Exercises for Elementary Differential Geometry

Transcription

1 Exercises for Elementary Differential Geometry Chapter Is γ(t) =(t 2,t 4 )aparametrizationoftheparabolay = x 2? Find parametrizations of the following level curves: (i) y 2 x 2 =1. (ii) x2 4 + y2 9 = Find the Cartesian equations of the following parametrized curves: (i) γ(t) =(cos 2 t, sin 2 t). (ii) γ(t) =(e t,t 2 ) Calculate the tangent vectors of the curves in Exercise Sketch the astroid in Example Calculate its tangent vector at each point. At which points is the tangent vector zero? Consider the ellipse x 2 p 2 + y2 q 2 =1, where p>q>0. The eccentricity of the ellipse is ɛ = 1 q2 p 2 and the points (±ɛp, 0) on the x-axis are called the foci of the ellipse, which we denote by f 1 and f 2. Verify that γ(t) =(p cos t, q sin t) isaparametrizationoftheellipse. Prove that: (i) The sum of the distances from f 1 and f 2 to any point p on the ellipse does not depend on p. (ii) The product of the distances from f 1 and f 2 to the tangent line at any point p of the ellipse does not depend on p. (iii) If p is any point on the ellipse, the line joining f 1 and p and that joining f 2 and p make equal angles with the tangent line to the ellipse at p A cycloid is the plane curve traced out by a point on the circumference of acircle as it rolls without slipping along a straight line. Show that, ifthestraightline is the x-axis and the circle has radius a>0, the cycloid can be parametrized as γ(t) =a(t sin t, 1 cos t) Show that γ(t) =(cos 2 t 1 2, sin t cos t, sin t) isaparametrizationofthecurve of intersection of the circular cylinder of radius 1 2 and axis the z-axis with the 1

2 2 sphere of radius 1 and centre ( 1 2, 0, 0). below. This is called Viviani s Curve -see The normal line to a curve at a point p is the straight line passing through p perpendicular to the tangent line at p. Find the tangent and normal lines to the curve γ(t) =(2cost cos 2t, 2sint sin 2t) atthepointcorrespondingto t = π/ Find parametrizations of the following level curves: (i) y 2 = x 2 (x 2 1). (ii) x 3 + y 3 =3xy (the folium of Descartes) Find the Cartesian equations of the following parametrized curves: (i) γ(t) =(1 + cos t, sin t(1 + cos t)). (ii) γ(t) =(t 2 + t 3,t 3 + t 4 ) Calculate the tangent vectors of the curves in Exercise For each curve, determine at which point(s) the tangent vector vanishes If P is any point on the circle C in the xy-plane of radius a>0andcentre(0,a), let the straight line through the origin and P intersect the line y =2a at Q, and let the line through P parallel to the x-axis intersect the line through Q parallel to the y-axis at R. AsP moves around C, R traces out a curve called the witch of Agnesi. Forthiscurve,find (i) a parametrization; (ii) its Cartesian equation Generalize Exercise by finding parametrizations of an epicycloid (resp. hypocycloid), the curve traced out by a point on the circumference of a circle

3 as it rolls without slipping around the outside (resp. inside) of a fixed circle For the logarithmic spiral γ(t) =(e t cos t, e t sin t), show that the angle between γ(t) andthetangentvectoratγ(t) isindependentoft. (There is a picture of the logarithmic spiral in Example ) Show that all the normal lines to the curve are the same distance from the origin. γ(t) =(cost + t sin t, sin t t cos t) Calculate the arc-length of the catenary γ(t) =(t, cosh t) starting at the point (0, 1). This curve has the shape of a heavy chain suspended at its ends - see Exercise Show that( the following curves are unit-speed: 1 (i) γ(t) = 3 (1 + t)3/2, 1 3 (1 t)3/2 t, 2 ). (ii) γ(t) = ( 4 5 cos t, 1 sin t, 3 5 cos t) A plane curve is given by γ(θ) =(r cos θ, r sin θ), where r is a smooth function of θ (so that (r, θ) arethepolarcoordinatesof γ(θ)). Under what conditions is γ regular? Find all functions r(θ) for which γ is unit-speed. Show that, if γ is unit-speed, the image of γ is a circle; what is its radius? This exercise shows that astraightlineistheshortestcurvejoiningtwogiven points. Let p and q be the two points, and let γ be a curve passing through both, say γ(a) =p, γ(b) =q, wherea<b.showthat,ifu is any unit vector, and deduce that (q p).u γ.u γ b a γ dt. By taking u =(q p)/ q p, showthatthelengthofthepartofγ between p and q is at least the straight line distance q p Find the arc-length of the curve starting at t =0. γ(t) =(3t 2,t 3t 3 ) 3

4 Find, for 0 x π, thearc-lengthofthesegmentofthecurve corresponding to 0 t x. γ(t) =(2cost cos 2t, 2sint sin 2t) Calculate the arc-length along the cycloid in Exercise 1.1.7correspondingtoone complete revolution of the circle Calculate the length of the part of the curve γ(t) =(sinht t, 3 cosh t) cut off by the x-axis Show that a curve γ such that... γ = 0 everywhere is contained in a plane Which of the following curves are regular? (i) γ(t) =(cos 2 t, sin 2 t)fort R. (ii) the same curve as in (i), but with 0 <t<π/2. (iii) γ(t) =(t, cosh t) fort R. Find unit-speed reparametrizations of the regular curve(s) The cissoid of Diocles (see below) is the curve whose equation in terms of polar coordinates (r, θ) is r =sinθ tan θ, π/2 <θ<π/2. Write down a parametrization of the cissoid using θ as a parameter and show that ( γ(t) = t 2 t 3 ),, 1 <t<1, 1 t 2 is a reparametrization of it.

5 1.3.3 The simplest type of singular point of a curve γ is an ordinary cusp: apointp of γ, correspondingtoaparametervaluet 0,say,isanordinarycuspif γ(t 0 )=0 and the vectors γ(t 0 )and... γ (t 0 )arelinearlyindependent(inparticular,these vectors must both be non-zero). Show that: (i) the curve γ(t) =(t m,t n ), where m and n are positive integers, has an ordinary cusp at the origin if and only if (m, n) =(2, 3) or (3, 2); (ii) the cissoid in Exercise has an ordinary cusp at the origin; (iii) if γ has an ordinary cusp at a point p, sodoesanyreparametrizationofγ Show that: (i) if γ is a reparametrization of a curve γ, thenγ is a reparametrization of γ; (ii) if γ is a reparametrization of γ, andˆγ is a reparametrization of γ, thenˆγ is areparametrizationofγ Repeat Exercise for the following curves: (i) γ(t) =(t 2,t 3 ), t R. (ii) γ(t) =((1+cost)cost, (1 + cos t)sint), π <t<π Show that the curve γ(t) = ( ) 2 2t,, t > 0, 1+t 2 is regular and that it is a reparametrization of the curve γ(t) = ( ) 2cost 1+sint, 1+sint, π 2 <t<π The curve γ(t) =(a sin ωt, b sin t), where a, b and ω are non-zero constants, is called a Lissajous figure. Showthat γ is regular if and only if ω is not the quotient of two odd integers Let γ be a curve in R n and let γ be a reparametrization of γ with reparametrization map φ (so that γ( t) =γ(φ( t))). Let t 0 be a fixed value of t and let t 0 = φ( t 0 ). Let s and s be the arc-lengths of γ and γ starting at the point γ(t 0 )= γ( t 0 ). Prove that s = s if dφ/d t >0forall t, and s = s if dφ/d t <0forall t Suppose that all the tangent lines of a regular plane curve pass through some fixed point. Prove that the curve is part of a straight line. Prove the same result if all the normal lines are parallel Show that the Cayley sextic γ(t) =(cos 3 t cos 3t, cos 3 t sin 3t), t R,

6 6 is a closed curve which has exactlyone self-intersection. What is its period? (The name of this curve derives from the fact that its Cartesian equation involves a polynomial of degree six.) Give an example to show that a reparametrization of a closed curve need not be closed Show that if a curve γ is T 1 -periodic and T 2 -periodic, it is (k 1 T 1 +k 2 T 2 )-periodic for any integers k 1,k Let γ : R R n be a curve and suppose that T 0 is the smallest positive number such that γ is T 0 -periodic. Prove that γ is T -periodic if and only if T = kt 0 for some integer k Suppose that a non-constant function γ : R R is T -periodic for some T 0. This exercise shows that there is a smallest positive T 0 such that γ is T 0 -periodic. The proof uses a little real analysis. Suppose for a contradiction that there is no such T 0. (i) Show that there is a sequence T 1,T 2,T 3,... such that T 1 >T 2 >T 3 > > 0 and that γ is T r -periodic for all r 1. (ii) Show that the sequence {T r } in (i) can be chosen so that T r 0asr. (iii) Show that the existence of a sequence {T r } as in (i) such that T r 0as r implies that γ is constant Let γ : R R n be a non-constant curve that is T -periodic for some T > 0. Show that γ is closed Show that the following curve is not closed and that it has exactly one selfintersection: ( t 2 ) 3 γ(t) = t 2 +1, t(t2 3). t Show that the curve γ(t) =((2 + cos t)cos ωt, (2 + cos t)sin ωt, sin t), where ω is a constant, is closed if and only if ω is a rational number. Show that, if ω = m/n where m and n are integers with no common factor, the period of γ is 2πn Show that the curve C with Cartesian equation y 2 = x(1 x 2 ) is not connected. For what range of values of t is γ(t) =(t, t t 3 )

7 aparametrizationofc? Whatistheimageofthisparametrization? State an analogue of Theorem for level curves in R 3 given by f(x, y, z) = g(x, y, z)= State and prove an analogue of Theorem for curves in R 3 (or even R n ). (This is easy.) Show that the conchoid (x 1) 2 (x 2 + y 2 )=x 2 is not connected, but is the union of two disjoint connected curves (consider the line x =1). Howdoyoureconcilethiswithits(single)parametrization γ(t) =(1+cost, sin t +tant)? Show that the condition on f and g in Exercise is satisfied for the level curve given by x 2 + y 2 = 1 4, x2 + y 2 + z 2 + x = 3 4 except at the point (1/2, 0, 0). Note that Exercise gives a parametrization γ of this level curve; is (1/2, 0, 0) a singular point of γ? Sketch the level curve C given by f(x, y) =0whenf(x, y) =y x. Note that f does not satisfy the conditions in Theorem because f/ x does not exist at the point (0, 0) on the curve. Show nevertheless that there is a smooth parametrized curve γ whose image is the whole of C. Istherearegular parametrized curve with this property? 7 Chapter Compute the ( curvature of the following curves: 1 (i) γ(t) = 3 (1 + t)3/2, 1 3 (1 t)3/2 t, 2 ). (ii) γ(t) = ( 4 5 cos t, 1 sin t, 3 5 cos t). (iii) γ(t) =(t, cosh t). (iv) γ(t) =(cos 3 t, sin 3 t). For the astroid in (iv), show that the curvature tends to as we approach one of the points (±1, 0), (0, ±1). Compare with the sketch found in Exercise Show that, if the curvature κ(t) ofaregularcurveγ(t) is> 0everywhere,then κ(t) isasmoothfunctionoft. Give an example to show that this may not be the case without the assumption that κ> Show that the curvature of the curve γ(t) =(t sinh t cosh t, 2cosht), t > 0,

8 8 is never zero, but that it tends to zero as t Show that the curvature of the curve γ(t) =(sect, sec t tan t), π/2 <t<π/2, vanishes at exactly two points on the curve Show that, if γ is a unit-speed plane curve, ṅn s = κ s t Show that the signed curvature of any regular plane curve γ(t) isasmooth function of t. (Compare with Exercise ) Let γ and γ be two plane curves. Show that, if γ is obtained from γ by applying an isometry M of R 2,thesignedcurvaturesκ s and κ s of γ and γ are equal if M is direct but that κ s = κ s if M is opposite (in particular, γ and γ have the same curvature). Show, conversely, that if γ and γ have the same nowhere-vanishing curvature, then γ can be obtained from γ by applying an isometry of R Let k be the signed curvature of a plane curve C expressed in terms of its arclength. Show that, if C a is the image of C under the dilation v av of the plane (where a is a non-zero constant), the signed curvature of C a in terms of its arc-length s is 1 a k( s a ). Aheavychainsuspendedatitsendshanginglooselytakestheform of a plane curve C. Showthat,ifs is the arc-length of C measured from its lowest point, ϕ the angle between the tangent of C and the horizontal, and T the tension in the chain, then T cos ϕ = λ, T sin ϕ = µs, where λ, µ are non-zero constants (we assume that the chain has constant mass per unit length). Show that the signed curvature of C is ) 1 (1+ s2, κ s = 1 a a 2 where a = λ/µ, anddeducethatc can be obtained from the catenary in Example by applying a dilation and an isometry of the plane Let γ(t) bearegularplanecurveandletλ be a constant. The parallel curve γ λ of γ is defined by γ λ (t) =γ(t)+λn s (t). Show that, if λκ s (t) 1forallvaluesoft, thenγ λ is a regular curve and that its signed curvature is κ s / 1 λκ s.

9 2.2.6 Another approach to the curvature of a unit-speed plane curveγ at a point γ(s 0 ) is to look for the best approximating circle at this point. We can then define the curvature of γ to be the reciprocal of the radius of this circle. Carry out this programme by showing that the centre of the circle which passes through three nearby points γ(s 0 )andγ(s 0 ± δs) onγ approaches the point ɛ(s 0 )=γ(s 0 )+ 1 κ s (s 0 ) n s (s 0 ) as δs tends to zero. The circle C with centre ɛ(s 0 )passingthroughγ(s 0 )iscalled the osculating circle to γ at the point γ(s 0 ), and ɛ(s 0 )iscalledthecentre of curvature of γ at γ(s 0 ). The radius of C is 1/ κ s (s 0 ) =1/κ(s 0 ), where κ is the curvature of γ -thisiscalledtheradius of curvature of C at γ(s 0 ) With the notation in the preceding exercise, we regard ɛ as the parametrization of a new curve, called the evolute of γ (if γ is any regular plane curve, its evolute is defined to be that of a unit-speed reparametrization of γ). Assume that κ s (s) 0forallvaluesofs (a dot denoting d/ds), say κ s > 0foralls (this can be achieved by replacing s by s if necessary). Show that the arc-length of ɛ is 1 κ s (s) (up to adding a constant), and calculate the signed curvature ofɛ. Show also that all the normal lines to γ are tangent to ɛ (for this reason, the evolute of γ is sometimes described as the envelope of the normal lines to γ). Show that the evolute of the cycloid 9 γ(t) =a(t sin t, 1 cos t), 0 <t<2π, where a>0isaconstant,is ɛ(t) =a(t +sint, 1+cost) (see Exercise 1.1.7) and that, after a suitable reparametrization, ɛ can be obtained from γ by a translation of the plane A string of length l is attached to the point γ(0) of a unit-speed plane curve γ(s). Show that when the string is wound onto the curve while being kept taught, its endpoint traces out the curve ι(s) =γ(s)+(l s) γ(s), where 0 <s<land a dot denotes d/ds. Thecurveι is called the involute of γ (if γ is any regular plane curve, we define its involute to be that of aunit-speed reparametrization of γ). Suppose that the signed curvature κ s of γ is never zero, say κ s (s) > 0foralls. Showthatthesignedcurvatureofι is 1/(l s).

10 Show that the involute of the catenary γ(t) =(t, cosh t) with l =0(seetheprecedingexercise)isthetractrix ( ) 1 x =cosh 1 1 y y 2. See 8.3 for a simple geometric characterization of this curve A unit-speed plane curve γ(s)rollswithoutslippingalongastraightlinel parallel to a unit vector a, andinitiallytouchesl at a point p = γ(0). Let q be a point fixed relative to γ. LetΓ(s) bethepointtowhichq has moved when γ has rolled adistances along l (note that Γ will not usually be unit-speed). Let θ(s) bethe angle between a and the tangent vector γ. Showthat Γ(s) =p + sa + ρ θ(s) (q γ(s)), where ρ ϕ is the rotation about the origin through an angle ϕ. Showfurtherthat Γ(s).ρ θ(s) (q γ(s)) = 0. Geometrically, this means that a point on Γ moves as if it is rotating about the instantaneous point of contact of the rolling curve with l. SeeExercise1.1.7for aspecialcase Show that, if two plane curves γ(t) and γ(t) havethesamenon-zerocurvature for all values of t, then γ can be obtained from γ by applying an isometry of R Show that if all the normal lines to a plane curve pass through some fixed point, the curve is part of a circle Let γ(t) =(e kt cos t, e kt sin t), where <t< and k is a non-zero constant (a logarithmic spiral see Example 1.2.2). Show that there is auniqueunitspeed parameter s on γ such that s>0forallt and s 0ast if ±k >0, and express s as a function of t. Show that the signed curvature of γ is 1/ks. Conversely, describe every curve whose signed curvature, as a function of arc-length s, is 1/ks for some non-zero constant k If γ is a plane curve, its pedal curve with respect to a fixed point p is the curve traced out by the foot of the perpendicular from p to the tangent line at a variable point of the curve. If γ is unit-speed, show that the pedal curve is parametrized by δ =p +((γ p).n s )n s,

11 where n s is the signed unit normal of γ. Showthatδ is regular if and only if γ has nowhere vanishing curvature and does not pass through p. Show that the pedal curve of the circle γ(t) =(cost, sin t) withrespecttothe point ( 2, 0) is obtained by applying a translation to the limaçon in Example A unit-speed plane curve γ has the property that its tangent vector t(s) makes afixedangleθ with γ(s) foralls. Showthat: (i) If θ =0,thenγ is part of a straight line. (ii) If θ = π/2, then γ is a circle. (iii) If 0 <θ<π/2, then γ is a logarithmic spiral Let γ λ be a parallel curve of the parabola γ(t) =(t, t 2 ). Show that: (i) γ λ is regular if and only if λ<1/2. (ii) If λ>1/2, γ λ has exactly two singular points. What happens if λ =1/2? This exercise gives another approach to the definition ofthe bestapproximating circle to a curve γ at a point γ(t 0 )ofγ -seeexercise2.2.6.weassumethatγ is unit-speed for simplicity. Let C be the circle with centre c and radius R, andconsiderthesmoothfunction F (t) = γ(t) c 2 R 2. Show that F (t 0 )= F (t 0 )=0ifandonlyifC is tangent to γ at γ(t 0 ). This suggests that the best approximating circle can be defined by the three conditions F (t 0 )= F (t 0 )= F (t 0 )=0. Showthat,if γ(t 0 ) 0, theuniquecirclec for which F satisfies these conditions is the osculating circle to γ at the point γ(t 0 ) Show that the evolute of the parabola γ(t) =(t, t 2 )isthesemi-cubical parabola ɛ(t) = ( 4t 3, 3t ) Show that the evolute of the ellipse γ(t) =(a cos t, b sin t), where a>b>0are constants, is the astroid ( a 2 b 2 ɛ(t) = cos 3 t, b2 a 2 ) sin 3 t. a b (Compare Example ) Show that all the parallel curves (Exercise 2.2.5) of agivencurvehavethesame evolute Let γ be a regular plane curve. Show that: (i) The involute of the evolute of γ is a parallel curve of γ. (ii) The evolute of the involute of γ is γ. (These statements might be compared to the fact that the integral of the derivative of a smooth function f is equal to f plus a constant, while the derivative of the integral of f is f.) 11

12 A closed plane curve γ is parametrized by the direction of its normal lines, i.e. γ(θ) isa2π-periodic curve such that θ is the angle between the normal line at γ(θ) andthepositivex-axis. Let p(θ) bethedistancefromtheorigintothe tangent line ( at γ(θ). Show that: (i) γ(θ) = p cos θ dp dp dθ sin θ, p sin θ + dθ ). cos θ (ii) γ is regular if and only if p + d2 p > 0forallθ (we assume that this condition dθ 2 holds in the remainder of this exercise). ( ) (iii) The signed curvature of γ is κ s = p + d2 p 1. dθ 2 (iv) The length of γ is 2π 0 p(θ)dθ. (v) The tangent lines at the points γ(θ) andγ(θ + π) areparallelandadistance w(θ) = p(θ)+p(θ + π) apart(w(θ) is called the width of γ in the direction θ). (vi) γ has a circumscribed square, i.e. a square all of whose sides are tangent to γ. (vii) If γ has constant width D, itslengthisπd; (viii) Taking p(θ) =a cos 2 (kθ/2) + b, wherek is an odd integer and a and b are constants with b>0, a + b>0, gives a curve of constant width a +2b. (ix) The curve in (viii) is a circle if k =1butnotif k > Show that if the parabola y = 1 2 x2 rolls without slipping on the x-axis, the curved traced out by the point fixed relative to the parabola and initially at (0, 1) can be parametrized by γ(t) = 1 (t +tanht, cosh t +secht) Show that, if γ(t) isaclosedcurveofperiodt 0,andt, n s and κ s are its unit tangent vector, signed unit normal and signed curvature, respectively, then t(t + T 0 )=t(t), n s (t + T 0 )=n s (t), κ s (t + T 0 )=κ s (t) Compute κ, τ, t, n and b for each of the following curves, and verify that the Frenet Serret ( equations are satisfied: 1 (i) γ(t) = 3 (1 + t)3/2, 1 3 (1 t)3/2 t, 2 ). (ii) γ(t) = ( 4 5 cos t, 1 sin t, 3 5 cos t). Show that the curve in (ii) is a circle, and find its centre, radius and the plane in which it lies Describe all curves in R 3 which have constant curvature κ>0andconstant torsion τ A regular curve γ in R 3 with curvature > 0iscalledageneralized helix if its tangent vector makes a fixed angle θ with a fixed unit vector a. Show that the

13 torsion τ and curvature κ of γ are related by τ = ±κ cot θ. Show conversely that, if the torsion and curvature of a regular curve are related by τ = λκ where λ is a constant, then the curve is a generalized helix. In view of this result, Examples and show that a circular helix is a generalized helix. Verify this directly Let γ(t) beaunit-speedcurvewithκ(t) > 0andτ(t) 0forallt. Show that, if γ is spherical, i.e.ifitliesonthesurfaceofasphere,then τ (2.22) κ = d ( ) κ. ds τκ 2 Conversely, show that if Eq holds, then ρ 2 +( ρσ) 2 = r 2 for some (positive) constant r, whereρ =1/κ and σ =1/τ, anddeducethat γ lies on a sphere of radius r. Verify that Eq holds for Viviani s curve (Exercise 1.1.8) Let P be an n n orthogonal matrix and let a R n,sothatm(v) =P v +a is an isometry of R 3 (see Appendix 1). Show that, if γ is a unit-speed curve in R n,thecurveγ = M(γ) isalsounit-speed. Showalsothat,ift, n, b and T, N, B are the tangent vector, principal normal and binormal of γ and Γ, respectively, then T = P t, N = P n and B = P b Let (a ij )beaskew-symmetric3 3 matrix(i.e. a ij = a ji for all i, j). Let v 1, v 2 and v 3 be smooth functions of a parameter s satisfying the differential equations v i = 3 a ij v j, j=1 for i =1, 2and3,andsupposethatforsomeparametervalues 0 the vectors v 1 (s 0 ), v 2 (s 0 )andv 3 (s 0 )areorthonormal. Showthatthevectorsv 1 (s), v 2 (s) and v 3 (s) areorthonormalforallvaluesofs Repeat Exercise ( for the following unit-speed curves: (i) γ(t) = sin 2 t 2, 1 2 sin t t 2, 2 ). ( ) 1 (ii) γ(t) = 3 cos t sin t, 3 1 cos t, 3 cos t 1 2 sin t Repeat Exercise for the curve γ(t) = 1 2 (cosh t, sinh t, t) 13 (which is not unit-speed).

14 Show that the curve ( 1+t 2 γ(t) = t is planar Show that the curvature of the curve,t+1, 1 t ) t γ(t) =(t cos(ln t),tsin(ln t),t), t > 0 is proportional to 1/t Show that the torsion of a regular curve γ(t) isasmoothfunctionoft whenever it is defined Let γ(t) beaunit-speedcurveinr 3, and assume that its curvature κ(t) is non-zero for all t. Defineanewcurveδ by δ(t) = dγ(t). dt Show that δ is regular and that, if s is an arc-length parameter for δ, then Prove that the curvature of δ is ds dt = κ. (1+ τ 2 κ 2 ) 1 2, and find a formula for the torsion of δ in terms of κ, τ and their derivatives with respect to t Show that the curve (shown below) on the cone σ(u, v) =(u cos v, u sin v, u) given by u = e λt, v = t, whereλ is a constant, is a generalized helix.

15 Show that the twisted cubic γ(t) =(at, bt 2,ct 3 ), where a, b and c are constants, is a generalized helix if and only if 3ac = ±2b A space curve γ is called a Bertrand mate of a space curve γ if, for each point P of γ, thereisapoint P of γ such that the line P P is parallel both to the principal normal of γ at P and to the principal normal of γ at P. If γ has a Bertrand mate it is called a Bertrand curve. Assume that γ and γ are unit-speed and let γ( s) bethepointof γ corresponding to the point γ(s) ofγ, where s is a smooth function of s. Showthat: (i) γ( s) = γ(s)+ an(s), where n is the principal normal of γ and a is a constant. (ii) There is a constant α such that the tangent vector, principal normal and binormal of γ and γ at corresponding points are related by t =cosαt sin αb, ñn = ±n, b = ±(sin αt +cosαb), where the signs in the last two equations are the same. (iii) The curvature and torsion of γ and γ at corresponding points are related by cos α d s ds =1 aκ, cos α ds d s =1+a κ, sin αd s ds = aτ, sin αds d s = a τ. (iv) aκ aτ cot α =1. (v) a 2 τ τ =sin 2 α,(1 aκ)(1 + a κ) =cos 2 α Show that every plane curve is a Bertrand curve Show that a space curve γ with nowhere vanishing curvature κ and nowhere vanishing torsion τ is a Bertrand curve if and only if there exist constants a, b such that aκ + bτ = Show that a Bertrand curve C with nowhere vanishing curvature and torsion has more than one Bertrand mate if and only if it is a circular helix, in which case it has infinitely-many Bertrand mates, all of which are circular helices with the same axis and pitch as C Show that a spherical curve of constant curvature is a circle.

16 The normal plane at a point P of a space curve C is the plane passing through P perpendicular to the tangent line of C at P.Showthat,ifallthenormalplanes of a curve pass through some fixed point, the curve is spherical Let γ be a curve in R 3 and let Π be a plane v.n = d, where N and d are constants with N 0, andv =(x, y, z). Let F (t) =γ(t).n d. Show that: ) F (t 0 )=0ifandonlyifγ intersects Π at the point γ(t 0 ). (ii) F (t 0 )= F (t 0 )=0ifandonlyifγ touches Π at γ(t 0 )(i.e. γ(t 0 )isparallel to Π). (iii) If the curvature of γ at γ(t 0 )isnon-zero,thereisauniqueplaneπsuch that F (t 0 )= F (t 0 )= F (t 0 )=0, and that this plane Π is the plane passing through γ(t 0 )parallelto γ(t 0 )and γ(t 0 )(Πiscalledtheosculating plane of γ at γ(t 0 ); intuitively, it is the plane which most closely approaches γ near the point γ(t 0 )). (iv) If γ is contained in a plane Π,thenΠ is the osculating plane of γ at each of its points. (v) If the torsion of γ is non-zero at γ(t 0 ), then γ crosses its osculating plane there. Compare Exercise Find the osculating plane at a general point of the circular helix γ(t) =(a cos t, a sin t, bt) Show that the osculating planes at any three distinct points P 1,P 2,P 3 of the twisted cubic γ(t) =(t, t 2,t 3 ) meet at a single point Q, andthatthefourpointsp 1,P 2,P 3,Qall lie in a plane Suppose that a curve γ has nowhere vanishing curvature and that each of its osculating planes pass through some fixed point. Prove that the curve lies in a plane Show that the orthogonal projection of a curve C onto its normal plane at a point P of C is a plane curve which has an ordinary cusp at P provided that C has non-zero curvature and torsion at P (see Exercise 1.3.3). Show, on the

17 other hand, that P is a regular point of the orthogonal projection of C onto its osculating plane at P Let S be the sphere with centre c and radius R. Letγ be a unit-speed curve in R 3 and let F (t) = γ(t) c 2 R 2. Let t 0 R. Showthat: (i) F (t 0 )=0ifandonlyifγ intersects S at the point γ(t 0 ). (ii) F (t 0 )= F (t 0 )=0ifandonlyifγ is tangent to S at γ(t 0 ). Compute F and F... and show that there is a unique sphere S (called the osculating sphere of γ at γ(t 0 )) such that F (t 0 )= F (t 0 )= F (t 0 )= F... (t 0 )=0. Show that the centre of S is c = γ + 1 κ n κ κ 2 τ b in the usual notation, all quantities being evaluated at t = t 0.Whatisitsradius? The point c(t 0 )iscalledthecentre of spherical curvature of γ at γ(t 0 ). Show that c(t 0 )isindependentoft 0 if and only if γ is spherical, in which case the sphere on which γ lies is its osculating sphere The osculating circle of a curve γ at a point γ(t 0 )istheintersectionofthe osculating plane and the osculating sphere of γ at γ(t 0 ). Show that the centre of the osculating circle is the centre of curvature γ + 1 κ n, and that its radius is 1/κ, allquantitiesbeingevaluatedatt = t 0. (Compare Exercise ) Show that Chapter 3 γ(t) =((1+a cos t)cost, (1 + a cos t)sint), where a is a constant, is a simple closed curve if a < 1, but that if a > 1its complement is the disjoint union of three connected subsets of R 2,twoofwhich are bounded and one is unbounded. What happens if a = ±1? Show that, if γ is as in Exercise 3.1.1, its turning angle ϕ satisfies dϕ dt =1+ a(cos t + a) 1+2a cos t + a 2.

18 18 Deduce that 2π 0 { a(cos t + a) 0 if a < 1, 1+2acos t + a 2 dt = 2π if a > Show that the length l(γ) andtheareaa(γ) areunchangedbyapplyingan isometry to γ By applying the isoperimetric inequality to the ellipse x 2 p 2 + y2 q 2 =1 (where p and q are positive constants), prove that 2π with equality holding if and only if p = q. 0 p 2 sin 2 t + q 2 cos 2 tdt 2π pq, What is the area of the interior of the ellipse where p and q are positive constants? γ(t) =(p cos t, q sin t), Show that the ellipse in Example is convex Show that the limac on in Example has only two vertices (cf. Example 3.1.3) Show that a plane curve γ has a vertex at t = t 0 if and only if the evolute ɛ of γ (Exercise 2.2.7) has a singular point at t = t Show that the vertices of the curve y = f(x) satisfy Show that the curve ( 1+ ( ) ) 2 df d 3 ( f d 2 ) 2 dx dx 3 =3df f dx dx 2. γ(t) =(at b sin t, a b cos t), where a and b are non-zero constants, has vertices at the points γ(nπ) forall integers n. Showthatthesearealltheverticesofγ unless a b b 2b a a + b, b

19 19 in which case there are infinitely-many other vertices. Chapter Show that any open disc in the xy-plane is a surface Define surface patches σ x ± : U R3 for S 2 by solving the equation x 2 +y 2 +z 2 =1 for x in terms of y and z: σ x ± (u, v) =(± 1 u 2 v 2,u,v), defined on the open set U = {(u, v) R 2 u 2 + v 2 < 1}. Define σ y ± and σ z ± similarly (with the same U) bysolvingfory and z, respectively.showthatthese six patches give S 2 the structure of a surface The hyperboloid of one sheet is Show that, for every θ, thestraightline S = {(x, y, z) R 3 x 2 + y 2 z 2 =1}. (x z)cosθ =(1 y)sinθ, (x + z)sinθ =(1+y)cosθ is contained in S, andthateverypointofthehyperboloidliesononeofthese lines. Deduce that S can be covered by a single surface patch, and hence is a surface. (Compare the case of the cylinder in Example )

20 20 Find a second family of straight lines on S, andshowthatnotwolinesofthe same family intersect, while every line of the first family intersects every line of the second family with one exception. One says that the surface S is doubly ruled Show that the unit cylinder can be covered by a single surface patch, but that the unit sphere cannot. (The second part requires some point set topology.) Show that every open subset of a surface is a surface Show that a curve on the unit cylinder that intersects the straight lines on the cylinder parallel to the z-axis at a constant angle must be a straight line, a circle or a circular helix Find a surface patch for the ellipsoid x 2 p 2 + y2 q 2 + z2 r 2 =1, where p, q and r are non-zero constants. (A picture of an ellipsoid can be found in Theorem ) Show that σ(u, v) =(sinu, sin v, sin(u + v)), π/2 <u,v<π/2 is a surface patch for the surface with Cartesian equation (x 2 y 2 + z 2 ) 2 =4x 2 z 2 (1 y 2 ) Show that, if f(x, y) isasmoothfunction,itsgraph {(x, y, z) R 3 z = f(x, y)}

21 is a smooth surface with atlas consisting of the single regular surface patch σ(u, v) =(u, v, f(u, v)). In fact, every surface is locally of this type - see Exercise Verify that the six surface patches for S 2 in Exercise are regular. Calculate the transition maps between them and verify that these maps are smooth Which of the following are regular surface patches (in each case, u, v R): (i) σ(u, v) =(u, v, uv). (ii) σ(u, v) =(u, v 2,v 3 ). (iii) σ(u, v) =(u + u 2,v,v 2 )? Show that the ellipsoid p + y2 2 q + z2 2 r =1, 2 where p, q and r are non-zero constants, is a smooth surface. x A torus (see above) is obtained by rotating a circle C in a plane Π around a straight line l in Π that does not intersect C. TakeΠtobethexz-plane, l to be the z-axis, a>0thedistanceofthecentreofc from l, andb<athe radius of C. Showthatthetorusisasmoothsurfacewithparametrization σ(θ, ϕ) =((a + b cos θ)cosϕ, (a + b cos θ)sinϕ, b sin θ).

22 A helicoid is the surface swept out by an aeroplane propeller, when both the aeroplane and its propeller move at constant speed (see the picture above). If the aeroplane is flying along the z-axis, show that the helicoid can be parametrized as σ(u, v) =(v cos u, v sin u, λu), where λ is a constant. Show that the cotangent of the angle that the standard unit normal of σ at a point p makes with the z-axis is proportional to the distance of p from the z-axis Let γ be a unit-speed curve in R 3 with nowhere vanishing curvature. The tube of radius a>0aroundγ is the surface parametrized by σ(s, θ) =γ(s)+a(n(s)cosθ +b(s)sinθ), where n is the principal normal of γ and b is its binormal. Give a geometrical description of this surface. Prove that σ is regular if the curvature κ of γ is less than a 1 everywhere. Note that, even if σ is regular, the surface σ will have self-intersections if the curve γ comes within a distance 2a of itself. This illustrates the fact that regularity is a local property: if (s, θ) is restricted to lie in a sufficiently small open subset U of R 2, σ : U R 3 will be smooth and injective (so there will be no selfintersections) - see Exercise We shall see other instances of this later (e.g. Example ). The tube around a circular helix Show that translations and invertible linear transformations of R 3 take smooth surfaces to smooth surfaces Show that every open subset of a smooth surface is a smooth surface Show that the graph in Exercise is diffeomorphic to an open subset of a plane.

23 Show that the surface patch in Exercise is regular Show that the torus in Exercise can be covered by three patches σ(θ, ϕ), with (θ, ϕ) lyinginanopenrectangleinr 2,butnotbytwo For which values of the constant c is z(z +4)=3xy + c 23 asmoothsurface? Show that x 3 +3(y 2 + z 2 ) 2 =2 is a smooth surface Let S be the astroidal sphere x 2/3 + y 2/3 + z 2/3 =1. Show that, if we exclude from S its intersections with the coordinate planes, we obtain a smooth surface S Show that the surface xyz =1 is not connected, but that it is the disjoint union of four connected surfaces. Find a parametrization of each connected piece Show that the set of mid-points of the chords of a circular helix is a subset of a helicoid If S is a smooth surface, define the notion of a smooth function S R. Show that, if S is a smooth surface, each component of the inclusion map S R 3 is asmoothfunctions R Let S be the half-cone x 2 + y 2 = z 2, z>0(seeexample4.1.5). Defineamap f from the half-plane {(0,y,z) y>0} to S by f(0,y,z)=(y cos z, y sin z, y). Show that f is a local diffeomorphism but not a diffeomorphism Find the equation of the tangent plane of each of the following surface patches at the indicated points: (i) σ(u, v) =(u, v, u 2 v 2 ), (1, 1, 0). (ii) σ(r, θ) =(r cosh θ, r sinh θ, r 2 ), (1, 0, 1) Show that, if σ(u, v) isasurfacepatch,thesetoflinearcombinationsofσ u and σ v is unchanged when σ is reparametrized.

24 Let S be a surface, let p Sand let F : R 3 R be a smooth function. Let S F be the perpendicular projection of the gradient F =(F x,f y,f z )off onto TpS. Showthat,ifγ is any curve on S passing through p when t = t 0,say, ( S F ). γ(t 0 )= d dt F (γ(t)). t=t0 Deduce that S F =0 if the restriction of F to S has a local maximum or a local minimum at p Let f : S 1 S 2 be a local diffeomorphism and let γ be a regular curve on S 1. Show that f γ is a regular curve on S Find the equation of the tangent plane of the torus in Exercise at the point corresponding to θ = ϕ = π/ Calculate the transition map Φ between the two surface patches for the Möbius band in Example Show that it is defined on the union of two disjoint rectangles in R 2,andthatthedeterminantoftheJacobianmatrixofΦisequal to +1 on one of the rectangles and to 1 ontheother Suppose that two smooth surfaces S and S are diffeomorphic and that S is orientable. Prove that S is orientable Show that for the latitude-longitude parametrization ofs 2 (Example 4.1.4) the standard unit normal points inwards. What about the parametrizations given in Exercise 4.1.2? Let γ be a curve on a surface patch σ, andletv be a unit vector field along γ, i.e. v(t) isaunittangentvectortoσ for all values of the curve parameter t, and v is a smooth function of t. Let ṽv be the result of applying a positive rotation through π/2 tov. Supposethat,forsomefixedparametervaluet 0, γ(t 0 )=cosθ 0 v(t 0 )+sinθ 0 ṽv(t 0 ). Show that there is a smooth function θ(t) suchthatθ(t 0 )=θ 0 and γ(t) =cosθ(t)v(t)+sinθ(t)ṽv(t) for all t The map F : R 3 \{(0, 0, 0)} R 3 \{(0, 0, 0)} given by F (v) = is called inversion with respect to S 2 (compare the discussion of inversion in circles in Appendix 2). Geometrically, F (v) is the point on the radius from the origin passing through v such that the product of the distances of v and F (v) v v.v

25 from the origin is equal to 1. Let S be a surface that does not pass through the origin, and let S = F (S). Show that, if S is orientable, then so is S.Show,in fact, that if N is the unit normal of S at a point p, thatofs at F (p) is N = 2(p.N) p 2 p N. 25 Chapter Show that the following are smooth surfaces: (i) x 2 + y 2 + z 4 =1; (ii) (x 2 + y 2 + z 2 + a 2 b 2 ) 2 =4a 2 (x 2 + y 2 ), where a>b>0areconstants. Show that the surface in (ii) is, in fact, the torus of Exercise Consider the surface S defined by f(x, y, z) =0,wheref is a smooth function such that f does not vanish at any point of S. Showthat f is perpendicular to the tangent plane at every point of S, anddeducethats is orientable. Suppose now that F : R 3 R is a smooth function. Show that, if the restriction of F to S has a local maximum or a local minimum at p then, at p, F = λ f for some scalar λ. (This is called Lagrange s Method of Undetermined Multipliers.) Show that the smallest value of x 2 + y 2 + z 2 subject to the condition xyz =1is 3, and that the points (x, y, z) thatgivethisminimumvaluelieatthevertices of a regular tetrahedron in R Write down parametrizations of each of the quadrics in parts (i) (xi) of Theorem (in case (vi) one must remove the origin) Show that the quadric x 2 + y 2 2z 2 2 xy +4z = c 3 is a hyperboloid of one sheet if c>2, and a hyperboloid of two sheets if c<2. What if c =2? (Thisexerciserequiresaknowledgeofeigenvaluesandeigenvectors.) Show that, if a quadric contains three points on a straight line, it contains the whole line. Deduce that, if L 1,L 2 and L 3 are non-intersecting straight lines in R 3,thereisaquadriccontainingallthreelines Use the preceding exercise to show that any doubly ruled surfaceis(partof) aquadricsurface. (Asurfaceisdoublyruledifitistheunionofeachoftwo families of straight lines such that no two lines of the same family intersect, but

26 26 every line of the first family intersects every line of the second family, with at most a finite number of exceptions.) Which quadric surfaces are doubly ruled? By setting u = x p y q, v = x p + y q, find a surface patch covering the hyperbolic paraboloid x 2 p 2 y2 q 2 = z. Deduce that the hyperbolic paraboloid is doubly ruled A conic is a level curve of the form ax 2 + by 2 +2cxy + dx + ey + f =0, where the coefficients a, b, c, d, e and f are constants, not all of which are zero. By imitating the proof of Theorem 5.2.2, show that any non-empty conic that is not a straight line or a single point can be transformed by applying a direct isometry of R 2 into one of the following: (i) An ellipse x2 + y2 =1. p 2 q 2 (ii) A parabola y 2 =2px. (iii) A hyperbola x2 p y2 2 q =1. 2 (iv) A pair of intersecting straight lines y 2 = p 2 x 2. Here, p and q are non-zero constants Show that: (i) Any connected quadric surface is diffeomorphic to a sphere, a circular cylinder or a plane. (ii) Each connected piece of a non-connected quadric surface isdiffeomorphicto aplane The surface obtained by rotating the curve x = cosh z in the xz-plane around the z-axis is called a catenoid (illustrated below). Describe an atlas for this surface Show that σ(u, v) =(sechu cos v, sech u sin v, tanh u) is a regular surface patch for S 2 (it is called Mercator s projection). Show that meridians and parallels on S 2 correspond under σ to perpendicular straight lines in the plane. (This patch is derived in Exercise )

27 Show that, if σ(u, v) isthe(generalized)cylinderinexample5.3.1: (i) The curve γ(u) = γ(u) (γ(u).a)a is contained in a plane perpendicular to a. (ii) σ(u, v) = γ(u)+ṽa, whereṽ = v + γ(u).a. (iii) σ(u, ṽ) = γ(u)+ ṽa is a reparametrization of σ(u, v). This exercise shows that, when considering a generalized cylinder σ(u, v) = γ(u)+va, wecanalwaysassumethatthecurveγ is contained in a plane perpendicular to the vector a Consider the ruled surface (5.5) σ(u, v) =γ(u)+vδ(u), where δ(u) =1and δ(u) 0 for all values of u (a dot denotes d/du). Show that there is a unique point Γ(u), say, on the ruling through γ(u) atwhich δ(u) is perpendicular to the surface. The curve Γ is called the line of striction of the ruled surface σ (of course, it need not be a straight line). Show that Γ. δ =0. γ. δ Let ṽ = v + δ,andlet σ(u, ṽ) bethecorrespondingreparametrizationofσ. 2 Then, σ(u, ṽ) =Γ(u)+ṽδ(u). This means that, when considering ruled surfaces as in (5.5), we can always assume that γ. δ =0. Weshallmakeuseofthisin Chapter A loxodrome is a curve on a sphere that intersects the meridians at a fixed angle, say α. Show that, in the Mercator surface patch σ of S 2 in Exercise 5.3.2, a unit-speed loxodrome satisfies u =cosα cosh u, v = ± sin α cosh u

28 28 (a dot denoting differentiation with respect to the parameter oftheloxodrome). Deduce that loxodromes correspond under σ to straight lines in the uv-plane A conoid is a ruled surface whose rulings are parallel to a given plane Πandpass through a given straight line L perpendicular to Π. If Π is the xy-plane and L is the z-axis, show that σ(u, θ) =(u cos θ, u sin θ, f(θ)), u 0 is a regular surface patch for the conoid, where θ is the angle between a ruling and the positive x-axis and f(θ) istheheightaboveπatwhichtherulingintersects L (f is assumed to be smooth) The normal line at a point P of a surface σ is the straight line passing through P parallel to the normal N of σ at P.Provethat: (i) If the normal lines are all parallel, then σ is an open subset of a plane. (ii) If all the normal lines pass through some fixed point, then σ is an open subset of a sphere. (iii) If all the normal lines intersect a given straight line, thenσ is an open subset of a surface of revolution Show that the line of striction of the hyperboloid of one sheet x 2 + y 2 z 2 =1 is the circle in which the surface intersects the xy-plane (recall from Exercise that this surface is ruled.) Which quadric surfaces are: (a) Generalized cylinders. (b) Generalized cones. (c) Ruled surfaces.

29 (d) Surfaces of revolution? Let S be a ruled surface. Show that the union of the normal lines (Exercise 5.3.7) at the points of a ruling of S is a plane or a hyperbolic paraboloid One of the following surfaces is compact and one is not: (i) x 2 y 2 + z 4 =1. (ii) x 2 + y 2 + z 4 =1. Which is which, and why? Sketch the compact surface Explain, without giving a detailed proof, why the tube (Exercise 4.2.7) around a closed curve in R 3 with no self-intersections is a compact surface diffeomorphic to a torus (provided the tube has sufficiently small radius) Show that the following are triply orthogonal systems: (i) The spheres with centre the origin, the planes containing the z-axis, and the circular cones with axis the z-axis. (ii) The planes parallel to the xy-plane, the planes containing the z-axis and the circular cylinders with axis the z-axis By considering the quadric surface F t (x, y, z) =0,where F t (x, y, z) = x2 p 2 t + y2 2z + t, q 2 t construct a triply orthogonal system (illustrated above) consisting of two families of elliptic paraboloids and one family of hyperbolic paraboloids. Find a parametrization of these surfaces analogous to (5.12) Show that the following are triply orthogonal systems: (i) xy = uz 2, x 2 + y 2 + z 2 = v, x 2 + y 2 + z 2 = w(x 2 y 2 ). (ii) yz = ux, x 2 + y 2 + x 2 + z 2 = v, x 2 + y 2 x 2 + z 2 = w What should be the definition of a (doubly) orthogonal system of curves in R 2? Give examples of such systems such that:

30 30 (i) Each of the two families of curves consists of parallel straight lines. (ii) One family consists of straight lines and the other consists of circles By considering the function F t (x, y) = x2 p 2 t + y2 q 2 t, where p and q are constants with 0 <p 2 <q 2,constructanorthogonalsystem of curves in which one family consists of ellipses and the other consists of hyperbolas. By imitating Exercise 5.5.2, construct in a similar way an orthogonal system of curves in which both families consist of parabolas Starting with an orthogonal system of curves in the xy-plane, construct two families of generalized cylinders with axis parallel to the z-axis which intersect the xy-plane in the two given families of curves. Show that these two families of cylinders, together with the planes parallel to the xy-plane, form a triplyorthogonal system Show that, if γ :(α, β) R 3 is a curve whose image is contained in a surface patch σ : U R 3,thenγ(t) =σ(u(t),v(t)) for some smooth map (α, β) U, t (u(t),v(t)) Prove Theorem and its analogue for level curves in R 3 (Exercise 1.5.1) Let σ : U R 3 be a smooth map such that σ u σ v 0 at some point (u 0,v 0 ) U. ShowthatthereisanopensubsetW of U containing (u 0,v 0 )such that the restriction of σ to W is injective. Note that, in the text, surface patches are injective by definition, but this exercise shows that injectivity near a given point is a consequence of regularity Let σ : U R 3 be a regular surface patch, let (u 0,v 0 ) U and let σ(u 0,v 0 )= (x 0,y 0,z 0 ). Suppose that the unit normal N(u 0,v 0 )isnotparalleltothexyplane. Show that there is an open set V in R 2 containing (x 0,y 0 ), an open subset W of U containing (u 0,v 0 )andasmoothfunctionϕ: V R such that σ(x, y) =(x, y, ϕ(x, y)) is a reparametrization of σ : W R 3. Thus, near p, the surface is part of the graph z = ϕ(x, y). What happens if N(u 0,v 0 )isparalleltothexy-plane? Let γ :(α, β) R n be a regular curve and let t 0 (α, β). Show that, for some ɛ>0, the restriction of γ to the subinterval (t 0 ɛ, t 0 + ɛ) of(α, β) isinjective. Chapter Calculate the first fundamental forms of the following surfaces:

31 (i) σ(u, v) =(sinhu sinh v, sinh u cosh v, sinh u). (ii) σ(u, v) =(u v, u + v, u 2 + v 2 ). (iii) σ(u, v) =(coshu, sinh u, v). (iv) σ(u, v) =(u, v, u 2 + v 2 ). What kinds of surfaces are these? Show that applying an isometry of R 3 to a surface does not change its first fundamental form. What is the effect of a dilation (i.e. a map R 3 R 3 of the form v av for some constant a 0)? Let Edu 2 +2Fdudv+Gdv 2 be the first fundamental form of a surface patch σ(u, v) of a surface S. Showthat,ifp is a point in the image of σ and v, w TpS, then v, w = Edu(v)du(w)+ F (du(v)dv(w)+ du(w)dv(v)) + Gdu(w)dv(w) Suppose that a surface patch σ(ũ, ṽ) isareparametrizationofasurfacepatch σ(u, v), and let Ẽdũ 2 +2 Fdũdṽ + Gdṽ 2 and Edu 2 +2Fdudv+ Gdv 2 be their first fundamental forms. Show that: (i) du = u u ũdũ + ṽ dṽ, v v dv = ũdũ + ṽ dṽ. (ii) If J = ( u ũ v ũ is the Jacobian matrix of the reparametrization map (ũ, ṽ) (u, v), and J t is the transpose of J, then ( Ẽ ) F F G u ṽ v ṽ ) ( ) = J t E F J. F G Show that the following are equivalent conditions on a surface patch σ(u, v) with first fundamental form Edu 2 +2Fdudv+ Gdv 2 : (i) E v = G u =0. (ii) σ uv is parallel to the standard unit normal N. (iii) The opposite sides of any quadrilateral formed by parameter curves of σ have the same length (see the remarks following the proof of Proposition 4.4.2). When these conditions are satisfied, the parameter curves of σ are said to form a Chebyshev net. Show that,in that case,σ has a reparametrization σ(ũ, ṽ) with first fundamental form dũ 2 +2cosθdũdṽ + dṽ 2,

32 32 where θ is a smooth function of (ũ, ṽ). Show that θ is the angle between the parameter curves of σ. Show further that, if we put û = ũ + ṽ, ˆv = ũ ṽ, the resulting reparametrization ˆσ(û, ˆv) of σ(ũ, ṽ) has first fundamental form where ω = θ/2. cos 2 ωdû 2 +sin 2 ωdˆv 2, Repeat Exercise for the following surfaces: (i) σ(u, v) =(u cos v, u sin v, ln u). (ii) σ(u, v) =(u cos v, u sin v, v). (iii) σ(u, v) =(coshu cos v, cosh u sin v, u) Find the length of the part of the curve on the cone in Exercise with 0 t π. Showalsothatthecurveintersectseachoftherulingsoftheconeat the same angle Let σ be the ruled surface generated by the binormals b of a unit-speed curve γ: σ(u, v) =γ(u)+vb(u). Show that the first fundamental form of σ is where τ is the torsion of γ. (1 + v 2 τ 2 ) du 2 + dv 2, If E, F and G are the coefficients of the first fundamental form of a surface patch σ(u, v), show that E u =2σ u.σ uu,andfindsimilarexpressionsfore v, F u, F v, G u and G v.deducethefollowingformulas: σ u.σ uu = 1 2 E u, σ u.σ uv = 1 2 E v, σ v.σ uu = F u 1 2 E v σ v.σ uu = 1 2 G u σ u.σ vv = F v 1 2 G u, σ v.σ uu = 1 2 G v By thinking about how a circular cone can be unwrapped ontotheplane,write down an isometry from σ(u, v) =(u cos v, u sin v, u), u > 0, 0 <v<2π, (a circular cone with a straight line removed) to an open subset of the xy-plane Is the map from the circular half-cone x 2 + y 2 = z 2, z>0, to the xy-plane given by (x, y, z) (x, y, 0) a local isometry?