University College Dublin Mathematics, Course MATH40060 (formerly called MATH4105) Differential Geometry url:- http://mathsci.ucd.ie/courses/math40060 September 23, 2009 Dr. J. Brendan Quigley comments to:- Dr J.Brendan.Quigley Department of Mathematics University College Dublin, Belfield ph. 716–2584, 716–2580; fax 716–1196 email [email protected]. prepared using LATEX running under Redhat Linux drawings prepared using gnuplot ii c jbquig-UCD September 23, 2009 Contents 0 Organization of math40060 semester1 yr 09-10 1 1 geometry and SO(3,R) 3 1.1 Euclidean three space R3 .................................... 3 1.1.1 notation and technicalities . 3 1.1.2 geometric concepts; distance, angle, volume, orientation . 4 1.1.3 Cauchy-Schwarz inequality (CSI) . 4 1.1.4 justifying our definition of distance xx(1.1) . 5 1.1.5 justifying our definition of angle, (formula(1.2)) . 6 1.1.6 justifying the definition of signed volume . 6 1.1.7 positively oriented orthonormal basis . 8 1.1.8 special orthogonal change of basis matrix . 8 1.2 projection, reflection, rotation . 9 1.2.1 line and plane,vector resolution . 9 1.2.2 Five linear mappings and their matrices . 9 1.2.3 Five linear mappings by similarity transform . 12 1.2.4 characteristic polynomial, traces, determinant . 14 1.3 Lie group SO(3;R) and Lie algebra so(3;R) .......................... 15 1.3.1 the Lie group SO(3;R) ................................. 15 1.3.2 equivalence of special orthogonal and rotation . 17 1.3.3 rotation continuously varying in time . 19 1.3.4 the Lie algebra so(3;R) ................................. 20 1.3.5 SO(3;R) and so(3;R), standard form, eigenvalues and eigenvectors . 21 1.4 exponential matrix . 21 1.4.1 exp(0) .......................................... 23 1.4.2 exp(A)exp(B) ...................................... 23 1.4.3 powers and exponentials of similar matrices . 24 d 1.4.4 exp(tA) ....................................... 24 dt 1.4.5 Convergence of the exponential matrix power series . 25 1.5 exponential of angular velocity, exp(tDn); 2 so(3;R) ..................... 25 1.6 Euler angles . 26 1.6.1 formulae for rotation . 27 1.7 topology of SO(3;R) ...................................... 28 1.7.1 the Quaternionic sphere as a Lie group . 28 1.7.2 the Lie group SU(2;C) ................................. 28 1.7.3 the Lie algebra su(2;C) ................................ 29 1.7.4 su(2;C) as an inner product space . 29 1.7.5 mapping F from SU(2;C) to SO(3;R) ......................... 30 1.7.6 SU(2;C) is a double cover of SO(3;R)......................... 30 1.8 problem set ........................................... 31 iii iv CONTENTS 2 curves 35 2.1 plane curves . 35 2.1.1 parametrization of regular curves . 35 2.1.2 re parametrization . 36 2.1.3 velocity, speed, acceleration . 37 2.1.4 arc-length . 38 2.1.5 arc-length parametrization . 39 2.2 curvature . 42 2.2.1 technicalities . 42 2.2.2 the Serret-Frenet frame . 43 2.2.3 classical definition of curvature . 43 2.2.4 curvature as angular speed . 45 2.2.5 a.l.p. of the osculating circle . 46 2.2.6 formulae for t;n;k in R2 ................................ 47 2.2.7 involute and evolute . 48 2.3 curves in R3 ........................................... 49 2.3.1 curves in R3, basic definitions . 50 2.3.2 Serret-Frenet formulae . 52 2.3.3 3-space formulae for t;n;b;k;t ............................ 54 2.3.4 curves in R3 with constant curvature and torsion . 58 2.4 problem set ........................................... 61 3 surfaces 65 3.1 surfaces in R3 .......................................... 65 3.2 surface presentation . 65 3.2.1 presentation as a level set or contour . 65 3.2.2 graphical presentation . 66 3.2.3 parametric presentation . 66 3.3 first example, the sphere . 67 3.4 saddle and monkey saddle . 69 3.5 surface of revolution . 69 3.5.1 the sphere revisited . 70 3.5.2 the torus as surface of revolution . 70 3.5.3 hyperboloids and cone as surfaces of revolution . 70 3.6 ruled surfaces . 72 3.6.1 cylinder as ruled surface . 72 3.6.2 cone as ruled surface . 73 3.6.3 single sheeted hyperboloid as ruled surface . 73 3.6.4 saddle surface is doubly ruled . 73 3.6.5 the right helicoid or screw surface is ruled . 74 3.6.6 the tangent developable helicoid ruled surface . 74 3.7 charts, affine linear approximate . 74 3.7.1 chart, surface element . 74 3.7.2 affine linear approximate . 75 3.7.3 the tangent plane . 75 3.7.4 normal vector . 76 3.8 Gauss and Weingarten maps . 79 3.9 self adjointness of L ....................................... 80 3.10 eigenvalues and eigenvectors of a self adjoint mapping . 81 3.11 invariants of the Weingarten mapping . 81 3.12 geometric meaning of Weingarten mapping . 82 3.13 problem set ........................................... 83 c jbquig-UCD September 23, 2009 CONTENTS v 4 bilinear forms 85 4.1 definition of bilinear forms I; II and III . 85 4.2 matrix representation of the three fundamental forms . 85 4.2.1 the matrix of a form . 86 4.2.2 g,h,e matrices representing I;II;III . 86 4.2.3 the matrix of the Weingarten mapping . 88 4.2.4 Examples, computing g;h;e;L ............................. 88 4.3 explicit presentation and curvature . 93 4.4 surface as a graph over the tangent plane at a point . 95 4.5 problem set ........................................... 98 I Answers to problem sets of part I 101 5 Answers to questions in Chapter1 103 5.1 question and answer . 103 5.1.1 answer . 103 5.1.2 answer . 103 5.1.3 answer . 103 5.1.4 answer . 104 5.1.5 answer . 104 5.2 question and answer . 104 5.2.1 answer . 104 5.2.2 answer . 104 5.2.3 answer . 105 5.2.4 answer . 105 5.2.5 answer . 105 5.3 question and answer . 107 5.3.1 answer . 107 5.4 question and answer . 107 5.4.1 answer . 107 5.4.2 answer . 107 5.4.3 answer . ..