32 8.1. Triangles 32 8.2. Geodesic lines and isometries 33 8.3. Balls 35 9. Riemannian metrics, area and volume 36 Last update: December 12, 2014. 1 1. Euclidean geometry 1.1. Metric spaces. A function d: X × X ! [0; +1[ is a metric in the nonempty set X if it satisfies the following properties (1) d(x; x) = 0 for all x 2 X and d(x; y) > 0 if x 6= y, (2) d(x; y) = d(y; x) for all x; y 2 X, and (3) d(x; y) ≤ d(x; z) + d(z; y) for all x; y; z 2 X (the triangle inequality). The pair (X; d) is a metric space. Open and closed balls in a metric space, continuity of maps between metric spaces and other “metric properties” are defined in the same way as in Euclidean space, using the metrics of X and Y instead of the Euclidean metric. Example 1.1. (a) The space Rn with the Euclidean distance is a metric space, see Section 1.2. (b) The circle S1 with the distance between two points defined as their angle as vectors in E2 is a metric space, see Section 2 for details and generalisations. (c) Let X 6= ;. The discrete metric d: X × X ! [0; 1[ is defined by setting d(x; x) = 0 for all x 2 X and d(x; y) = 1 for all x; y 2 X if x 6= y. α (d) It is easy to check that for any α > 1, the expression hα(st; ) = js − tj does not define a metric on R as it fails to satisfy the triangle inequality: hα(0; 2) > 2 = 1 + 1 = hα(0; 1) + hα(1; 2) : On the other hand, it can be shown that hα is a metric if 0 < α ≤ 1. If (X1; d1) and (X2; d2) are metric spaces, then a map i: X ! Y is an isometric embedding, if d2(i(x); i(y)) = d1(x; y) for all x; y 2 X1. If the isometric embedding i is a bijection, then it is called an isometry between X and Y . An isometry i: X ! X is called an isometry of X. The isometries of a metric space X form a group Isom(X), the isometry group of X, with the composition of mappings as the group law. A map i: X ! Y is a locally isometric embedding if each point x 2 X has a neighbourhood U such that the restriction of i to U is an isometric embedding. A (locally) isometric embedding i: I ! X is (1)a (locally) geodesic segment, if I ⊂ R is a (closed) bounded interval, (2)a (locally) geodesic ray, if I = [0; +1[, and (3)a (locally) geodesic line, if I = R. 1.2. Euclidean space. Let us denote the Euclidean inner product of Rn by n X (xjy) = xiyi : i=1 The Euclidean norm kxk = p(xjx) defines the Euclidean distance d(x; y) = kx−yk. The triple En = (Rn; (·|·); k · k) is n-dimensional Euclidean space. Proposition 1.2. The Euclidean distance is a metric. Proof. The first two properties of a metric are clear from the expression of the metric. It suffices to show that the triangle inequality holds. Let x; y; z 2 En. Using the linearity and symmetry properties of the inner product and Cauchy’s inequality, we 2 12/12/2014 get 2 dEn (x; y) = (x − yjx − y) = (x − z + z − yjx − z + z − y) = (x − zjx − z) + 2(x − zjz − y) + (z − yjz − y) 2 2 ≤ dEn (x; z) + 2dEn (x; z)dEn (z; y) + dEn (z; y) 2 = (dEn (x; z) + dEn (z; y)) ; which implies the triangle inequality because all the terms in the inequality are positive. Euclidean space is a geodesic metric space: For any two distinct points x; y 2 En, the map y − x t 7! x + t ; ky − xk n is a geodesic line that passes through the points x and y. Indeed, for any x0 2 E and n−1 n any u 2 S , let jx0;u : R ! E be the map jx0;u(t) = x0 + tu : For any s; t 2 R, we have n 1 dE (jx0;u(t); jx0;v(s)) = kx0 + tu − (x0 + su)k = k(t − s)uk = jt − sjkuk = dE (t; s) : The restriction jx;yj[0;kx−yk] is a geodesic segment that connects x to y: j(0) = x and j(kx − yk) = y. In fact, this is the only geodesic segment that connects x to y up to replacing the interval of definition [0; kx−yk] of the geodesic by [a; a+kx−yk] for some a 2 R. More precisely: A metric space (X; d) is uniquely geodesic, if for any x; y 2 X there is exactly one geodesic segment j : [0; d(x; y)] ! X such that j(0) = x and j(d(x; y)) = y. Proposition 1.3. Euclidean space is uniquely geodesic. Proof. If g is a geodesic segment that connects x to y and z is a point in the image of g, then, by definition, kx−zk+kz −yk = kx−yk. But, using Cauchy’s inequality, it is easy to see that the Euclidean triangle inequality becomes an equality if and only if z is in the image of the linear segment jj[0;kx−yk]. Even if the proof of the above proposition appears obvious, it uses the connection of the Euclidean metric with the inner product in an essential way. There are plenty of examples of metric spaces arising from vector spaces endowed with a norm that are not uniquely geodesic. For example, the expression d1(x; y) = maxfjx1 − y1j; jx2 − y2jg defines a metric on R2. It is easy to check that, among many others, the mappings 2 j1; j2 : [0; 1] ! (R ; d1) defined by j1(t) = t(1; 0) and ( 1 t(1; 1); if 0 ≤ t ≤ 2 ; j2(t) = 1 (t; 1 − t); if 2 ≤ t ≤ 1 2 are both geodesic segments in (R ; d1) connecting 0 to (1; 0). If a metric space X is uniquely geodesic and x; y 2 X, x 6= y, we denote the (image of the) unique geodesic segment connecting x to y by [x; y]. 3 12/12/2014 1.3. Isometries. We will now study the isometries of Euclidean space more closely. The (Euclidean) orthogonal group of dimension n is n O(n) = fA 2 GLn(R):(AxjAy) = (xjx) for all x; y 2 E g T = fA 2 GLn(R): A A = Ing : Recall the following basic result from linear algebra: Lemma 1.4. An n × n-matrix A = (a1; : : : ; an) is in O(n) if and only if the vectors n a1; : : : ; an form an orthonormal basis of E . It is easy to check that elements of O(n) give isometries on En for any n 2 N: Let A 2 O(n) and let x; y 2 En. Now d(Ax; Ay)2 = (Ax − AyjAx − Ay) = (A(x − y)jA(−y)) = (AT A(x − y)jx − y) = (x − yjx − y) = d(x − y)2 : n For any b 2 R , let tb(x) = x + b be the translation by b. Again, it is easy to see that translations are isometries of En. The translation group is n T(n) = ftb : b 2 R g Orthogonal maps and translations generate the Euclidean group n E(n) = fx 7! Ax + b : A 2 O(n); b 2 R g which consists of isometries of En. If a group G acts on a space X, and x is a point in X, the set G(x) = fg(x): g 2 gg is the G-orbit of x. The action of a group is said to be transitive if G(x) = X for some (and therefore for any) x 2 X. A more elementary way to express this is that a group G acts transitively on X if for all x; y 2 X there is some g 2 G such that g(x) = y. Proposition 1.5. E(n) acts transitively by isometries on En. In particular, Isom(En) acts transitively on En. Proof. The Euclidean group of En contains the group of translations T(n) as a subgroup. This subgroup acts transitively because for any x; y 2 Rn, we have Ty−x(x) = y. An affine hyperplane of En is a subset of the form H = H(P; u) = P + u? ; where P; u 2 En and kuk = 1. The reflection in H is the map rH (x) = x − 2(x − P ju)u : Reflections are very useful isometries, the following results give some of their basic properties.
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