Curvesw3.impa.br/~l.ambrozio/listagrande.pdf · 2021. 1. 26. · IMPA, Ver~ao 2021 - Prof. Lucas...

Geometria Diferencial de Curvas e Superf́ıciesIMPA, Verão 2021 - Prof. Lucas Ambrozio

Material complementar: lista de exerćıcios (versão 25/02/2020)

A lista de exerćıcios abaixo está organizada em seis partes.O modo de usar sugerido é: antes de mais nada, leiam os enunciados e procurem entender

o que eles dizem e como eles se relacionam ao que foi dito em aula. Então, escolhamfazer aqueles que lhes parecem ser mais interessantes/importantes/curiosos/desafiadores paravocês. Ocasionalmente, será proveitoso seguir adiante e retornar a certos exerćıcios quandomais experiência tiver sido acumulada.

Dúvidas serão tiradas preferencialmente durante nossos encontros semanais ao vivo.Não deixem de olhar os exerćıcios da bibliografia sugerida também! Quanto mais exerćıcios

vocês fizerem, mais vocês aprenderão.Os exerćıcios estão em inglês porque eu os escrevi para uso em outro curso, e pretendo

utilizá-los muitas vezes ainda. Em particular, a última parte da lista crescerá ao longo destecurso...

1. Curves

Exercise 1.1. Let α : I → Rn be a regular parametrised curve and p be a point in Rn. Thedistance from points in the trace of α to p is constant if and only if the vector α(t) − p isorthogonal to α′(t) for all t ∈ I.

Exercise 1.2. Let L ⊂ Rn be a straight line and p ∈ Rn a point that is not contained in L.The distance between p and L is the number

d(p, L) := inf{|p− q| ∈ R; q ∈ L}.

Prove that the above infimum is positive and attained at some point q0 ∈ L. Using someparametrisation of L, prove that the straight line passing through p and q0 is orthogonal toL, and show that q0 is the only point in L such that |p− q0| = d(p, L).

In the particular case n = 2, show that

|p− q0| = |〈p− q0, N〉|,

where N is a choice of unit vector normal to the line L.

Exercise 1.3. Let p, q ∈ Rn. Let α : I → Rn be a regular parametrised curve, [a, b] ⊂ Ia compact interval, and assume that p = α(a) and q = α(b). In short, α([a, b]) is a curvejoining p to q.

i) Show that the length of α([a, b]) satisfies the inequality

|p− q| ≤ `(α; [a, b]).

ii) If the length of α([a, b]) is equal to the distance between p and q, show that α([a, b])is the straight line segment joining p to q.

Conclude that the distance between any two given points of the Euclidean space can becomputed as the minimum of the length of all curves joining these points.

Geometria Diferencial de Curvas e Superf́ıcies - IMPA, verão 2021 - Prof. Lucas Ambrozio

Exercise 1.4. Let C = {(x, y) ∈ R2; y = |x|}. Please draw this set. Then, justify thefollowing assertions:

i) The map

α1 : t ∈ R 7→ (t, |t|) ∈ C ⊂ R2

is not a regular parametrised curve.ii) Construct a smooth map α2 : I → R2 such that α2(I) = C. (Hint: define parametriza-

tions of both half-lines that constitute C in such way they reach the endpoint (0, 0) withzero derivatives of all orders. The well-known function t ∈ (0,+∞) 7→ exp(−1/t) ∈R, extended to be 0 at t = 0, might be useful).

iii) The tangent lines of the parametrised curve you have defined in the previous item donot change locally at points where they are defined. However, the trace of the curveis not contained inside a straight line (To think about: how is this example related tothe usual assumption that a regular parametrised curve has non-zero tangent vectoreverywhere? ).

iv) The set C is not the trace of a regular parametrised curve. (Suggestion: first show thatC cannot be the trace of a smooth graphical curve β(u) = (u, f(u)) near (0, 0) ∈ C.Then reach a contradiction by using the Inverse Function Theorem to reparametrisea hypothetical regular parametrised curve α with trace C, in a neighbourhood of t ∈ Isuch that α(t) = (0, 0), so that it becomes written in the above graphical form).

Exercise 1.5. Let a > 0, b < 0 be two real numbers. The logarithmic spiral is the trace ofthe regular parametrised curve

α : t ∈ R 7→ (aebt cos(t), aebt sin(t)) ∈ R2.

Sketch this curve. Show that α(t) and α′(t) converge to zero as t goes to +∞, and that, forany given t0 ∈ R,

limt→+∞

ˆ tt0

|α′(t)|dt < +∞.

Compute the curvature of α.

Exercise 1.6. Mark a point on the circumference of a circular disc of radius R > 0. Thefigure described by that point as the disk rolls without slipping on the x−axis is called acycloid.

i) What can be said about the distance between any two points belonging to the inter-section of the cycloid and the x−axis?

ii) Find a smooth map α : R→ R2 describing the trajectory of the marked point. Is αa regular parametrised curve?

iii) Compute the length of a piece of a cycloid between two consecutive instants t1 <t2 ∈ R when α(t1) and α(t2) are points in the x−axis.

Exercise 1.7. Let α : R→ R3 be given by

α(t) =

(a cos

(t

c

), a sin

(t

c

),b

ct

)for all t ∈ R,

where a, b, c ∈ R, c 6= 0, are given constants.2


i) Give conditions on a, b and c so that α is a regular parametrised curve not containedin a plane. Under these conditions, the trace of α is called a circular helix.

ii) Give conditions on a, b and c so that α is parametrised by arc-length.

From now on, assume α is a circular helix parametrised by arc-length.

iii) Show that the tangent lines to α make a constant angle θ with the z−axis, andexpress θ in terms of the constants a and b.

iv) Show that normal vector to α at t ∈ I is orthogonal to the z−axis.v) Compute the curvature and the torsion of α.vi) Show that, for all t ∈ I, the line passing through the point α(t) pointing in the

direction of the normal vector of α at t ∈ I intersects the z−axis.vii) Show that the intersection point Z(t) defined in the previous item moves on the

z−axis with constant non-zero speed.viii) Compute the distance between Z(t) and α(t).

Exercise 1.8. Let α : I → R2 be a map given by

α(u) = (u, f(u)) for all u ∈ I

for some smooth function f : I → R. Such α is said to be a curve written as a graph (overthe x−axis).

i) Show that α is a regular parametrised curve.ii) Write the formula for the length of α([a, b]), [a, b] ⊂ I a compact interval.iii) Show that the curvature of α at a point u ∈ I is given by:

k(u) =f ′′(u)

(1 + (f ′(u))2)3/2.

How does the graph of f look like at a point u ∈ I where the curvature of α ispositive? And where it is negative?

iv) Compute the curvature of the catenary :

α : u ∈ R 7→ (u, cosh(u)) ∈ R2.

Exercise 1.9. Let α : I → R2 \ {0} be a map given by

α(θ) = (r(θ) cos(θ), r(θ) sin(θ))

for some smooth positive function r : I → (0,+∞). Such α is said to be a curve written inpolar coordinates.

i) Show that α is a regular parametrised curve.ii) Compute the length of α([a, b]), [a, b] ⊂ I a compact interval.iii) Show that the curvature of α at a point θ ∈ I is given by:

k(θ) =2(r′(θ))2 − r(θ)r′′(θ) + r(θ)2

((r(θ))2 + (r′(θ))2)3/2.

iv) Consider the parabola P = {(x, y) ∈ R2; y = (x2 − 1)/4}. Describe this set as thetrace of a curve written in polar coordinates.

3


Exercise 1.10. Let α : I → R2 be a regular parametrised plane curve. The straight linepassing through α(t) that is orthogonal to the tangent line of α at t ∈ I is called the normalline of the curve α at t ∈ I.

Assume that all normal lines to a regular parametrised curve pass through a fixed point.Prove that the trace of the curve is contained in a circle.

Exercise 1.11. Let α : I → R2 be a plane curve parametrised by arc-length. Assume that itscurvature never vanishes. The evolute of α is the map

e : s ∈ I 7→ α(s) + 1k(s)

N(s) ∈ R2.

i) Show that e is a regular parametrised curve if and only if k′ never vanishes.ii) Under the condition of the previous item, compute the length of e([a, b]), where

[a, b] ⊂ I is a compact interval.iii) Still under the conditions of the previous items, show that the tangent line to e at

s ∈ I is parallel to the normal line of α at s ∈ I.iv) Show that e is a constant map if and only if the trace of α is contained in a circle.v) Compute the evolute of the logarithmic spiral (see Exercise 5).

Exercise 1.12. (Rigid motions - ver material complementar à Aula 1).We say that a map M : Rn → Rn preserves the Euclidean distance when

|M(p)−M(q)| = |p− q| for all p, q ∈ Rn.i) Show that translations and orthogonal transformations preserve the Euclidean dis-

tance.ii) Define v = M(0). Show that M̃ = T−v◦M is a map that preserves Euclidean distances

and fixes the origin. In particular, conclude that |M̃(p)| = |p| for all p ∈ Rn.iii) Keeping the notation of the previous item, show that M̃ is an orthogonal transfor-

mation, and that the derivative of M at any point is precisely the linear map M̃ .

iv) Conclude that M = Tv ◦ M̃ , where Tv is a translation and M̃ is an orthogonaltransformation.

v) Show that the set of maps that preserve the Euclidean distance is a group whenendowed with the operation of composition of maps.

Exercise 1.13. Let α : I → R3 be a regular curve parametrised by arc-length, and M amap that preserves the Euclidean distance. Show that M ◦ α : I → R3 is a regular curveparametrised by arc-length. Distinguishing between the cases where the derivative of M isorientation-preserving or not, study the relations between the length, the Frenet frames, thecurvature and the torsion of the curves α and M ◦α. Perform a similar study for plane curves(in the case of closed curves, study also the area of the region bounded by the curves).

Exercise 1.14. (cf. do Carmo, 1.5, 10). Let α : (−√

2/3,√

2/3)→ R3 be the map given by

α(t) =

(t, e−1/t2 , 0) for all t ∈ (−

√2/3, 0),

(0, 0, 0) for t = 0,

(t, 0, e−1/t2) for all t ∈ (0,

√2/3).

Show that α is a regular parametrised curve whose curvature is positive except at t = 0.Observe that the osculating planes of α do not change on (0,

√2/3) or on (−

√2/3, 0), where

4


they are well-defined. However, α is not a plane curve. (To think about: how is this examplerelated to the usual assumption that a regular parametrised curve in R3 has positive curvatureat all points? ).

Exercise 1.15. Let α : I → R2 be a curve parametrised by arc-length, and pick q0 = α(s0),s0 ∈ I, a point on the trace of α. Let H+ be the half-plane determined by the tangent lineof α at s0 ∈ I into which the normal vector N(s0) points, i.e.

H+ = {p ∈ R2; 〈p− q0, N(s0)〉 ≥ 0}.

The other half-plane H− is defined similarly.The function f : I → R2 given by

f(s) = 〈α(s)− q0, N(s0)〉 for all s ∈ I

measures the “signed” distance between the point α(s) and the tangent line (cf. Exercise3), and the sign of f(s) determines whether the point α(s) belongs to H+ or to H−.

i) Show that f(s0) = 0, f′(s0) = 0 and f

′′(s0) = k(s0).ii) Prove that if k(s0) > 0 then there exists a neighbourhood J of s0 ∈ I such that α(J)

is contained in H+.iii) Show by examples that the same conclusion does not necessarily hold if k(s0) = 0.iv) Prove that if there exists a neighbourhood J of s0 in I such that α(J) is contained

in H+, then k(s0) ≥ 0.v) State and prove similar assertions about the case k(s0) < 0 and the half-space H

−.

Exercise 1.16. (The osculating circle).Let α : I → R2 be a curve parametrised by arc-length, and pick q0 = α(s0), s0 ∈ I, a point

in the trace of α where the curvature does not vanish. Assume without loss of generalitythat k(s0) > 0. Let Bλ be the (closed) ball of radius λ > 0 that is tangent to α at q0 andsuch that the unit normal N(s0) of α at s0 ∈ I points inside it.

i) Show that the centre of the ball Bλ is the point cλ = q0 + λN(s0) ∈ R2.ii) The function fλ : I → R2 defined by

fλ(s) = |α(s)− cλ|2 for all s ∈ I

gives the square of the distance between a point in the trace of α and the centre ofthe ball Bλ. Show that fλ(s0) = λ

2, f ′λ(s0) = 0 and f′′λ (s0) = 2(1− λk(s0)).

iii) Prove that if λ < 1/k(s0), then there exists a neighbourhood J of s0 in I such thatα(J \ {s0}) is contained outside the ball Bλ.

iv) Prove that if λ > 1/k(s0), then there exists a neighbourhood J of s0 in I such thatα(J \ {s0}) is contained inside the ball Bλ.

v) Give an example to show that the behaviour described in the previous two items doesnot necessarily occur if λ = 1/k(s0).

vi) When λ = 1/k(s0), we say that the circle centred at cλ with radius λ is the osculatingcircle of α at the point s0 ∈ I. Conclude from the above the following characteriza-tion: the osculating circle of α at a point s0 ∈ I where k(s0) > 0 is the circle with thelargest radius among those circles satisfying the following properties: it is tangent tothe curve α at s0, is such that the unit normal N(s0) of α points inside it, and issuch that the trace of α lies outside it in a neighbourhood of α(s0).

5


vii) Conclude the following characterisation of the evolute (cf Exercise 11): the evoluteof a curve α with non-vanishing curvature is the curve described by the centres ofthe osculating circles of α.

Exercise 1.17. Let α : I → R2 be a regular parametrised plane curve. Assume that thedistance from α(t) to the origin attains a maximum value at t0 ∈ I. Show that

|k(t0)| ≥1

|α(t0)|.

Exercise 1.18. If the trace of a closed plane regular curve is contained in a disk of radiusR > 0, then it contains a point whose curvature is in absolute value greater than or equalto 1/R.

Exercise 1.19. Let α : [a, b]→ R2 be a closed plane regular curve parametrised by arc-length.Show that

`(α; [a, b]) =

ˆ ba

k(s)(x(s)y′(s)− y(s)x′(s))ds.

Suggestion: integrate (x′)2 + (y′)2 ≡ 1 by parts.

Exercise 1.20. Let α : [a, b]→ R2 be a closed plane curve parametrised by arc-length. Showthat there exists an integer m ∈ Z such that

ˆ ba

k(s)ds = 2πm.

Draw examples of curves where m is an arbitrary integer. If the curve is simple, what do youexpect the number m to be? (We will discuss this question thoroughly later in the course).

Exercise 1.21. Let α : [0, `] → R2 be a closed simple regular curve parametrised by arc-length. As always, orient α so that its unit normal points inside the region it bounds. For agiven r ∈ R, consider the curve

αr : s ∈ [0, `] 7→ α(s) + rN(s) ∈ R2.

i) Show that, if r is small enough, then αr is also a simple closed regular parametrisedcurve.

ii) Prove that

`(αr) = `−(ˆ `

0

k(s)ds

)r.

iii) If Ωr is the (closed) region enclosed by αr, and Ω is the region enclosed by α, provethat

A(Ωr) = A(Ω)− `r +(ˆ `

0

k(s)ds

)r2

2.

iv) For r > 0 sufficiently small, the tube of radius r > 0 around α([0, `]) is the setΩ−r \ int(Ωr). Show that the area of the tube of sufficiently small radius r > 0around α([0, L]) is degree one polynomial expression in r that does not depend onthe curvature of α.

6


Exercise 1.22. (cf. do Carmo, 1.5, 17) A regular parametrised curve α : I → R3 with positivecurvature and nowhere vanishing torsion is called a helix if the tangent lines to α make aconstant angle with a fixed direction.

i) α is a helix if and only if k/τ is constant.ii) α is a helix if and only if the normal lines of α are all parallel to a fixed plane.iii) α is a helix if and only if the straight lines containing α(s) and parallel to the binormal

B(s) makes a constant angle with a fixed direction.iv) Describe a helix that is not any of the circular helices described in Exercise 8.

Exercise 1.23. Let α : I → R2 be a curve parametrised by arc-length. Assume that thereexists a point p ∈ R2 such that the distance between p and the normal lines of α is constant.Prove that, for some a, b ∈ R with a2 + b2 6= 0,

k(s) = ± 1√as+ b

.

Can you prove the converse as well?

Exercise 1.24. (The aim of this exercise is to show that the curvature of a plane curve isrelated to how the length of a curve varies when we deform it, cf. Exercise 21).

Let p, q ∈ R2 be two distinct points on the plane, and α : [a, b] → R2 be a curveparametrised by arc-length such that α(a) = p and α(b) = q.

Let V : [a, b]→ R2 be a smooth map such that V (a) = V (b) = 0. Let F : [a, b]× (−�, �)→R2 be the smooth map given by

F (s, t) = α(s) + tV (s) for all (s, t) ∈ [a, b]× (−�, �).Notice that F (a, t) = p and F (b, t) = q for all t ∈ (−�, �). F is called a variation of α, andthe vector field V is called the variational vector field associated to the variation F . We wantto compute how the length of the curve α varies infinitesimally under such deformations.

i) Show that, if � > 0 is small enough, then for every fixed t ∈ (−�, �), the mapαt := s ∈ [a, b] 7→ F (s, t) ∈ R2

is a regular parametrised curve joining p to q.ii) Show that the length of αt([a, b]), t ∈ (−�, �), is given by

`(αt; [a, b]) :=

ˆ ba

√(1 + 2t〈α′(s), V ′(s)〉+ t2|V ′(s)|2)ds,

and therefore is a smooth function of t ∈ (−�, �).iii) Show that

d

dt

∣∣∣∣t=0

`(αt; [a, b]) = −ˆ ba

k(s)〈N(s), V (s)〉ds.

(Hint: integration by parts).iv) Using arbitrary variations V (s) = φ(s)N(s), where φ : [a, b]→ R is a smooth function

vanishing at a and at b, prove that

d

dt

∣∣∣∣t=0

`(αt; [a, b]) = 0

for all variations of α if and only if α([a, b]) is the line segment joining p to q.7


Exercise 1.25. Let X1, . . . , Xn : I → Rn be smooth functions. Let A : I → Rn2

be the mapthat assigns to each s ∈ I the n× n matrix A(s) where the entry in the i-th row and in thej-th column is A(s)ij := 〈Xi(s), Xj(s)〉 for each i, j = 1, . . . , n.

Assume that, for every s ∈ I, the set {X1, . . . , Xn} is an orthonormal basis of Rn, i.e.〈Xi(s), Xj(s)〉 = δij for all i, j = 1, . . . , n.

(Recall that the symbol δij is equal to 1 if i = j and zero otherwise.). Show that, for all s ∈ I,the n× n matrix

B(s) :=d

dsA(s)

is an anti-symmetric matrix, i.e. B(s)ij = −B(s)ji.

Exercise 1.26. Let α : I → Rn be a curve parametrised by arc-length. Under appropriateconditions, can you define a “Frenet frame” for α, satisfying similar “Frenet equations”involving “curvature” functions k1, k2, . . . , kn−2, kn−1 : I → R, ki > 0 on I for each i =1, . . . , n− 1, and prove a “Fundamental Theorem of Curves in Rn”? (The answer is “yes”!Test your ideas first in R4. Generalise).

Exercise 1.27. (On the Wirtinger Inequality - ver material complementar à Aula 2 ).

i) Show that the Wirtinger Inequality stated in class is equivalent to the following(more general) statement: if f : R → R is a smooth periodic function of period Land f = 1

L

´ L0f(s)ds denotes its mean value on the interval [0, L], thenˆ L

0

(f ′(s))2ds ≥ 4π2

L2

ˆ L0

(f(s)− f)2ds.

(Hint: change of variables).ii) Which functions satisfy the equality?

2. Surfaces

Exercise 2.1. In analogy to the definition of regular surfaces in R3, define a regular curvein Rn as a subset of Rn satisfying further properties. Give examples of subsets of Rn thatare regular curves, and examples of subsets of Rn that are not regular curves (in each caseproviding the reasons of why this is the case). Show that the trace of simple closed regularparametrised curves, as defined in the lectures, are regular curves according to your proposeddefinition.

Exercise 2.2. By exhibiting a collection of parametrisations whose images cover the givenset explicitly, show that:

i) The cylinder{(x, y, z) ∈ R3; x2 + y2 = 1}

is a regular surface.ii) More generally, let Z ⊂ R2 be a regular curve as defined in Exercise 1. The right

cylinder over Z, i.e. the setC := {(x, y, z) ∈ R3; (x, y) ∈ Z}

is a regular surface in R3.8


Exercise 2.3. By exhibiting a collection of parametrisations whose images cover the givenset explicitly, show that:

i) The catenoid

{(x, y, z) ∈ R3; cosh(z) =√x2 + y2}

is a regular surface. Draw this set.ii) More generally, let C ⊂ R2 be a regular curve (as defined in Exercise 1) that does

not intersect the z-axis and is contained in the plane orthogonal to the y-axis. Showthat the set obtained by rotation of C around the z-axis is a regular surface in R3.(This surface is called the surface of revolution generated by the curve C.)

iii) Using the above construction, define a regular surface in R3 that is a torus of revo-lution.

Exercise 2.4. Justify the following assertions: the set {(x, y, z) ∈ R3; x = 0 and y2 + z2 < 1}is a regular surface; its closure {(x, y, z) ∈ R3; x = 0 and y2 + z2 ≤ 1} is not a regularsurface.

Exercise 2.5. Draw a picture of the cone

{(x, y, z) ∈ R3; z =√x2 + y2}.

Show that the above set is not a regular surface. Now, remove the point p = (0, 0, 0) (thetip of the cone). Show that the set

{(x, y, z) ∈ R3; z =√x2 + y2, z 6= 0}

is a regular surface.

Exercise 2.6. Given a, b, c three positive real numbers, consider the ellipsoid

E(a, b, c) ={

(x, y, z) ∈ R3; x2

a2+y2

b2+z2

c2= 1

}.

Prove that E(a, b, c) is a regular surface in R3. Show that the map

X : (u, v) ∈ (0, π)× (0, 2π) 7→ (a sin(u) cos(v), b sin(u) sin(v), c cos(u)) ∈ R3

is a parametrisation of E(a, b, c). Describe in geometric terms what are the coordinate curvesof the above parametrisation X (i.e. describe the curves u = constant, and v = constant).

Exercise 2.7. As any given subset of R3, a regular surface can be compact, closed but notcompact, connected or disconnected. Give examples of regular surfaces satisfying each one ofthese properties. (Remark: no open subset of R3 is a regular surface. Why? ).

Exercise 2.8. Show that a regular surface in R3 can be covered by countably many coordinateneighbourhoods. (Hint: the topology of the Euclidean space is second-countable).

Exercise 2.9. Consider the following set

S =+∞⋃n=1

{(x, y, z) ∈ R3; z = 1/n},

9


which is a countable union of horizontal planes. Show that S is a regular surface. Show thatits closure,

S = {(x, y, z) ∈ R3; z = 0} ∪+∞⋃n=1

{(x, y, z) ∈ R3; z = 1/n},

which is also a countable union of horizontal planes, is not a regular surface.

Exercise 2.10. (The stereographic projection). Let S ⊂ R3 be the sphere of radius one centredat the origin, and p0 a point in S. The stereographic projection based at the point p0 is afunction πp0 that maps each point p ∈ S \ {p0} to the point where the straight line joiningp0 and p intersects the plane through the origin that is orthogonal to p0.

i) Show that the stereographic projection πN based at the north pole (0, 0, 1) is suchthat its inverse π−1N can be written as

π−1N : (u, v) ∈ R2 7→

(2u

1 + u2 + v2,

2v

1 + u2 + v2,u2 + v2 − 11 + u2 + v2

)∈ S \ {p0}.

(Notice that we have identified R2 and the plane z = 0 in R3).ii) Show that the inverse of the two stereographic projections πN and πS based at the

north pole (0, 0, 1) and at the south pole (0, 0,−1), respectively, are two parametri-sations of the sphere whose images cover all of its points.

iii) Compute the change of coordinates

πS ◦ π−1N : R2 → R2.

Denoting by C the circle {(u, v) ∈ R2; u2 + v2 = 1}, show that πS ◦ π−1N swaps theexterior region to C and the region bounded by C, minus the origin.

Exercise 2.11. Given a, b, c real numbers, compute the regular values of the function

F : (x, y, z) ∈ R3 7→ ax2 + by2 + cz2 ∈ R.

(Remark: your answer will depend on the parameters a, b, c. Analyse at least the cases i)a = b = c = 1; ii) a = b = 1 and c = 0; iii) a = b = 1 and c = −1; and iv) a = 1and b = c = −1). What are the regular surfaces that are the inverse image of such regularvalues?

Exercise 2.12. Let X : U ⊂ R2 → R3 be a regular parametrised surface, as defined in thelectures. Show that for every point p ∈ U there exists an open neighbourhood U ′ of p suchthat the set X(U ′) ⊂ R3 is a regular surface. Give an example of a regular parametrisedsurface X : U ⊂ R2 → R3 such that the full image of the map X, the set X(U) ⊂ R3, is nota regular surface.

(Suggestion: use the same map Φ that appeared in the proof that changes of parametrisa-tions are smooth maps).

Exercise 2.13. Let S be a regular surface, and p a point in S. Prove that there exists anopen set W of R3, W containing the point p, and a smooth function f : U ⊂ R2 → R definedon an open subset U of one the coordinate xy-, xz- or yz-planes, such that

W ∩ S = graph(f).10


(Hint: given a parametrisation X of a neighbourhood of p, compose it with the orthogonalprojection onto one of the coordinate planes to obtain a smooth map between open subsets ofR2. Can you invert this map?).

Exercise 2.14. Consider the following two parametrisations of the cylinder S = {(x, y, z) ∈R3; x2 + y2 = 1}:

X : (u, v) ∈ (−1, 1)× R 7→ (u,√

1− u2, v) ∈ S,

X̃ : (s, θ) ∈ R× (−π, π) 7→ (cos(θ), sin(θ), s) ∈ S.

Compute the domains of the change of parametrisations X−1 ◦ X̃ and X̃−1 ◦ X explicitly,and check that they are smooth maps.

Exercise 2.15. Let S be a regular surface in R3, and f : S ⊂ R3 → R be a smooth function.The gradient of f at a point p is defined to be the unique vector

grad(f)(p) ∈ TpS

such that

Df(p) · v = 〈grad(f)(p), v〉 for all v ∈ TpS.

i) Fix a non-zero vector w ∈ R3, and define the height function:

f : p ∈ S 7→ 〈p, w〉 ∈ R.

Show that f is smooth and describe what is the tangent plane of S at a point wherethe gradient of f vanishes (in other words, at a critical point of f).

ii) Let F : V ⊂ R3 → R be a smooth function, S a regular surface in R3, V ∩ S 6= ∅,and f : V ∩ S → R the restriction of F to V ∩ S. Show that, at each pointp ∈ V ∩ S, the vector grad f(p) ∈ TpS ⊂ R3 is the orthogonal projection of thevector gradF (p) ∈ R3 onto the tangent plane TpS.

Exercise 2.16. Recall the following theorem from Topology:

Let X be a topological space that is locally path-connected. X is connected if and only if Xis path-connected.

Show that a regular surface S in R3 is a locally path-connected topological space. In partic-ular, conclude that

a regular surface in R3 is connected if and only if it is path-connected.Prove that if S is a connected regular surface in R3, then a smooth function f : S → R isconstant if and only if its gradient vanishes at all points of S.

Exercise 2.17. Let F : R3 → R be a smooth function, and t ∈ F (R3) a regular value of F .Let gradF (p) ∈ R3 denote the gradient vector of F at a point p ∈ F−1(t) (cf. Exercise 15;observe that gradF (p) 6= 0 since p ∈ F−1(t) is a regular point of F ).

Show that the regular surface S = F−1(t) is such that, at all points p ∈ S,

TpS = {v ∈ R3; 〈gradF (p), v〉 = 0}.11


Exercise 2.18. Let S be a regular surface in R3 and p0 a point in R3 that does not belong tothe surface. Show that the map

f : p ∈ S 7→ |p− p0| ∈ Ris smooth and compute its derivative at a point p ∈ S. Show that the tangent plane at acritical point p of f is orthogonal to the vector p− p0.

Exercise 2.19. Let S be a regular surface in R3, andγ : (−�, �)→ R3

a map such thatγ(t) ∈ S for all t ∈ (−�, �).

Show that γ is a smooth map if and only if, for every parametrisation X : U ⊂ R2 →V ∩ S ⊂ R3 such that γ(t) ∈ V ∩ S for all t ∈ (−�′, �′) ⊂ (−�, �), the map

γ̃ := X−1 ◦ γ : (−�′, �′)→ U ⊂ R2

is smooth.(Suggestion: when proving the implication ⇒, recall the map Φ defined in the proof that

changes of parametrisations are smooth).

Exercise 2.20. Let S1 and S2 be regular surfaces in R3. Check that the derivative of a smoothmap φ : S1 → S2 at a point p ∈ S1, defined as the map

Dφ(p) : v ∈ TpS1 7→ (φ ◦ γ)′(0) ∈ Tφ(p)S2,where γ : (−�, �) → S is a smooth map such that γ(0) = p and γ′(0) = v, is well-defined,and indeed a linear map between the tangent planes of S1 at p and of S2 at φ(p). Usingparametrisations of coordinates neighbourhoods of p in S1 and of φ(p) in S2, express it incoordinates.

Exercise 2.21. Let S1 = {p ∈ R3; |p| = 1} be the unit sphere centred at the origin, S2 ={(x, y, z) ∈ R3; x2 + y2 = 1, |z| < 1} the piece of the cylinder over the unit circle on the xyplane centred at the origin whose points are at height −1 < z < 1, and

φ : S1 \ {(0, 0,±1)} → S2the map that sends each point p in the sphere minus the two poles to the point of intersectionof the cylinder and the half-line that issues from the z-axis horizontally (i.e. parallel to thexy plane) and passes through the point p.

Show that φ is a smooth bijective map, and show that its derivative at any given pointp in S1 \ {(0, 0,±1)} is a linear isomorphism between the tangent planes TpS1 and Tφ(p)S2.Conclude that φ : S1 \ {(0, 0,±1)} → S2 is a diffeomorphism.

Exercise 2.22. Let S be a regular surface in R3, and denote by S21(0) the sphere of radiusone centred at the origin. Assume that there exists a continuous map

N : S → S21(0)such that N(p) is orthogonal to TpS for all p in S. Show that N is smooth. In particular,conclude that S must be orientable.

Exercise 2.23. (On orientation-preserving maps):12


i) Let (V,P) and (W,Q) be oriented real vector spaces of finite dimension n. A linearmap L : V → W is said to be orientation-preserving when it sends positive bases ofV into positive bases of W , and orientation-reversing otherwise. Show that the set oforientation-preserving linear endomorphisms of an oriented vector space (V,P) forma group under composition of maps. Prove that the linear map L : V → V given by

L : v ∈ V → −v ∈ V

preserves the orientation of (V,P) if and only if n is an even integer number.ii) Let S1 and S2 be orientable regular surfaces in R3, oriented by collections of parametriza-

tions C1 and C2, respectively. A smooth map φ : S1 → S2 is said to be orientation-preserving when, for every p in S, the derivativeDφ(p) : TpS1 → Tφ(p)S2 is orientation-preserving. Write this definition as a condition on the expression of φ in parametriza-tions X1 ∈ C, X2 ∈ C2.

iv) Let S11(0) ⊂ R2 be the unit circle centred at the origin, and choose an orientation forit. Show that the antipodal map,

A : p ∈ S11(0) 7→ −p ∈ S11(0),

preserves the chosen orientation of S11(0).iv) Let S21(0) ⊂ R3 be the unit sphere centred at the origin, and choose an orientation

for it. Show that the antipodal map,

A : p ∈ S21(0) 7→ −p ∈ S21(0),

reverses the chosen orientation of S21(0).

Exercise 2.24. Let S ⊂ R3 be a connected regular surface. Is S orientable when:i) there exists a single parametrisation X : U → S with S = X(U)?ii) there exists two parametrisations X : U → S ∩ V and X̃ : Ũ → S ∩ Ṽ , none of them

covering the entire surface S, such that S ⊂ X(U) ∪ X̃(Ũ)?iii) there exists two parametrisations X : U → S ∩ V and X̃ : Ũ → S ∩ Ṽ , none of

them covering the entire surface S, such that S ⊂ X(U)∪ X̃(Ũ) and the intersectionX(U) ∩ X̃(Ũ) is a non-empty connected proper subset of S?

If you answer to an item is yes, please justify your answer. If it is no, please give a counter-example.

Exercise 2.25. Considering smooth functions defined on (open subsets of) Rm and takingvalues in a regular surface in R3, or defined on (open subsets of) regular surfaces in R3 andtaking values on regular surfaces in R3, or defined on (open subsets of) regular surfaces in R3and taking values in Rn, can you extend the usual operations and theorems of Differentiationto that setting? (Hint: the answer is “yes”.) For example, please think for a while on thefollowing concrete questions: a) is the product of two smooth functions f1, f2 : S → Ra smooth function, and what is its derivative? b) what would be the statement (and theproof) of the Chain Rule? c) what can be said about smooth maps f : S1 → S2 such thatDf(p) : TpS1 → Tφ(p)S2 is a linear isomorphism? d) What is a regular value t ∈ R of afunction f : S → R, and what do you expect the set f−1(t) to be (when it is not the emptyset)?

13


Exercise 2.26. In analogy to the definition of regular surfaces in R3, how would you define a“k-dimensional surface in Rn”, its “tangent space” at a given point, “smooth functions” andtheir “derivatives” etc etc? (Remark: The jargon is “a k-dimensional submanifold of Rn”).

3. Geometry of the Gauss Map

Exercise 3.1. Consider the following regular surfaces in R3:i) The paraboloid of revolution {(x, y, z) ∈ R3; z = x2 + y2};ii) The ellipsoid {(x, y, z) ∈ R3; x2 + 2y2 + 3z2 = 1};iii) The hyperboloid of revolution {(x, y, z) ∈ R3; x2 + y2 − z2 = 1}; andiv) The catenoid {(x, y, z) ∈ R3; x2 + y2 = cosh2(z)}.

In each case, describe the subset of the unit sphere centred at the origin that is covered bythe image of the (chosen) Gauss map of the surface.

Exercise 3.2. (Geometry of graphs):Let S be a regular surface in R3 described by the graph of a smooth function φ : U ⊂

R2 → R,S = {(x, y, z) ∈ R3; (x, y) ∈ U, z = φ(x, y)}.

Orient S by the unit normal associated to its graphical parametrisation

X : (u, v) ∈ U 7→ (u, v, φ(u, v)) ∈ S.Express, in terms of the function φ, the coefficients of the first and second fundamental formsof S in this coordinate chart, and give a formula for the mean and the Gaussian curvaturesof S at an arbitrary point p = X(u, v).

Exercise 3.3. Let a, b be real numbers, and consider the surface

S = {(x, y, z) ∈ R3; z = ax2 + by2}.Classify the origin (0, 0, 0) ∈ S as an elliptic, hyperbolic, parabolic or planar point, accordingto the values of a and b. When (0, 0, 0) ∈ S is an umbilic point of S?Exercise 3.4. Without doing any computations, answer the following question (and explainwhy you have answered that way): given arbitrary constants a, b, c and d, what is the meancurvature and the Gaussian curvature of the graphical surface

S = {(x, y, z) ∈ R3; z = ax3 + bx2y + cxy2 + dy3}at the origin (0, 0, 0) ∈ S?Exercise 3.5. Let S1 and S2 be two regular surfaces in R3 that intersect tangentially alonga regular curve C, i.e. for every point p in C = S1 ∩ S2, the tangent planes of S1 and S2coincide.

Let p be a point in C. Orient (locally) the surfaces S1 and S2 near p so that the respectiveunit normal vectors coincide at p. What can be said about the normal curvatures of C atthe point p when viewed as a curve in S1 and as a curve in S2?

Exercise 3.6. Let S be a regular surface in R3. Prove that its Gaussian curvature and thesquare of its mean curvature satisfy the inequality

H2 ≥ K.Show that equality holds at a point p ∈ S if and only if p is an umbilical point of S.

14


Exercise 3.7. Let S be a non-orientable regular surface in R3. Show that the mean curvaturevector of S vanishes at some point p ∈ S.

Exercise 3.8. Let S be a regular surface, p a point in S and N a local smooth unit normalvector field on S in a neighbourhood of p. Considering the associated second fundamentalform II, describe the sets

{v ∈ TpS; IIp(v) = 1}when p is an elliptic, a hyperbolic, a parabolic, a planar, and an umbilical but not planarpoint, respectively.

Exercise 3.9. Let S be a regular surface and λ a positive number. Consider the dilatedsurface,

λ · S = {p ∈ R3; p = λp0 for some p0 ∈ S},obtained by multiplying all points in S by λ (notice that λ · S is a regular surface as well).

What is the relation between the Gaussian curvatures of S and λ ·S? What is the relationbetween the mean curvature vectors of S and λ · S?

Exercise 3.10. Let S be a regular surface in R3 such that p = (0, 0, 0) belongs to S and TpSis the xy plane.

i) If p is an elliptic point, then there exists a neighbourhood V ⊂ S of p such that V \{p}is contained either in the half-space open {z > 0} or in the open half-space {z < 0}.If you also know the mean curvature vector of S at 0, can you tell in which half-spaceV \ {p} is contained?

ii) If p is a hyperbolic point, then every neighbourhood V ⊂ S of p intersects both openhalf-spaces {z > 0} and {z < 0}.

iii) What happens when K(p) = 0? Give examples of several possible behaviours of Snear the point p.

Exercise 3.11. Let w ∈ R3 be a non-zero unit vector, and S a regular surface.Consider the function

φ : p ∈ R3 7→ 〈p, w〉 ∈ R,whose level sets φ−1(t), t ∈ R3, foliate R3 by parallel planes that are orthogonal to w.

Suppose that the restriction of φ to S attains a local minimum at a point p0 ∈ S. Showthat K(p0) ≥ 0 and 〈 ~H(p), w〉 ≥ 0. Similarly, what can be said about a point of localmaximum?

Exercise 3.12. Let S be a compact regular surface in R3. Let p0 be an arbitrary point in R3.Show that there exists a point p1 6= p0 in S such that

|p1 − p0| ≥ |p− p0| for all p ∈ S.Prove that p1 is an elliptic point of S.

Exercise 3.13. Let S be a compact connected regular surface in R3. Let A be the set of allpositive real numbers r such that S is strictly contained inside a sphere of radius r.

Prove that the infimum of A is a positive real number r0 > 0. Moreover, show that thereexists a point p0 ∈ R3 such that S is contained in the closed ball of radius r0 centred at p0,and such that the sphere Sr0(p0) of radius r0 centred at p0 intersects S tangentially at eachintersection point.

15


Show that S intersects Sr0(p0) at two different points at least. Give an example of aregular surface in R3 such that its intersection with Sr0(p0) consists of exactly two points.Exercise 3.14. Let S1 and S2 be regular surfaces in R3 that intersect tangentially at a pointp ∈ S1 ∩ S2. Assume that S1 lies on one side of S2 near the point p, and choose the unitnormal Np of S2 at p that points towards the side of S2 where S1 lies. Show that

〈 ~HS1 , Np〉 ≥ 〈 ~HS2 , Np〉.Exercise 3.15. A regular surface in R3 is called minimal when its mean curvature vanishesat every point (e.g. planes). Prove that there exists no compact minimal regular surfaces inR3.Exercise 3.16. A regular surface in R3 is said to have constant mean curvature when itsmean curvature vector has constant norm. Give examples of non-compact surfaces of con-stant mean curvature. Prove that a connected compact regular surface with constant meancurvature and positive Gaussian curvature must be a sphere.

(Remark : a theorem by A. D. Alexandrov improves this result and asserts that the onlycompact regular surfaces in R3 with constant mean curvature are spheres !).Exercise 3.17. Consider the regular parametrised surface of revolution

X : (s, θ) ∈ I × R 7→ (φ(s) cos(θ), φ(s) sin(θ), ψ(s)) ∈ R3,where φ(s) > 0 and (φ′(s))2 + (ψ′(s))2 = 1 for all s ∈ I. Compute its mean curvature andits Gaussian curvature.

Exercise 3.18. Consider the regular parametrised surface of revolution

X : (s, θ) ∈ I × R 7→ (φ(s) cos(θ), φ(s) sin(θ), ψ(s)) ∈ R3,where φ(s) > 0 and (φ′(s))2 + (ψ′(s))2 = 1 for all s ∈ I. Compute the normal curvature ofthe coordinate curves s = const. and θ = const. (Caution: the formula given in the lecturesis the formula that is valid for curves parametrised by arc-length).

Exercise 3.19. (Surfaces of revolution with constant Gaussian curvature, c.f. do Carmo,3.3., Exercise 7 - beware the sketch, it is not quite right in the case K = 1! See do Carmo5.2, Example 1 ):

Considering a regular parametrised surface of revolution described by a map X as in theprevious exercise, prove that its Gaussian curvature is equal to a constant c ∈ R if and onlyif the function φ > 0 satisfy the second order ODE

(∗) φ′′ + cφ = 0 on I.Assuming for simplicity that 0 ∈ I, show that ψ is then determined by the formula

(∗∗) ψ(s) =ˆ s

0

√1− (φ′(u))2du for all s ∈ I.

Let us then suppose that φ(s) > 0 and (φ′(s))2 + (ψ′(s))2 = 1 for all s ∈ I, and that I is themaximal interval containing 0 where solutions to (∗) and (∗∗) satisfying these two furtherconditions exist.

Solving (∗) for an initial admissible condition φ(0) = A > 0, φ′(0) = B ∈ [−1, 1], on themaximal interval I, and considering the corresponding ψ given by (∗∗), study the surfacesobtained on each case c = 0, c = 1 and c = −1. In particular, conclude that

16


i) If c = 0, then X parametrises either a right circular cylinder, or a cone, or a plane.ii) If c = 1, show that all solutions φ attain a maximum value at a point in I. Changing

parameters so that 0 ∈ I corresponds to that maximum point, show that the solutionsare given by

φ(s) = A cos(s) and ψ(s) =

ˆ s0

√1− A2 sin2(u)du.

Determine the domain of a solution φ. Which value of A correspond to a sphere ofcurvature 1? For values of 0 < A < 1, show that the curve (φ(s), 0, ψ(s)), s ∈ I,converges to the rotation axis as s approaches the boundary of the interval I, at adefinite angle 6= π/2. For values A > 1, show that (φ(s), 0, ψ(s)), s ∈ I, is such thatφ(s) >

√A2 − 1/A > 0. Finally, show that for all positive A 6= 1, the surface of

revolution generated by the above curves are regular surfaces in R3 with no umbilicpoints that, moreover, are not proper subsets of any regular surface of R3.

iii) If c = −1, show that solutions φ are such that |φ′(t)| eventually grows monotonicallyto 1 as t approaches a finite number t. (This means that the image of X can not becontinued further as a regular surface). Possibly after changing the parameter, showthat solutions fall into one of the following types

φ(s) = A cosh(s) and ψ(s) =

ˆ s0

√1− A2 sinh2(u)du, or

φ(s) = D sinh(s) and ψ(s) =

ˆ s0

√1−D2 cosh2(u)du, or

φ(s) = es and ψ(s) =

ˆ s0

√1− e2udu.

(in the second case, D is a constant such that 0 < D < 1).

Draw sketches of the solutions obtained (or plot them using a computer).

Exercise 3.20. (Non-planar minimal surfaces of revolution are catenoids):

i) Show that the catenoid is a minimal surface.ii) Let S be the surface of revolution

S = {(x, y, z) ∈ R3; z ∈ R,√x2 + y2 = f(z)},

where f : R→ R is a smooth positive function. Prove that, if S is a minimal surface,then it is a catenoid (up to a dilation and vertical translation).

Exercise 3.21. (The helicoid):Let S be the set obtained by taking a horizontal straight line (say, the x-axis), and moving

it with constant speed in the z direction while the line rotates horizontally at constant speed.Show that S is a regular surface in R3 parametrised by the map

X : (t, θ) ∈ R3 7→ (t cos(θ), t sin(θ), θ) ∈ R3.Prove that S is a minimal surface.

Exercise 3.22. Let S be a regular surface in R3. A line of curvature is a curve C in S thatis tangent to a principal direction of S at every point p in C.

Assume that S is orientable, with Gauss map NS. Show that a regular parametrised curve17


α : I → S is a line of curvature if and only if there exists a smooth function λ : I → R suchthat

(NS ◦ α)′(t) = λ(t)α′(t) for all t ∈ I.

Exercise 3.23. Let S be a regular surface in R3. An asymptotic curve is a curve in S whosenormal curvature vanishes identically.

i) Show that asymptotic curves do not contain elliptic points of S.ii) If S contains a straight line L, then L is an asymptotic curve of S.iii) If L is an asymptotic curve in S that is tangent to a principal direction at all of its

points, then L is a plane curve.iv) Give an example of a regular surface in R3 that contains straight lines that are not

tangent to a principal direction at every point.

Exercise 3.24. (An exercise in Differentiation):Let F : U ⊂ Rm → R be a smooth function, and p be a critical point of F . Let

γ : (−�, �)→ U be a smooth map such that γ(0) = p. If γ′(0) = (a1, . . . , am) ∈ Rn, then

(F ◦ γ)′′(0) =m∑

i,j=1

aiaj∂2F

∂xi∂xj(0).

In particular, conclude that if γ1, γ1 : (−�, �) → U are smooth maps such that γ1(0) =γ2(0) = p and γ

′1(0) = γ

′2(0), then

(F ◦ γ1)′′(0) = (F ◦ γ2)′′(0).

Show that above conclusion does not remain true if the same computation is done at a pointp that is a regular point of F .

Exercise 3.25. (On the Spectral Theorem):Let S1 denote the unit circle in R2 centred at the origin.i) If f : S1 → R is a smooth map such that

f(−v) = f(v) for all v ∈ S1,

then f has at least four critical points.ii) Let L : R2 → R2 be a self-adjoint linear map. Let E be the function given by

E : v ∈ S1 7→ 〈L(v), v〉 ∈ R.

Show that v ∈ S1 is a critical point of L if and only if v is an eigenvector of L. Whatis the corresponding eigenvalue?

iii) Show that eigenvectors of a self-adjoint linear map L : R2 → R2, corresponding todifferent eigenvalues, are mutually orthogonal.

iv) Combining the topological lemma in item i) with the other two items, can you provethe Spectral Theorem? (Discussion: item i) can be generalised to higher dimensions(and beyond) via a topological theory called the Lusternik-Schnirelmann Theory, whichover years has inspired many remarkable results in Geometry).

18


4. Intrinsic Geometry

Exercise 4.1. Compute the coefficients E, F and G of the first fundamental form of the unitsphere centred at the origin in the coordinate system given by the inverse of the stereographicprojection from the north pole. In particular, verify that E = G and F = 0 (Remark: suchparametrisations are called conformal, or isothermal).

Exercise 4.2. Let S1 be the unit sphere in R3 centred at the origin and S2 be the cylinder{(x, y, z) ∈ R3; x2 +y2 = 1}. Show that the map φ : S1\{(0, 0,±1)} → S2 defined on Section2, Exercise 21, is an area-preserving map: for every compact subset K of S1 \ {(0, 0,±1)},the area of K is equal to the area of φ(K).

Exercise 4.3. (Area of spheres and tori of revolution):

i) Describe the sphere of radius R > 0 centred at the origin as a surface of revolutionaround the z-axis, and compute its area.

ii) Compute the area of the torus Tr,R obtained by revolution around the z-axis of thecircle on the plane xz of radius r centred at the point (R, 0, 0), where R > r > 0.

Exercise 4.4. Let f : U → R be a smooth function, and Ω a compact smooth domaincontained in U . Show that the area of K = f(Ω) ⊂ graph(f) is given byˆ

Ω

√1 + |∇f |2dudv,

where ∇f(u, v) = grad f(u, v) = (fu(u, v), fv(u, v)) ∈ R2 denotes the gradient of f at thepoint (u, v) ∈ R2.

Exercise 4.5. (Integral of real functions defined on surfaces):Let S be a connected regular surface in R3, f : S → R a continuous function, and K a

compact subset of S that is contained in a coordinate neighbourhood V ∩ S = X(U). Theintegral of f over the set K is the numberˆ

K

fdS :=

ˆX−1(K)

(f ◦X)√EG− F 2dudv.

(Remark: this definition is independent of choices of parametrisations).Let S = Tr,R be the torus of revolution

Tr,R = {(x, y, z) ∈ R3; (√x2 + y2 −R)2 + z2 = r2},

and

X : (u, v) ∈ (0, 2π)× (0, 2π) 7→ ((r cos(u) +R) cos(v), (r cos(u) +R) sin(v), r sin(u)) ∈ Tr,Ra parametrisation of S. Show that, in these coordinates, the Gaussian curvature of Tr,R isgiven by

K =cos(u)

r(r cos(u) +R),

and compute the integral ˆS

KdS.

19


(Suggestion: use the formula K = (eg−f 2)/(EG−F 2). Convince yourself that the subsetof S missed by the image of parametrisation X is negligible for the calculus of the aboveintegral).

Exercise 4.6. Let S1 and S2 be two regular surfaces in R3, and assume that there exists twoparametrisations X1 : U → V1 ∩ S1 and X2 : U → V2 ∩ S2, defined over the same opensubset U of R2, such that the coefficients of the first fundamental form of S1 and S2 in thecoordinate systems X1 and X2 coincide at each point of U :

E1 = E2, F1 = F2, and G1 = G2.

Show that there exists a local isometry φ : V1 ∩ S1 → V2 ∩ S2. Use this criterion to producelocal isometries between: a plane and the right cylinder over an horizontal circle, and a planeand a cone. (Extra: and between the helicoid and the catenoid - ver Parte 3 ).

Exercise 4.7. Let S be a regular surface in R3 and X : U ⊂ R2 → V ∩ S be a parametri-sation such that the coefficient F of the first fundamental form vanishes identically on U .

Compute the Christoffel symbols in this parametrisation and show that

K = − 12√EG

((Gu√EG

)u

+

(Ev√EG

)v

).

Use the above formula to compute the Gaussian curvature in coordinates given by a parametri-sation such that

E(r, θ) = 1, F (r, θ) = 0, and G(r, θ) = φ2(r),

for all (r, θ) in its domain, in the following cases:

a) φ(r) = sin(r).b) φ(r) = r.c) φ(r) = sinh(r).

Exercise 4.8. Let S be a regular surface in R3 and X : U ⊂ R2 → V ∩S be a parametrisationsuch that the coefficients of first fundamental satisfy

E = G = λ and F = 0

on U , where λ is a smooth positive function on U . Parametrisations satisfying the aboveproperty are called isothermal parametrisations.

Show that

K = − 12λ

∆(log(λ)),

where ∆ denotes the Laplacian operator in U ,

∆λ :=∂2λ

∂u2+∂2λ

∂v2.

In particular, conclude that, if the coefficients of the first fundamental form are given by

E(u, v) = G(u, v) =1

v2and F (u, v) = 0,

for all (u, v) in its domain, then the Gaussian curvature is constant and equal to −1.20


Exercise 4.9. A regular surface S in R3 is called homogeneous when for every two points pand q in S there exists an isometry φ : S → S such that φ(p) = q. Prove that a homogeneoussurface has constant Gaussian curvature. What are the connected compact homogeneousregular surfaces in R3?

Exercise 4.10. Let S be a regular surface in R3 and let α : I → S be a regular parametrisedcurve in S. Show that the following statements are equivalent:

i) The tangent α′ is a parallel vector field along α.ii) For every t ∈ I, the vector α′′(t) is orthogonal to S.iii) |α′| is constant and the geodesic curvature of α vanishes identically on I.

(Remark: notice that i) can be expressed as an instrinsic condition, whereas ii) is an extrinsiccondition).

Exercise 4.11. Let S be a regular surface in R3 and C a regular curve in S. We may call Ca geodesic when it is covered by parametrisations by arc-length β : I → C whose geodesiccurvature vanishes identically. Let α : I → S be a parametrisation of C, not necessarily byarc-length. Show that the regular curve C0 = α(I) ⊂ C is a geodesic in S if and only if α′′(t)belongs to the plane generated by the tangent vector α′(t) and a unit normal to the surfaceat α(t) ∈ S for all t ∈ I.

(Suggestion: to solve this exercise, reparametrise β = α◦φ : J → S by arc-length, computeits geodesic curvature and check what the condition kg(s) = 0, s ∈ J , for the parametrisedcurve β, imposes on the vector α′′(t), t = φ(s) ∈ I. Compare with the previous exercise).

Exercise 4.12. Let S1 and S2 be two regular surfaces in R3 that intersect tangentially alonga regular curve C (i.e. TpS1 = TpS2 for every point p ∈ C ⊂ S1 ∩ S2). Prove that C is ageodesic in S1 if and only if C is a geodesic in S2.

Exercise 4.13. (Exercise on geodesics, and their extrinsic geometry):Let S be a regular surface in R3.i) A curve in S is simultaneously a geodesic and an asymptotic line if and only if it is

contained in a straight line.ii) Show that if a curve C in S is a geodesic and a line of curvature, then it must be

contained in a plane.iii) If a geodesic C is a plane curve, then either C is contained in a straight line or is a

line of curvature.iv) Give an example of a regular surface in R3 that contains line of curvatures that are

plane curves but not geodesics.

Exercise 4.14. Let S be a regular surface of revolution in R3 parametrised by

X : (s, θ) ∈ I → (φ(s) cos(θ), φ(s) sin(θ), ψ(s)),

where φ(s) > 0 and (φ′(s))2 + (ψ′(s))2 = 1 for all s ∈ I. Prove that the coordinate curvesθ = const. are geodesics. When are the coordinate curves s = const. geodesics?

Exercise 4.15. (Geodesics of the sphere): Let S be a sphere in R3. A great circle in S is theregular curve in S determined by the intersection of S with a plane that contains the origin.

i) Prove that great circles are geodesics.21


ii) Show that for each point p in S and for each tangent vector v 6= 0 in TpS there existsa unique great circle passing through p and tangent to v.

iii) Conclude that the geodesics of a sphere are great circles.

Exercise 4.16. Let S be a regular surfaces in R3. Show that the set of isometries φ : S → S,endowed with the operation of composition of maps, is a group. If S is orientable, show thatthe subset of those isometries of S that preserve a given orientation of S form a subgroup ofthe group of isometries.

Exercise 4.17. (Isometries of the sphere) Let S = S21(0) be the unit sphere centred at theorigin.

i) Let R : R3 → R3 be an orthogonal transformation. Show that R(S) = S and thatthe restriction R : S → S is an isometry of S.

ii) Let φ : S → S be an isometry such that φ(p) = p andDφ(p) · v = v for all v ∈ TpS.

Show that φ(p) = p for all p ∈ S (i.e. φ : S → S must be the identity map).iii) Let φ : S → S be an isometry of S, and denote by p0 the north pole. Show that there

exists a unique orthogonal transformation R : R3 → R3 such that φ(p0) = R(p0) andDφ(p0) · v = R(v) for every v ∈ TpS ⊂ R3.

iv) Conclude that isometries of the unit sphere centred of the origin are restrictions oforthogonal transformations of R3.

v) What are the orientation-preserving isometries of S21(0)? Give an example of anorientation-reversing isometry of S21(0).

Exercise 4.18. Let S be a regular surface in R3, α : I → S a regular parametrised curve inS starting at p = α(0), 0 ∈ I, and W1, W2 : I → R3 two tangent vector fields to S along α.Prove that

〈W1,W2〉′ =〈DW1

dt,W2

〉+〈W1,

DW2dt

〉.

Next, suppose that W1 and W2 are parallel along α. If W1(0) is orthogonal to W2(0), showthat W1(t) is orthogonal to W2(t) for every t ∈ I.

Exercise 4.19. Let S be a regular surface in R3, α : I → S a regular parametrised curve inS, and β = α ◦ φ : J → S a reparametrisation of α. If W : I → R3 is a parallel tangentvector field along α, then Y = W ◦ φ : J → R3 is a parallel tangent vector field along β.

Exercise 4.20. (The holonomy group):Let S be a regular surface in R3, p a point in S and α : [0, 1] → S a piecewise smooth

map into S that starts and finishes at the point p, that is, α(0) = p and α(1) = p. LetPα : TpS → TpS be the map that assigns, to each vector v ∈ Tα(0)S = TpS, the vectorPα(v) ∈ Tα(1)S = TpS that is the parallel transport of v along α at t = 1.

Show that Pα : TpS → TpS is a linear map that preserves the first fundamental form of Sat p:

〈Pα(v), Pα(w)〉 = 〈v, w〉 for all v, w ∈ TpS.Notice that the composition of maps Pβ ◦ Pα corresponds to do parallel transport along thecurve γ : [0, 1]→ S that on [0, 1/2] traverses α with double speed, and on [1/2, 1] traversesβ with double speed. The set of all such transformation Pα, endowed with composition, is a

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group known as the holonomy group of the surface S at the point p.Let S = S21(0) be the unit sphere centred at the origin, and let p0 = (0, 0, 1) be the

north pole. Show that the holonomy group of S21(0) at p0 is the set of orientation-preservingrotations of the (oriented) plane Tp0S.

(Hint: it is convenient to use curves consisting of geodesics, since their tangent vectorsare parallel. Exercise 18 is also useful. Use paths consisting of a geodesic that goes from p0until the equator, then move along the equator, and then return to the the north pole as ageodesic).

Exercise 4.21. (Geometry of regular parametrised curves in spheres):Let S = S21(0) denote the unit sphere in R3, centred at the origin. Orient S21(0) by the

outward pointing Gauss map

NS : p ∈ S 7→ p ∈ S.Let α : I → S be a regular curve in S parametrised by arc-length. Consider the orthonor-

mal frame at s ∈ I consisting of the vectors

{T := α′(s), NS(s) := NS(α(s)), D(s) := NS(s)× α′(s)}.

(Remark: this frame is in general different from the Frenet frame).Show that the above adapted frame satisfy the equations T ′N ′S

D′

= 0 −1 kg1 0 0−kg 0 0

· TNS

D

,where kg = 〈α′′, NS × α′〉 is the geodesic curvature of α.

Can you propose a statement and a proof for the Fundamental Theorem of Curves inS21(0)?

Classify those curves in the unit sphere that have constant geodesic curvature.

5. The Gauss Bonnet Theorem

Exercise 5.1. Draw several closed piecewise regular parametrised plane curves and computethe total angular variation of their tangent lines (please draw at least four of them with totalangular variation equal to 2π, 4π, 0 and −6π, respectively).

Exercise 5.2. Explain the precise meaning of each term in the Gauss-Bonnet formula for acompact piecewise regular region Ω in an oriented surface S in R3:

ˆΩ

K +

ˆ∂Ω

kg +k∑i=1

φi = 2πχ(Ω).

Exercise 5.3. Check that a sphere S of radius R > 0 in R3 is such thatˆS

K = 4π.

Exercise 5.4. Compute the area of a geodesic triangle T in a unit sphere in R3 in terms ofthe interior angles of T .

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Exercise 5.5. Let S be the unit sphere in R3, centred at the origin. Use the Gauss-BonnetTheorem to show that the area of the region bounded by the parallels at height z = h1 andz = h2, where −1 ≤ h1 < h2 ≤ 1, is equal to

2π(h2 − h1).

Exercise 5.6. Let S be a compact connected regular surface in R3 that has non-negativeGaussian curvature. What is the genus of S?

Exercise 5.7. Let S be a connected compact oriented regular surface in R3 with positiveGaussian curvature. Any two disjoint closed simple curves in S bound an annular region.Conclude that any two closed simple geodesics in S must intersect.

Exercise 5.8. Let S be an oriented regular surface in R3 with non-positive Gaussian cur-vature, and assume that every simple closed piecewise regular curve in S bounds a regionhomeomorphic to a disc. Given p and q two distinct points in S, show that there can be nomore than one geodesic joining p to q.

Exercise 5.9. Compute the Euler characteristic of at least ten different (not homeomorphic)piecewise regular regions on arbitrary regular surfaces in R3.

6. Other exercises

Exercise 6.1. (Cf. Part 1, Exercise 1.16 ) Let α : I → R2 be a curve parametrised by arc-length. Suppose that Cλ is a circle of radius R > 0 that is tangent to α(I) at the pointα(s0), and that α is oriented so that its normal at s0 points inside the circle.

i) If α(I) lies inside the circle, then k(s0) ≥ 1/R.ii) If α(I) lies outside the circle, then k(s0) ≤ 1/R.

Exercise 6.2. (The sign of the torsion, cf. do Carmo section 1.6 ).Let α : I → R3 be a curve parametrized by arc-length with non-vanishing curvature.

Assume that 0 ∈ I, α(0) = 0 and that the Frenet trihedron at 0 is the canonical basis of R3.In particular, its osculating plane at 0 ∈ I is the plane z = 0.

Writing α(s) = (x(s), y(s), z(s)) in Euclidean coordinates, write the Taylor expansion ofα at s = 0 up to order three,

α(s) = α(0) + sα′(0) +s

2α′′(0) +

s3

6α′′′(0) + o(s3),

in terms of the curvature k, its derivative k′, and the torsion τ of α at 0. Analysing the signof z(s), describe what happens with the point α(s) as s varies from a negative number to apositive number, according to the sign of τ(0).

Exercise 6.3. (Curvature of curves in R3 and osculating planes, cf. do Carmo section 1.6 ).Keeping the notation of the previous exercise, show that the curvature of the orthogonal

projection of α on the plane z = 0 at the point s = 0 (i.e., the osculating plane of α ats = 0) is equal to the curvature of the curve α at s = 0.

Exercise 6.4. Let α : I → R3 be a curve parametrised by arc-length and fix s0 ∈ I. Supposeits curvature is never zero, and let B : I → R3 denote the bi-normal vector field. If α has

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constant torsion τ 6= 0, then

α(s) = α(s0)−1

τ

ˆ ss0

B(s)× dBds

(s)ds for all t ∈ I.

Exercise 6.5. Let S be a regular surface in R3, and p a point in S. Show that there is asufficiently small neighbourhood V of p in R3 such that S∩V is a graph of a smooth functionon the tangent plane of S at p. (Hint: without loss of generality, after a rigid motion youmay assume p = 0 and TpS is the xy-plane. Use the Inverse Function Theorem).

Exercise 6.6. Let f : U → R be a smooth function defined on an open neighbourhood of theorigin. Assume that f(0, 0) = 0 and Df(0, 0) = 0.

i) Consider the regular surface S = graph(f). Show that T(0,0)S is the xy plane.ii) Consider the regular surfaces

λ · S = {λ · p ∈ R3; p ∈ S},obtained by dilation based at the origin by the factor λ > 0. Show that the regularsurfaces λ · S are the graphs of smooth functions fλ on open subsets of the xy plane.

iii) As λ goes to infinity, the domains of the functions fλ eventually contain a open discthat is the intersection of an open ball Br in R3 of radius r centred at the originwith the xy-plane. Show that functions fλ and its first partial derivatives convergeto zero uniformly on each Br as λ goes to infinity. (Comment: actually, each partialderivative of any order converge to zero uniformly. This exercise can be interpretedas saying that, as one dilates S around a point p ∈ S by a factor λ > 0, dilatedsurfaces converge locally smoothly to TpS as λ goes to infinity. Thus, in a geometricsense, TpS is the best infinitesimal approximation of S).

Exercise 6.7. (Suggested by discussion in the lectures) Show that there exists a subset S of

R2 and maps X, X̃ : (−1, 1)→ R2 that satisfy the following properties:i) X and X̃ are bijections onto S.

ii) X and X̃ are smooth.

iii) X ′(t) 6= 0 and X̃ ′(t) 6= 0 for every t ∈ (−1, 1).iv) None of the compositions X−1◦X̃ : (−1, 1)→ (−1, 1) and X̃−1◦X : (−1, 1)→ (−1, 1)

is continuous.

How does this exercise relate to the proof that change of parametrisations of a regular curvesin R2 are difeomorphims?

(Hint: consider the figure of the number 8, map 0 ∈ (−1, 1) to the self-intersection point,and try to imagine different ways to map (−1, 0] and [0, 1) continuously onto the figure 8 insuch way that near −1 and 1 the maps are converging to the self-intersection point).

Exercise 6.8. (Suggested by discussion in the lectures) Consider a piece of plane regular curvethat can be described as the graph of a smooth positive function f : (−1, 1) → R on theinterval (−1, 1), and that touches the points (−1, 0) and (0, 1) tangentially to the unit circlecentred at the origin. Suppose that this piece of curve lies always above the upper semi-circle with radius one centred at the origin, and that its curvature is everywhere non-negative(when the curve is oriented appropriately). Show that this plane regular curve contains apoint with curvature at least one.

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Exercise 6.9. Let L : Rn → Rn be a self-adjoint endomorphism. Show thatˆSn−1〈L(v), v〉dAcan(v) = αntr(L),

where Sn−1 is the unit sphere in Rn centred at the origin, and αn is a constant dependingonly on the dimension n.

(Hint: try first in dimension n = 2. Use a basis diagonalising L and then compute theintegral over the circle S1 directly. In higher dimensions, the vector field p ∈ Rn 7→ L(p) ∈ Rnmight be useful).

Exercise 6.10. (Suggested by discussion in the lectures). Let L, M : Rn → Rn be self-adjointendomorphisms. Let Q, R : Rn → R be the associate quadratic forms,

Q(v) = 〈Lv, v〉, R(v) = 〈Mv, v〉 for all v ∈ Rn,and let λ1 ≤ . . . ≤ λn, µ1 ≤ . . . ≤ µn be the corresponding eigenvalues of L and M .

In this exercise, we assume that:

Q(v) ≥ R(v) for all v ∈ Rn.i) Show that trace(L) ≥ trace(M).ii) Suppose the dimension is n = 2. Give an example of L and M as above such that

det(L) < det(M).iii) Show that λ1 ≥ µ1 and λn ≥ µn.iv) Suppose the dimension is n = 3. What can be said about the diference of the middle

eigenvalues of L and M , that is, about the number λ2 − µ2?(Comment: the reader should try to make the connection between this exercise and the dis-cussion on regular surfaces that intersect tangentially at a point p).

Exercise 6.11. Let α : [0, `] → R3 be a closed curve parametrized by arc-length whosecurvature never vanishes. Let B : [0, `] → R3 be the binormal. If α has constant torsion,show that B is a regular parametrised curve whose trace lies in the unit sphere centred atthe origin in R3. What can you say about the geodesic curvature of this curve?

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Curvesw3.impa.br/~l.ambrozio/listagrande.pdf · 2021. 1. 26. · IMPA, Ver~ao 2021 - Prof. Lucas...

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