Technische Universität Berlin Fakultät II Institut für Mathematik

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1 Technische Universität Berlin Fakultät II Institut für Mathematik Springborn, Kouřimská!!Trial exam!! Written Examination Geometry I Surname: First name: Matr. Nr.: Studiengang: Apart from pens and rulers you are allowed to bring with you sheet of paper, DIN A4. one hand-written Solutions written with a pencil will not be corrected. Your final answers should fit into the framed boxes. After every exercise there is empty space for your calculations and notes. Additional paper will be provided upon request. You may rip off the last two sheets, which contain a summary of all problems. Please leave the other sheets stapled together. You have 90 minutes for your work. The total amount of points for this exam is Σ Note

2 . Question (2+2+6)= 0 points Let A, B, C, D S 2 = {x R 3 x = }. Two spherical triangles with vertices A, B, C and A, B, D are colunar if they share an edge and their other two edges belong to the same great circles. a) Triangles with vertices A, B, C and A, B, D are colunar, and C = (0, 0, ). Determine D. b) How many triangles are colunar to a given spherical triangle? c) Let γ be the parametrization of the shorter great circle arc connecting 0 A = 0 and B = Find an explicit formula for γ and show that its length satisfies cos(l(γ)) = A, B.. Overview of results: a) D = ( ). b) The number of triangles colunar to a given spherical triangle: c) [ ] γ : S 2, γ(t) = Your. work: 2

3 . Notes and calculations: 3

4 2. Question (4+2+4)= 0 points Let H 2 = {x R 3 x, x 2, = } be the hyperboloid model of the hyperbolic plane, l and l 2 be two hyperbolic lines and h i = {y H 2 y, n i 2, 0}, i {, 2}, be halfplanes bounded by l and l 2. A hyperbolic rotation of H 2 with centre x H 2 is a map T SO + (2, ) with T (x) = x. Let x = 0 n = 0, n 2 =. a) A hyperbolic rotation T SO + (2, ) with centre x maps h to h 2. Find T. Show your work. Let C be a Euclidean circle with radius 0 < r < and centre at the origin in the open unit disc D. Interpreting D as the Poincaré disc model makes C a hyperbolic circle. b) Parametrize C by a curve γ. c) The Riemannian metric on the Poincaré disc model is g p (v, w) = 4 ( p 2 p 2 2) 2 v, w R 2, p = (p, p 2 ), v, w D. Use the Riemannian metric on D to calculate the length of γ as a curve in the Poincaré disc model. Overview of results: a) T = Your work: 4

5 . 5

6 [ ] b) γ : C, γ(t) = c). 6

7 Notes and calculations: 7

8 3. Question (3++3+4)= points q : R 3 R, q(x) = x 2 + 2x x 2 + x x 3 7x 2 3. a) Find the symmetric bilinear form b : R 3 R 3 R, (x, y) b(x, y) such that q(x) = b(x, x). b) What is the signature of q? c) Is q non-degenerate? Show your work. d) Let e, e 2, e 3 denote the canonical basis of R 3. Let U = span{e, e + e 2 }. Find U with respect to b. Overview of results: x y a) b x 2, y 2 = x 3 y 3 b) The signature of q is c) Is q degenerate? Your work: 8

9 d) U = Notes and calculations: 9

10 4. Question (9)= 9 points Decide if the following statements are true or false. requested. Briefly justify your answer if. ( point) The side lengths of the polar triangle are the interior angles of the original triangle and vice versa. (Justify your work.) 2. ( point) The great circles connecting (, 0, 0) and (, 0, 0) in the unit sphere S 2 = {x R 3 x = } correspond to lines through the origin under the stereographic projection. (Justify your work.) 3. ( point) A symmetric bilinear form on a vector space V is non-degenerate if and only if the set {v V b(v, v) = 0} = {0}. (Justify your work.) 0

11 4. ( point) Let L R 2, be the light cone. Then {[x] R 3 x L} is a conic. (Justify your work.) 5. ( point) A hyperbolic line in the hyperboloid model of the hyperbolic space H n R n, is the shortest path in the Lorentz space R n, connecting two vectors x, y H n. (Justify your work.) 6. ( point) The Klein-Beltrami model of hyperbolic geometry is conformal. 7. ( point) The projective linear group P GL(n, R) is a projective space.(justify your work.)

12 8. ( point) Let l,..., l 4 be four lines in a projective plane Π intersecting in a point P. Let a, b Π be two lines not containing P. Then cr(a l, a l 2, a l 3, a l 4 ) = cr(b l, b l 2, b l 3, b l 4 ). (Justify your work.) 9. ( point) For any five points in RP 2, there is a unique conic section containing them. (Justify your work.) Notes and calculations: 2

13 Overview of the exercises. Question (2+2+6)= 0 points Let A, B, C, D S 2 = {x R 3 x = }. Two spherical triangles with vertices A, B, C and A, B, D are colunar if they share an edge and their other two edges belong to the same great circles. a) Triangles with vertices A, B, C and A, B, D are colunar, and C = (0, 0, ). Determine D. b) How many triangles are colunar to a given spherical triangle? c) Let γ be the parametrization of the shorter great circle arc connecting 0 0 A = 0 and B =. Find an explicit formula for γ and show that its length satisfies 2 2 cos(l(γ)) = A, B. 2. Question (4+2+4)= 0 points Let H 2 = {x R 3 x, x 2, = } be the hyperboloid model of the hyperbolic plane, l and l 2 be two hyperbolic lines and h i = {y H 2 y, n i 2, 0}, i {, 2}, be halfplanes bounded by l and l 2. A hyperbolic rotation of H 2 with centre x H 2 is a map T SO + (2, ) with T (x) = x. Let x = 0 n = 0, n 2 =. a) A hyperbolic rotation T SO + (2, ) with centre x maps h to h 2. Find T. Show your work. Let C be a Euclidean circle with radius 0 < r < and centre at the origin in the open unit disc D. Interpreting D as the Poincaré disc model makes C a hyperbolic circle. b) Parametrize C by a curve γ. c) The Riemannian metric on the Poincaré disc model is g p (v, w) = 4 ( p 2 p 2 2) 2 v, w R 2, p = (p, p 2 ), v, w D. Use the Riemannian metric on D to calculate the length of γ as a curve in the Poincaré disc model.

14 3. Question (3++3+4)= points q : R 3 R, q(x) = x 2 + 2x x 2 + x x 3 7x 2 3. a) Find the symmetric bilinear form b : R 3 R 3 R, (x, y) b(x, y) such that q(x) = b(x, x). b) What is the signature of q? c) Is q non-degenerate? Show your work. d) Let e, e 2, e 3 denote the canonical basis of R 3. Let U = span{e, e + e 2 }. Find U with respect to b.

15 4. Question (9)= 9 points Decide if the following statements are true or false. requested. Briefly justify your answer if. ( point) The side lengths of the polar triangle are the interior angles of the original triangle and vice versa. (Justify your work.) 2. ( point) The great circles connecting (, 0, 0) and (, 0, 0) in the unit sphere S 2 = {x R 3 x = } correspond to lines through the origin under the stereographic projection. (Justify your work.) 3. ( point) A symmetric bilinear form on a vector space V is non-degenerate if and only if the set {v V b(v, v) = 0} = {0}. (Justify your work.) 4. ( point) Let L R 2, be the light cone. Then {[x] R 3 x L} is a conic. (Justify your work.) 5. ( point) A hyperbolic line in the hyperboloid model of the hyperbolic space H n R n, is the shortest path in the Lorentz space R n, connecting two vectors x, y H n. (Justify your work.) 6. ( point) The Klein-Beltrami model of hyperbolic geometry is conformal. 7. ( point) The projective linear group P GL(n, R) is a projective space.(justify your work.) 8. ( point) Let l,..., l 4 be four lines in a projective plane Π intersecting in a point P. Let a, b Π be two lines not containing P. Then (Justify your work.) cr(a l, a l 2, a l 3, a l 4 ) = cr(b l, b l 2, b l 3, b l 4 ). 9. ( point) For any five points in RP 2, there is a unique conic section containing them. (Justify your work.)