New Diff Geom

8/12/2019 New Diff Geom

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Problems and Solutions

in

Differential Geometry

byWilli-Hans SteebInternational School for Scientific ComputingatUniversity of Johannesburg, South Africa

Yorick HardyDepartment of Mathematical SciencesatUniversity of South Africa, South Africa


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Preface

The purpose of this book is to supply a collection of problems in differential

geometry.

[email protected]

[email protected]

Home page of the author:

http://issc.uj.ac.za

v


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vi


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Contents

Preface v

1 Curves, Surfaces and Manifolds 1

2 Vector Fields and Lie Series 16

3 Metric Tensor Fields 25

4 Differential Forms and Applications 34

5 Lie Derivative and Applications 50

6 Killing Vector Fields 65

7 Lie-Algebra Valued Differential Forms 68

8 Lie Symmetries and Differential Equations 77

9 Integration 82

10 Lie Groups and Lie Algebras 85

11 Miscellaneous 87

Bibliography 94

Index 103

vii


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ix

Notation

:= is defined as belongs to (a set)/ does not belong to (a set) intersection of sets union of sets empty setN set of natural numbersZ set of integersQ set of rational numbersR set of real numbers

R+ set of nonnegative real numbersC set of complex numbersRn n-dimensional Euclidian space

space of column vectors withn real componentsCn n-dimensional complex linear space

space of column vectors withn complex componentsM manifoldH Hilbert spacei

1z real part of the complex numberzz imaginary part of the complex numberz|z| modulus of complex number z

(

|x + iy

|= (x2 + y2)1/2, x , y

R

T S subset Tof set SS T the intersection of the setsSand TS T the union of the setsSandTf(S) image of setSunder mapping ff g composition of two mappings (f g)(x) = f(g(x))x column vector inCn

xT transpose ofx (row vector)0 zero (column) vector . normx yxy scalar product (inner product) inCnx y vector product in R3Sn symmetric groupA

n alternating group

Dn n-th dihedral group


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x

A, B, C m nmatricesdet(A) determinant of a square matrix Atr(A) trace of a square matrixArank(A) rank of matrix A

AT transpose of matrix AA conjugate of matrixAA conjugate transpose of matrix AA conjugate transpose of matrix A

(notation used in physics)A1 inverse of square matrix A (if it exists)In n n unit matrixI unit operator0n n n zero matrixAB matrix product ofm n matrixA

and n p matrix BV vector field

ofm n matricesA and B[A, B] := AB BA commutator for square matricesA and B[A, B]+ := AB+ BA anticommutator for square matricesA and B tensor product exterior product, Grassmann product, wedge productjk Kronecker delta withjk = 1 for j = k

and jk = 0 for j=k eigenvalue real parametert time variable

H Hamilton operator


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Chapter 1

Curves, Surfaces and

Manifolds

Problem 1. Consider the compact differentiable manifold

S2 :={ (x1, x2, x3) : x21+ x22+ x23 = 1 }.An elementS2 can be written as

= (cos sin , sin sin , cos )

where[0, 2) and [0, ]. The stereographic projectionis a map : S2 \ { (0, 0, 1) } R2

given by

x1(, ) =2 sin() cos()

1 + cos() , x2(, ) =

2 sin() sin()

1 + cos() .

(i) Let = 0 and arbitrary. Findx1, x2. Give a geometric interpretation.(ii) Find the inverse of the map, i.e., find

1 : R2 S2 \ { (0, 0, 1) }.

Problem 2. The parameter representation for thetorusis given by

x1(u1, u2) = (R+ r cos u1)cos u2

x2(u1, u2) = (R+ r cos u1)sin u2

x3(u1, u2) = r sin u1

1


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Curves, Surfaces and Manifolds 3

for the equivalence relation

(u0, u1, . . . , un)(v0, v1, . . . , vn) C : uj =vj 0jn

where C

:= C\ { 0 }. Show that P1

(C) is a one-dimensional complexmanifold.

Problem 6. Let

Sn :={ (x1, x2, . . . , xn+1) : x21+ x22+ + x2n+1 = 1 } .(i) Show thatS3 can be considered as a subset ofC2 (C2= R4)

S3 ={ (z1, z2) C2 :|z1|2 + |z2|2 = 1 } .(ii) The Hopf map : S3 S2 is defined by

(z1, z2) := (z1z2+ z2z1, iz1z2+ iz2z1, |z1|2 |z2|2) .

Find the parametrization ofS3

, i.e. find z1(, ), z2(, ) and thus showthat indeed maps S3 onto S2.(iii) Show that (z1, z2) =(z1, z

2) if and only ifz

j =e

izj (j = 1, 2) andR.

Problem 7. Then-dimensional complex projective space CPn is the setof all complex lines on Cn+1 passing through the origin. Letfbe the mapthat takes nonzero vectors in C2 to vectors in R3 by

f(z1, z2) =

z1z2+ z1z2z1z1+ z2z2

, z1z2 z1z2i(z1z1+ z2z2)

,z1z1 z2z2z1z1+ z2z2

The map f defines a bijection between CP1 and the unit sphere in R3.Consider the normalized vectors in C2

10

,

01

,

12

11

,

12

11

, 1

2

ii

.

Applyfto these vectors in C2.

Problem 8. The stereographic projection is the map : S2 \N Cdefined by

(x, y, z) = x

1 z + i y

1 z.Show that the inverse of the stereographic projection takes a complex num-ber u + iv (u, v R)

2u1 + u2 + v2 , 2v1 + u2 + v2 ,1 u2 v21 + u2 + v2


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4 Problems and Solutions

to the unit sphere.

Problem 9. Show that the projective space Pn(C) is a compact manifold.

Problem 10. Consider the solid torus M = S1 D2, where D2 is theunit disk in R2. On it we define coordinates (,x,y) such thatS1 and(x, y) D2, that is, x2 +y2 1. Using these coordinates we define themap

f :MM, f(,x,y) =

2, 1

10x +

1

2cos ,

1

10y+

1

2sin

.

(i) Show that this map is well-defined, that is, f(M)M.(ii) Show that f is injective.

Problem 11. Show that a parameter representation of the hyperboloid

x21 x22 x23 = 1

is given by

x1(t) = cosh(t), x2(t) = sinh(t) cos(), x3(t) = sinh(t) sin()

where 0t 0 } .

Find a parametrization for v.

Problem 13. Find the stereographic projection of the two-dimensionalsphere

S2 :={v R3 : v2 =v20+ v21+ v22 = 1 }

Problem 14. Consider the curve

(t) =

t

cosh(t)

, t R

Show that the curvatureis given by

(t) = 1

cosh2(t) .


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Problem 15. Consider the unit ball

S2 :={(x, y, z) R3 : x2 + y2 + z2 = 1 } .

Let (t) = (x(t), y(t), z(t)) be a parametrized differentiable curve on S2

.Show that the vector (x(t), y(t), z(t)) (t fixed) is normal to the sphere atthe point (x(t), y(t), z(t)).

Problem 16. A generic superquadric surface can be defined as a closedsurface in R3

r(, )x(, )y(, )

z(, )

= a1cos1()cos2()a2cos1()sin2()

a3sin1()

, /2/2, < .There are five parameters1,2,a1,a2,a3. Here1 and 2are the deforma-tion parameters that control the shape with 1, 2(0, 2). The parametera1, a2, a3 define the size in x, y and z direction. Find the implicit repre-sentation.

Problem 17. Let

x1(z, z) = sech

z+ z

2

cosh

z z

2

x2(z, z) = isech

z+ z

2

sinh

z z

2

x3(z, z) = tanh

z+ z

2

Findx21+ x

22+ x

23. Note that

sech(z) := 2

ez + ez.

Problem 18. Let d = (d0, d1, . . . , dn) be an (n+ 1)-tuple of integersdj >1. We define

V(d) :={ z= (z0, z1, . . . , zn) Cn+1 : f(z) := zd00 + zd11 + + zdnn = 0 } .Let S2n+1 denote the unit sphere in Cn+1, i.e.

z0z0+ z1z1+ + znzn= 2 .We define

(d) := V(d) S2n+1 .


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Show that (d) is a smooth manifold of dimension 2n 1. The manifolds(d) are called Brieskorn manifolds.

Problem 19. Consider the stereographic projection

r(w) =

2(w)|w|2 + 1 ,

2(w)|w|2 + 1 ,

|w|2 1|w|2 + 1

.

Letw = i. Find r(w).

Problem 20. (i) Consider the rational curve in the plane

y2 =x2 + x3 .

Find the parameter representation.(ii) Consider the rational curve in the plane

x2 + y2 = 1 .

Find the parameter representation.

Problem 21. Let a > 0. Consider the transformation Minkowski coor-dinates (t, z and Rindler coordinates(, )

t(, ) = 1

aexp(a)sinh(a), z(, ) =

1

aexp(a) cosh(a) .

Find the inverse transformation.

Problem 22. Show that thehelicoid

x(u, v) = (a sinh(v) cos(u), a sinh(v) sin(u), au)

is a minimal surface.

Problem 23. LetA be a symmetricn nmatrix over R. Let 0=b R.Show that the surface

M={ x Rn : xTAx= b }

is an (n 1) dimensional submanifold of the manifold Rn.

Problem 24. LetCbe the topological space given by the boundary of

[0, 1]n := [0, 1] [0, 1] .


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This means Cis the surface of the n-dimensional unit cube. Show thatCcan be endowed with the structure of a differential manifold.

Problem 25. Find the Gaussian curvature for the torus given by theparametrization

x(u, v) = ((a + r cos(u)) cos(v), (a + r cos(u)) sin(v), r sin(u))

where 0< u < 2 and 0< v


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For fixed t the curve(x(), y(), z())

can be considered as a solution of a differential equation. Find this differ-

ential equation. Then t plays the role of a bifurcation parameter.

Problem 28. LetMbe a differentiable manifold. Suppose thatf :MM is a diffeomorphism with Nm(f)


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Extend the transformation to Rn.

Problem 31. A fixed chargeQ is located on the z-axis with coordinatesra = (0, 0, d/2), where d is interfocal distance of the prolate spheroidalcoordinates

x(, , ) =1

2d((1 2)(2 1))1/2 cos

y(, , ) =1

2d((1 2)(2 1))1/2 sin

z(, , ) =1

2d

where1 +1, 1 , 0 2. Express the Coulombpotential

V = Q

|r ra|in prolate spheroidal coordinates.

Problem 32. Let, ,, R. Consider the vector in R5

x(,,,) =

cosh sin cos cosh sin sin

cosh cos sinh cos sinh sin

.Find

x21+ x22+ x

23 x24 x25 .

This vector plays a role for the Lie group S O(3, 2). The invariant measureis

cosh2 sinh sin dddd .

Problem 33. Show that the surface Cof the unit cube

C={(x1, x2, x3) : 0x11, 0x21, 0x31 }can be made into a differentiable manifold.

Problem 34. The equation of the monkey saddle surface in R3 is givenby

x3 = x1(x21 3x22)

with the parameter representation

x1(u1, u2) = u1, x2(u1, u2) = u2, x3(u1, u2) = u31 3u1u22 .


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Find the mean and Gaussian curvature.Let

g= dx1 dx1+ dx2 dx2+ dx3 dx3 .

Findg restricted to the monkey saddle surface. Find the curvature scalar.

Problem 35. Leta >0 and consider the surface

x1(u1, u2) = a1 u221 + u22

cos(u1)

x2(u1, u2) = a1 u221 + u22

sin(u1)

x3(u1, u2) = 2au21 + u22

.

Findx21+ x22+ x

23.

Problem 36. Show that anopen disc

D2 :={ (x1, x2) R2 : x21+ x22 < 1 }

is homeomorphic to R2.

Problem 37. Let r > 0. The Klein bagel is a specific immersion of theKlein bottle manifold into three dimensions with the parameter represen-tation

x1(u1, u2) = (r+ cos(u1/2) sin(u2) sin(u1/2) sin(2u2)) cos(u1)x2(u1, u2) = (r+ cos(u1/2) sin(u2)

sin(u1/2) sin(2u2)) sin(u1)

x3(u1, u2)=sin(u1/2) sin(u2) + cos(u1/2) sin(2u2)

where 0 u1 < 2 and 0 u2 < 2. Find the mean curvature andGaussian curvature.

Problem 38. Consider the circle

S1 :={(x1, x2) R2 : x21+ x22 = 1 }

and the square

I2 ={ (x1, x2) R : (|x1|= 1, |x2| 1), (|x1| 1, |x2|= 1) } .

Find a homeomorphism.


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Problem 39. The transformation between the orthogonal ellipsoidal co-ordinates (,,) and the Cartesian coordinates (x1, x2, x3) is

x2

1=

222

h2k2

x22=(2 2)(2 h2)(h2 2)

h2(k2 h2)x23=

(2 k2)(k2 2)(k2 2)k2(k2 h2)

wherek2 =a21 a23, h2 =a21 a22 and a1 > a2 > a3 denote the three semi-axes of the ellipsoid. The three surfaces in R3, = constant, (k ), = constant, (h k) and = constant, (0 h, representellipsoids and hyperboloids of one and two sheets, respectively. Find theinverse transformation.

Problem 40. Letx1, x2, x3 R andx21+ x

22+ x

23 = 1 .

Letw C withw=

x1+ ix21 + x3

.

Findx1, x2, x3 as functions ofw andw.

Problem 41. (i) LetM be a manifold and f : M M, g : M M.Assume thatfis invertible. Then we say that the map f is a symmetryofthe map g if

f g f1

=g .LetM= R and f(x) = sinh(x). Find all g such thatf g f1 =g.(ii) Letfand g be invertible maps. We say that g has a reversing symmetryf if

f g f1 =g1 .LetM= R and f(x) = sinh(x). Find all g that satisfy this equation.

Problem 42. Consider the map f : R R2 defined by

f(x) = (2 cos(x /2), sin(2(x /2))) .

Show that (f,R) is an immersed submanifold of the manifold R2, but notan imbedded submanifold.


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Problem 43. Use GNU-plot to plot the curve

x1(t) = cos(3t), x2(t) = sin(5t)

in the (x1, x2)-plane with t

[0, 2].

Problem 44. A special set of coordinates onSn called spheroconical (orelliptic spherical) coordinates are defined as follows: For a given set of realnumbers 1 < 2


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of classCr (this means thatisrtimes continuously differentiable) is calleda parametric curve of class Cr of the curve . t is called the parameter ofthe curve . The parameter t may represent time and the curve (t) asthe trajectory of a moving particle in space. IfIis a closed interval [a, b],

then (a) the starting point and (b) is the endpoint of the curve . If : (a, b) Rn is injective, we call the curve simple. Ifis a parametriccurve which can be locally described as a power series, we call the curveanalytic or of class C. A Ck-curve

: [a, b] Rn

is called regular of order m if for any t in interval I

{d(t)/dt,d2(t)/dt2,...,dm(t)/dtm} mkare linearly independent in the vector space Rn. AFrenet frameis a movingreference frame ofnorthonormal vectorsej(t) (j = 1, . . . , n) which are usedto describe a curve locally at each point (t). Using the Frenet frame wecan describe local properties (e.g. curvature, torsion) in terms of a localreference system than using a global one like the Euclidean coordinates.Given aCn+1-curve inRn which is regular of order n the Frenet frame forthe curve is the set of orthonormal vectors

e1(t), . . . , en(t)

called Frenet vectors. They are constructed from the derivatives of (t) usingthe GramSchmidt orthogonalization algorithm with

e1(t) = d(t)/dt

d(t)/dt , ej(t) = ej(t)

ej(t) , j = 2, . . . , n

where

ej(t) = (j)(t)

j1

i=1(j)(t), ei(t) ei(t)where (j) denotes the j derivative with respect to t and , denotes thescalar product in the Euclidean space Rn. The Frenet frame is invariantunder reparametrization and are therefore differential geometric propertiesof the curve. Find the Frenet frame for the curve ( t R)

(t) =

cos(t)tsin(t)

.Problem 47. Show that the Lemniscate of Geronox41 = x

21 x22 can be

parametrized by

(x1(t), x2(t)) = (sin(t), sin(t)cos(t))


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where 0t.

Problem 48. Study the curve

x1(t)= cosc0t + c1

sin(t)x2(t) = sin

c0t +

c1

sin(t)

in the plane with c0, c1, > 0, where c0, c1, have the dimension of afrequency and t is the time.

Problem 49. The Hammer projectionis an equal-area cartographic pro-jections that maps the entire surface of a sphere to the interior of an ellipseof semiaxis

8 and

2. The Hammer projection is given by the transfor-

mation between (, ) and (x1, x2)

x1(, ) = 8 sin() sin(/2)1 + sin()cos(/2)

, x2(, ) = 2 cos()1 + sin()cos(/2)

where 0 and 0 0. Study the manifold

x21R2e

+ x22R2e

+ x23R2e2

= 1

where is a deformation parameter.(ii) Show that the volume Vof the spheroid is given by V = (4/3)R3.

Problem 52. Plot the graph

r() = 1 + 2 cos(2).


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Problem 53. Leta >0. Consider

x1(u, v) = a1 v21 + v2

cos(u), x2(u, v) = a1 v21 + v2

sin(u), x3(u, v) = a 2v

1 + v2.

(i) Show thatx21(u, v) + x

22(u, v) + x

23(u, v) = a

2.

(ii) Calculatex

u x

v

wheredenotes the vector product. Discuss.


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Chapter 2

Vector Fields and Lie

Series

Problem 1. Consider the vector fields

V =x

x+ y

y, W=x

y y

x

defined on R2.

(i) Do the vector fields V, Wform a basis of a Lie algebra? If so, what typeof Lie algebra do we have.

(ii) Express the two vector fields in polar coordinates x(r, ) = r cos ,

y(r, ) = r sin .(iii) Calculate the commutator of the two vector fields expressed in polarcoordinates. Compare with the result of (i).


V1 = d

dx, V2 = x

d

dx, V3 = x

2 d

dx.

(i) Show that the vector fields form a basis of a Lie algebra under thecommutator.

(ii) Find the adjoint representation of this Lie algebra.

(iii) Find the Killing form.

(iv) Find the Casimir operator.

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Vector Fields and Lie Series 17


V1= cos

+

sin

sin

cot sin

V2= sin

+cos sin

cot cos

V3=

.

Calculate the commutators and show that V1, V2, V3 form a basis of a Liealgebra.

Problem 4. Let X1, X2, . . . , X r be the basis of a Lie algebra with thecommutator

[Xi, Xj ] =r

k=1

CkijXk

where the Ckij are the structure constants. The structure constants satisfy(third fundamental theorem)

Ckij= Ckjir

m=1

Cmij C

mk+ C

mjkC

mi+ C

mkiC

mj

= 0 .

We replace the Xis byc-number differential operators (vector fields)

XiVi =r

=1

rk=1

xkCki

x, i= 1, 2, . . . , r .

Let

Vj =

rn=1

rm=1

xmCmjn xn.

Show that

[Vi, Vj ] =n

k=1

CkijVk

where

Vk =r

n=1

rm=1

xmCmkn

xn.

Problem 5. Consider the vector fields (differential operators)

E= x y , F=y x , H=x x y y .


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Show that these vector fields form a basis of a Lie algebra, i.e. calculatethe commutators. Consider the basis forn Z

{xjyk : j, k

Z, j+ k= n

}.

FindE(xjyk), F(xjyk), H(xjyk).

Problem 6. Show that the sets of vector fields

x, x

x, x2

x

x, x

x+ u

u, x2

x+ 2xu

u

x+

u, x

x+ u

u, x2

x+ u2

u

form each a basis of the Lie algebra s(2,C

) under the commutator.

Problem 7. Consider the Lie algebrao(3, 2). Show that the vector fieldsform a basis of this Lie algebra

V1 =

t, V2 = t

t+

1

2x

x, V3 = t

2

t+ tx

x+

1

4x2

u

V4=

x, V5 = t

x+

1

2x

u, V6 =

u

V7 =1

2x

x+ u

u, V8=

1

2xt

t+ (tu +

1

4x2)

x+

1

2xu

u

V9 =1

4

x2

t

+ ux

x

+ u2

u

, V10 =1

2

x

t

+ u

x

.

Show that the vector fields V1, . . . , V 7 form a Lie subalgebra.

Problem 8. Let X1, X2, . . . , X r be the basis of a Lie algebra with thecommutator

[Xi, Xj ] =r

k=1

CkijXk

where the Ckij are the structure constants. The structure constants satisfy(third fundamental theorem)

Ckij= Ckjir

m=1

Cmij Cmk+ CmjkCmi+ CmkiCmj= 0 .


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We replace theXis byc-number differential operators (linear vector fields)

XiVi =r

=1r

k=1 xkCki

x, i= 1, 2, . . . , r .

which preserve the commutators.Consider the Lie algebra with r = 3 and the generators X1, X2, X3 andthe commutators

[X1, X3] = X1, [X2, X3] =X2 .

All other commutators are 0. The Lie algebra is solvable. Find the corre-sponding linear vectors fields. Find the smooth functionsf such that

Vjf(x) = 0 for all j = 1, 2, 3 .

Problem 9. LetV , Wbe two smooth vector fields

V = f1

u1+ f2

u2+ f3

u3

W= g1

u1+ g2

u2+ g3

u3

defined on R3. Let du/dt= f(u) and du/dt = g(u) be the correspondingautonomous system of first order differential equations. The fixed pointsofVare defined by the solutions of the equations fj(u

1, u

2, u

3) = 0 (j =

1, 2, 3) and the fixed points ofWare defined as the solutions of the equationsgj(u1, u

2, u

3) = 0 (j = 1, 2, 3). What can be said about the fixed points of

[V, W]?

Problem 10. Consider the nonlinear differential equations

du

dt =u2 u, du

dt = sin(u) .

with the corresponding vector fields

V = (u2 u) ddu

, W = sin(u) ddu

.

(i) Show that both differential equations admit the fixed pointu = 0.(ii) Consider the vector field given by the commutator of the two vectorfieldsV andW, i.e. [V, W]. Show that the corresponding differential equa-tion of this vector field also admits the fixed point u = 0.


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Problem 11. Letz C. Consider the vector field

Ln:= zn+1 d

dz, n Z

Calculate the commutator [Lm, Ln].


ujk, ujm

ujk, uk

ujk, ujmuk

ujk

where j = 1, 2, . . . , p; k = 1, 2, . . . , n; m= 1, 2, . . . , n; = 1, 2, . . . , p. Findthe commutators. Do the vector fields form a basis of a Lie algebra. Discuss.


V1 =

r, V2=

1

r

, V3 =

1

r sin

.

Find the commutators

[V1, V2], [V2, V3], [V3, V1] .

Problem 14. Show that the differential operators (vector fields)

y+ x

z, y

y+ z

z, (xy z)

x+ y2

y+ yz

z

generate a finite-dimensional Lie algebra.

Problem 15. Consider smooth vector fields inR3

V = V1(x)

x1+ V2(x)

x2+ V3(x)

x3

W= W1(x)

x1+ W2(x)

x2+ W3(x)

x3.

Now

curl

V1V2V3

= V3x2 V2x3V1

x3 V3x1

V2x1

V1x2

, curlW1W2

W3

= W3x2 W2x3W1

x3 W3x1

W2x1

W1x2

.We consider now the smooth vector fields

Vc = ( V3x2 V2x3 ) x1 + ( V

1x3

V3x1 ) x2 + ( V2

x1 V1x2 ) x3


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Wc = (W3x2

W2x3

)

x1+ (

W1x3

W3x1

)

x2+ (

W2x1

W1x2

)

x3.

Note that if= V1(x)dx1+ V2(x)dx2+ V3(x)dx3

then

d=

V2x1

V1x2

dx1dx2+

V1x3

V3x1

dx3dx1+

V3x2

V2x3

dx2dx3 .

(i) Calculate the commutator [Vc, Wc]. Assume that [V, W] = 0. Can weconclude [Vc, Wc] = 0 ?(ii) Assume that [V, W] = R. Can we conclude that [Vc, Wc] = Rc?

Problem 16. Consider the first order ordinary differential equation

du

dt =u + 1

with the corresponding vector field

V = (u + 1) d

du.

Calculate the mapuexp(tV)u .

Solve the inital value problem of the differential equation and compare.


V1= (u2+ u1u3)

u1+ (u1+ u2u3)

u2+ (1 + u23)

u3

V2= (1 + u21)

u1

+ (u1u2+ u3) u2

+ (u2+ u1u3) u3

V3= (u1u2 u3) u1

+ (1 + u22)

u2+ (u1+ u2u3)

u3.

Find the commutators [V1, V2], [V2, V3], [V3, V1] and thus show that we havea basis if the Lie algebra so(3,R).

Problem 18. Let { ,} denote thePoisson bracket. Consider the functions

S1 =1

4(x21+p

21 x22 p22), S2=

1

2(p1p2+ x1x2, S3 =

1

2(x1p2 x2p1) .

Calculate{

S1

, S2}

,{

S2

, S3}

,{

S3

, S1}

so thus estabilish that we have a basisof a Lie algebra. Classify the Lie algebra.


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Problem 19. Consider the vector fields inR2

V1 =

x, V2 = x

x+ y

y, V3= (x

2 y2) x

+ 2xy

y.

Find the fixed points of the corresponding autonomous systems of first orderdifferential equations. Study their stability.

Problem 20. Consider in R3 the vector fields

V12 = x2

x1 x1

x2, V23 = x3

x2 x2

x3, V31 = x1

x3 x3

x1

with the commutators

[V12, V23] = V31, [V23, V31] = V12, [V31, V12] = V23 .

Thus we have a basis of the simple Lie algebra so(3,R).(i) Find the curl of these vector fields.

(ii) Let= dx1 dx2 dx3

be the volume form in R3. Find the differential two-forms

V12, V23, V31 .(iii) Let * be the Hodge star operator. Find the one forms

(V12), (V23), (V31) .

Problem 21. TheKustaanheimo-Stiefel transformationis defined by themap from R4 (coordinatesu1, u2, u3, u4) to R3 (coordinatesx1, x2, x3)

x1(u1, u2, u3, u4) = 2 (u1u3 u2u4)x2(u1, u2, u3, u4) = 2 (u1u4+ u2u3)

x3(u1, u2, u3, u4) = u21+ u

22 u23 u24

together with the constraint

u2du1 u1du2 u4du3+ u3du4 = 0 .(i) Show that

r2 =x21+ x22+ x

23 = u

21+ u

22+ u

23+ u

24 .

(ii) Show that

3 = 14r

4 14r2

V2


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Problem 24. Consider the smooth vector fields inRn

V =n

j,k=1ajkxj

xk, W=

n

j,k,=1cjkxjxk

x

whereajk , cjk R. Find the conditions onajk and cjk such that [V, W] =0.

Problem 25. Find two smooth vector fieldsV andW in Rn such that

[[W, V], V] = 0 but [W, V]= 0 .Find two n n matrices A and B such that

[[B, A], A] = 0 but [B, A]= 0 .

Problem 26. Letf : R R be an analytic function. Calculate

expi dd, expi d

d2, expi d

d f() .

Problem 27. Do the vector fields

x,

t, t

x+ x

t

form a basis of a Lie algebra under the commutator?

Problem 28. Give a vector fieldV in R3 such that

V curlV=0.Give a vector field V in R3 such that

V curlV =0.

Problem 29. Let f1 : R2 R, f2 : R2 R be analytic function.

Consider the analytic vector fields

V =f1(x1, x2)

x1+

x2, W=

x1+ f2(x1, x2)

x2

in R2.(i) Find the conditions on f1 and f2 such that [V, W] = 0.(ii) Find the conditions on f1 andf2 such that [V, W] = V + W.


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Chapter 3

Metric Tensor Fields

Problem 1. Leta > b >0 and definef : R2 R3 byf(, ) = ((a + b cos )cos , (a + b cos )sin , b sin ) .

The functionfis a parametrized torus T2 in R3. Consider the metric tensorfield

g= dx1 dx1+ dx2 dx2+ dx3 dx3.(i) Calculate g |T2 .(ii) Calculate the Christoffel symbols mab from g |T2 .(iii) Calculate the curvature.(iv) Give the differential equations for the geodesics.

Problem 2. The two-dimensional de Sitter spaceV with the topologyR S may be visualized as a one-sheet hyperboloid Hr0 embedded in 3-dimensional Minkowski space M, i.e.

Hr0 ={(y0, y1, y2) M : (y2)2 + (y1)2 (y0)2 =r20, r0> 0 }where r0 is the parameter of the one-sheet hyperboloid Hr0 . The inducedmetric,g (, = 0, 1), on Hr0 is the de Sitter metric.(i) Show that we can parametrize (parameters and ) the hyperboloid asfollows

y0(, ) =r0cos(/r0)sin(/r0)

, y1(, ) = r0cos(/r0)

sin(/r0) , y2(, ) =

r0sin(/r0))

sin(/r0)

where 0< < r0 and 0


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(ii) Using this parametrization find the metric tensor field induced on Hr0 .

Problem 3. Consider the metric tensor field

g=dZ dZ dT dT+ dW dW.

Consider the parametrization

Z(z, t) = cosh(z) cos(t)

T(z, t) = cosh(z) sin(t)

W(z, t) = sinh(z).

(i) Find Z2 + T2 W2.(ii) Express g using this parametrization.

Problem 4. The anti-de Sitter spaceis defined as the surface

X2 + Y2 + Z2 U2 V2 =1

embedded in a five-dimensional flat space with the metric tensor field

g= dX dX+ dY dY + dZ dZ dU dU dV dV.

This is a solution of Einsteins equations with the cosmological constant =3. Its intrinsic curvature is constant and negative. Find the metrictensor field in terms of the intrinsic coordinates (,,,t) where

X(,,,t) = 2

1 2 sin cos

Y(,,,t) = 21 2 sin sin

Z(,,,t) = 2

1 2 cos

U(,,,t) =1 + 2

1 2 cos t

V(,,,t) =1 + 2

1 2 sin t

where 0


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Metric Tensor Fields 27

with metric tensor field

g=1

ydx 1

ydx +

1

ydy 1

ydy

which is conformal with the standard inner product. Find the curvatureforms.

Problem 6. Consider the manifoldMof the upper space x2 > 0 ofR2

endowed with the metric tensor field

g=dx1 dx1+ dx2 dx2

x22.

Show that the metric tensor field admits the symmetry (x1, x2)(x1, x2)and the transformation (z = x1+ ix2)

zz = az+ bcz+ d

, a, b, c, dR, ad bc= 1

preserve the metric tensor field. Find the Gaussian curvature ofg .

Problem 7. Consider the manifoldMof the upper space xn >0 ofRn

endowed with the metric tensor field

g=dx1 dx1+ + dxn dxn

x2n.

Find the Gaussian curvature.

Problem 8. The Klein bagel(figure 8 immersion) is a specific immersionof the Klein bootle manifold into three dimensions. The figure 8 immersionhas the parametrization

x(u, v) = (r+ cos(u/2) sin(v) sin(u/2) sin(2v)) cos(u)y(u, v) = (r+ cos(u/2) sin(v) sin(u/2) sin(2v)) sin(u)z(u, v)=sin(u/2)sin(v) + cos(u/2) sin(2v)

where r is a positive constant and 0 u < 2, 0 v < 2. Find theRiemann curvature of the Klein bagel.

Problem 9. Consider the compact differentiable manifold S3

S3 :={ (x1, x2, x3, x4) : x21+ x22+ x23+ x24 = 1 }and the metric tensor field

g = dx1 dx1+ dx2 dx2+ dx3 dx3+ dx4 dx4.


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(i) Express g using the following parametrization

x1(, , )= cos cos

x2(, , )= si n cos

x3(, , )= cos sin

x4(, , )= si n sin

where 0/2, 0, 2.(ii) NowS3 is the manifold of the compact Lie group S U(2). Thus we candefine the vector fields (angular momentum operators)

L1=1

2cos( + )(tan

cot

) sin( + )

L2=1

2sin( + )(tan

cot

) + cos( + )

L3= + .Find the commutation relation [Lj , Lk] for j, k= 1, 2, 3.(iii) Find the dual basis ofL1, L2, L3.


g= dx1 dx1+ dx2 dx2+ dx3 dx3.

The parabolic set of unit-less coordinates (u, v, ) is defined by a transfor-mation of Cartesian coordinates (0u , 0v and 02)

x1(u, v, ) = auv cos , x2(u,v,) = auv sin , x3(u,v,) =

1

2 a(u

2

v2

).

Expressg using this parabolic coordinates.


g= cdt0 cdt0 dx0 dx0 dy0 dy0 dz0 dz0and the transformation

t0= t

x0= r cos( + t)

y0= r sin( + t)

z0= z.


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Expressg in the new coordinates t, r , z , .

Problem 12. Consider the upper half-plane{(x1, x2) : x2 > 0}endowedwith the metric tensor field

g = 1

x22(dx1 dx1+ dx2 dx2)

defines a two-dimensional Riemann manifold.(i) Show that the Gaussian curvature is given by R =1.(ii) Find the surface element dSand the Laplace operater .(iii) Consider the conformal mapping from the upper half-plane{z = x1+ix2 : x2 > 0} to the unit disk{w= rei : r1}

w(z) = iz+ 1

z+ i.

Expressg in r and .

Problem 13. (i) Consider the metric tensor field

g(u1, u2) = du1 du1+ e2u1du2 du2, < u1, u2 < +.Show that Gaussian curvature K(u1, u2) has the value1.(ii) Consider the transformation

x1(u1, u2) = u2, x2(u1, u2) = eu2 .

Expressg using the coordinates x1, x2.(iii) Consider the transformation

x1(, ) = x10+ cos(), x2(, ) = sin()

wherex10 is a constant. Express g in and

Problem 14. Let N 2 and a > 0. An N-dimensional Riemann man-ifold of constant negative Gaussian curvature K =1/a2 is described bythe metric tensor field

g = dr dr+ a2 sinh r

a

dN1 dN1

where r[0, ) measures the distance to the origin and dN1 dN1denotes the metric tensor field of the unit sphere SN1.(i) Show that volume element dV is covariantly defined as

dVN= a sinh raN1 drdN1


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in oblate spheroidal coordinates.

Problem 17. Consider the manifoldR3. Leta,b,c >0 anda=b, a=c,b=c. The sphero-conical coordinatess2, s3 are defined to be the roots ofthe quadratic equation

x21a + s

+ x22b + s

+ x23c + s

= 0.

The first sphero-conical coordinate s1 is given as the sum of the squares

s1 = x21+ x

22+ x

23.

The formula that expresses the Cartesian coordinates x1, x2, x3 throughs1, s2, s3 are

x21=s1(a + s2)(a + s3)

(a

b)(a

c)

x22=s1(b + s2)(b + s3)

(b a)(b c)x23=

s1(c + s2)(c + s3)

(c a)(c b) .

Given the metric tensor field

g= dx1 dx1+ dx2 dx2+ dx3 dx3.Express this metric tensor field using sphero-conical coordinates.


g=dT dT+ dX dX+ dY dY + dZ dZand the invertible coordinates transformation (b > 0)

T(t,x,y,z) =1

b(ebz cosh(bt)1), X(t,x,y,z) = x, Y(t,x,y,z) = y, Z(t,x,y,z) = 1

bebz sinh(bt).

Express the metric tensor field in the new coordinates. Given the inversetransformation.


g = dT dT dX dXwhere 0< X


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(0< r


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where a > 0 has the dimension of a length. Show that this metric tensorfield satisfies the vacuum Einstein equationwith a positive cosmologicalconstant

R

1

2Rg

+ g

= 0

wherea = 1/

().


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Chapter 4

Differential Forms and

Applications

We denote bythe exterior product. It is also called the wedge product orGrassmann product. The exterior product is associative. We denote by dthe exterior derivative. The exterior derivative d is linear.

Problem 1. Let f, g be two smooth functions defined on R2. Find thedifferential two-form df dg.

Problem 2. Consider the complex numberz = rei. Calculate

dz dzz

.

Problem 3. (i) Consider the differential one form

= x1dx2 x2dx1on R2. Show that is invariant under the transformation

x1x2

=

cos sin sin cos

x1x2

.

Show that = dx1 dx2 is invariant under this transformation.

34


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Differential Forms and Applications 35

(ii) Let be the (n 1) differential form on Rn given by

=n

j=1(1)j1xjdx1

dxj dxn

where indicates omission. Show that is invariant under the orthogonalgroup ofRn. Show that = dx1 dx2 dxn is invariant under theorthogonal group.

Problem 4. Letf : R2 R2 be a smooth planar mapping with constantJacobian determinant J= 1, written as

Q= Q(p, q), P =P(p, q).

For coordinates in R2 the (area) differential two-form is given as

= dp dq.(i) Find f.(ii) Show that pdq f(pdq) = dFfor some smooth function F : R2 R.

Problem 5. Consider the differential one-form inR3

= x1dx2+ x2dx3+ x3dx1.

Find d. Find the solutions of the equation d= 0.

Problem 6. Consider the differential one-form inR3

= dx3 x2dx1 dx2.

Show that d= 0.Problem 7. Let j,k, { 1, 2, . . . , n}. Consider the differential one-forms

jk :=dzj dzk

zj zk .

Calculatejk k+ k j+ j jk .

Problem 8. Consider all 2 2 matrices with U U = I2, det U= 1 i.e.,USU(2). ThenUcan be written as

U= a bb a , a, bC


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Problem 10. Consider the differential one-form

= i

4

n

j=0(zjdzjzjdzj) .

Letzj =xj+ iyj . Find . Find d.

Problem 11. Consider the vector spaceR3 and the smooth vector field

V =V1(x)

x1+ V2(x)

x2+ V3(x)

x3.

Given the differential two forms

1 = x1dx2 dx3, 2 = x2dx3 dx1, 3 = x3dx1 dx2 .Find the conditions on V1, V2, V3 such that the following three conditionsare satisfied

LV1 Vd1+ d(V1) = 0LV2 Vd2+ d(V2) = 0LV3 Vd3+ d(V3) = 0 .

Then solve the initial value problem of the autonomous system of first orderdifferential equations corresponding to the vector field V .

Problem 12. Letz = x + iy (x, y R). Finddz dz and dz dz.

Problem 13. Consider the vector spaceR3. Find a differential one-form such that d= 0 but d= 0.

Problem 14. In vector analysis inR3 we have the identity

( A B) Bcurl A Acurl B .Express this identity using differential forms, the exterior derivative andthe exterior product.

Problem 15. Consider the differentialn + 1 form

= df+dtdfn

j=1

(1)j+1Vj(x, t)dx1dxjdxn+(divV)f dtwhere the circumflex indicates omission and = dx

1 dx

n. Here

f : Rn+1 Ris a smooth function ofx, t and Vis a smooth vector field.


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(i) Show that the sectioned form

= df(x, t) + dt df(x, t) n

j=1(1)j+1Vj(x, t)dx1 dxj dxn+(divV(x, t))f(x, t)dt

where we distinguish between the independent variables x1, . . . , xn, t andthe dependent variable f leads using the requirement that = 0 to thegeneralized Liouville equation.(ii) Show that the differential form is closed, i.e. d= 0.

Problem 16. LetM= Rn andp Rn. LetTp(Rn) be the tangent spaceat p. A differential one-form at p is a linear map from Tp(R

n) into R.This map satisfies the following properties

(Vp) R, for all Vp Rn

(aVp+ bWp) = a(Vp) + b(Wp) for all a, bR, Vp, WpTp(Rn)

A differential one-form is a smooth choice of a linear map defined abovefor each point p in the vector space Rn. Let f : R R be a real-valuedC(Rn) function. One defines the dfof the function f as the differentialone-form such that

df(V) = V(f)

for every smooth vector fieldV inRn. Thus at any point p, the differentialdfof a smooth functionfis an operator that assigns to a tangent vector Vpthe directional derivative of the function fin the direction of this vector,i.e.

df(V)(p) = Vp

(f) =

f(p)

V(p) .

If we apply the differential of the coordinate functions xj (j = 1, . . . , n) weobtain

dxj

xk

xjdxk = xj

xk=jk.

(i) Let f : R2 R bef(x1, x2) = x

21+ x

22

and

V =x1

x1+ x2

x2.

Finddf(V).(ii) Letf : R2

R be

f(x1, x2) = x21+ x

22


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and

V =x1

x2 x2

x1.

Finddf(V).

Problem 17. Consider the manifoldM = R4 and the differential two-form

= dq1 dp1+ dq2 dp2 .Let

= (a2 +p21)dq1 dp2p1p2(dq1 dp1 dq2 dp2) (b2 +p22)dq2 dp1where a and b are constants. Find d. Can d be written in the formd= , where is a differential one-form?

Problem 18. A necessary and sufficient condition for the Pfaffian system

of equationsj = 0, j = 1, . . . , r

to be completely integrable is

dj0 mod (1, . . . , r), j = 1, . . . , rLet

P1(x)dx1+ P2(x)dx2+ P3(x)dx3 = 0 (1)be a total differential equation inR3, whereP1,P2,P3are analytic functionson R3. Complete integrabilty of means that in every sufficiently smallneighbourhood there exists a smooth function f such that

f(x1

, x2

, x3

) = const

is a first integral of (1). A necessary and sufficient condition for (1) to becompletely integrable is

d = 0 .

Problem 19. Consider the differential one-form in space-time

= a1(x)dx1+ a2(x)dx2+ a3(x)dx3+ a4(x)dx4

withx = (x1, x2.x3, x4) (x4 = ct).(i) Find the conditions on the aj s such thatd = 0.(ii) Find the conditions on the aj s such that d= 0 and d= 0.(iii) Find the conditions on the a

js such that

d

= 0 andd

d= 0.

(iv) Find the conditions on the aj s such that d d= 0.


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(v) Consider the metric tensor field

g= dx1 dx1+ dx2 dx2+ dx3 dx3 dx4 dx4 .

Find the condition such that d() = 0, where denotes the Hodge staroperator.

Problem 20. Letz = x + iy, x, y R. Calculate

idz dz .

Problem 21. Consider the manifoldM= R2 and the metric tensor fieldg= dx1 dx1+ dx2 dx2. Let

= 1(x)dx1+ 2(x)dx2

be a differential one-form inMwith 1, 2C

(R2

). Show that can bewritten as

= d + +

where is aC(R2) function,is a two-form given by = b(x)dx1 dx2(b(x)C(R2)) and= 1(x)dx1 + 2(x)dx2 is a harmonic one-form, i.e.(d+ d)= 0. We define

:= (1) d .

Problem 22. Given a Lagrange functionL. Show that the Cartan formfor a Lagrange function is given by

= L(x, v, t)dt +n

j=1

Lvj

(dxj vjdt) . (1)Let

H=n

j=1

vjL

vj L, pj = L

vj. (2)

Find the Cartan form for the Hamilton function.

Problem 23. Letx1,x2,. . .,xnbe the independent variables. Letu1(x),u2(x), . . . um(x) be the dependent variables. There arem n derivativesuj(x)/xi. We introduce the coordinates

(xi, uj , uji)(x1, x2, . . . , xn, u1, u2, . . . , ym, u11, u12, . . . , umn) .


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Consider the n-differential form (called the Cartan form) can be writtenas

= L ni=1

mj=1

Luj,i uj,i +

ni=1

mj=1

L

uj,iduj xi

where

:= dx1 dx2 . . . dxn .Let

H:=

ni=1

mj=1

L

uj,iuj,i

L, pji := Luj,i .Show that we find the Cartan form for the Hamilton

:=

Hdx1

dx2 . . . d xn1

dxn

+n

i=1

mj=1

pjidxju dx1 . . . d xi1dxi dxi+1 . . . dxn

where the hat indicates that this term is omitted.

Problem 24. Consider the differential 2-form

= 4dz dz(1 + |z|2)2

and the linear fractional transformations

z=

aw+ b

cw+ d , ad bc= 1 .What is the conditions ona,b,c,dsuch thatis invariant under the trans-formation?

Problem 25. Consider the two-dimensional sphere

S21 + S22 + S

23 =S

2

whereS >0 is the radius of the sphere. Consider the symplectic structureon this sphere with the symplectic differential two form

:= 1

2S2

3

jk=1

jkSjdSk dS


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(123 = 1) and the Hamilton vector fields

VSj :=3

k=1jkSk

S.

The Poisson bracket is defined by

[Sj , Sk]PB:=VSjSk.(i) Calculate [Sj , Sk]PB.(ii) CalculateVSj.(iii) CalculateVSjVSk.(iv) Calculate the Lie derivative LVSj .

Problem 26. Consider the system of partial differential equations (con-tinuity and Euler equation of hydrodynamics in one space dimension)

ut

+ u ux

+ 2c cx

= Hx

, ct

+ u cx

+ 12

c ux

= 0

where u and c are the velocities of the fluid and of the disturbance withrespect to the fluid, respectively. Hthe depth is a given function ofx. Showthat the partial differential equations can be written in the forms d = 0andd = 0, where and are differntial one-forms. Owing tod = 0 andd = 0 one can find locally (Poincare lemma) zero-forms (functions) (alsocalled potentials) such that

= d, = d .

Problem 27. Leta >0. Toroidal coordinatesare given by

x1(,,) = a sinh cos

cosh cos , x2(,,) = a sinh sin

cosh cos , x3(,,) = a sin

cosh cos where

0<


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(iii) Letbe the Hodge star operator in R3 with metric tensor fieldg= dx1 dx1+ dx2 dx2+ dx3 dx3 .

Find(V12),(V23),(V31).(iv) Find

d((V12)), d((V23)), d((V31)) .

Problem 33. (i) LetV, Wbe smooth vector fields in Rn (n2) and , be differential one-forms. Calculate

L[V,W]( ) .(ii) Assume thatLV= 0 and LW= 0. Simplify the result from (i).(iii) Assume that LV = f and LW = g, where f, g are smoothfunctions. Simplify the result from (i).(iv) Let LV= andLW= . Simplify the result from (i).

Problem 34. A symplectic structure on a 2n-dimensional manifold Mis a closed non-degenerate differential two-form such that d = 0 andn does not vanish. Every symplectic form is locally diffeomorphic to thestandard differential form

0 = dx1 dx2+ dx3 dx4+ + dx2n1 dx2non R2n. Consider the vector field

V =x1

x1+ x2

x2+ + x2n

x2n

in R2n. Find V

0 andLV0.

Problem 35. Leta > b >0. Consider the transformation

x1(, ) = (a + b cos )cos , x2(, ) = (a + b cos )sin .

Finddx1 dx2 anddx1 dx1+ dx2 dx2.

Problem 36. Consider the differential one-form

= (2xy x2)dx + (x + y2)dy(i) Calculate d.(ii) Calculate


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with the closed pathC1C2starting from (0, 0) moving along via the curveC1 : y = x

2 to (1, 1) and back to (0, 0) via the curve C2 : y =

x. Let Dbe the (convex) domain enclosed by the two curves C1 andC2.(iii) Calculate the double integral

D

d

whereD is the domain given in (i), i.e. C1C2 is the boundary ofD. Thusverify the theorem of Gauss-Green.

Problem 37. Consider the differential one form in the plane

= x22dx1+ x21dx2

Calculate the integral

Cwhere C is the closed curve which the boundary of a triangle with ver-tices (0, 0), (1, 1), (1, 0) and counterclockwise orientation. Apply Greenstheorem

C

f(x1, x2)dx1+ g(x1, x2)dx2 =

D

g

x1 f

x2

dx1dx2 .

Problem 38. (i) The lemniscate of Gerono is described by the equation

x4 =x2 y2 .

Show that a parametrization is given by

x(t) = sin(t), y(t) = sin(t)cos(t)

witht[0, ].(ii) Consider the differential one-form

= xdy

in the plane R2. Let x(t) = sin(t), y(t) = sin(t) cos(t). Find(t).(iii) Calculate

0

x(t)dy(t) .

Disucss.


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(iv) Consider the metric tensor field

g= dx1 dx1+ dx2 dx2+ dx3 dx3+ dx4 dx4 .

Using the parametrization show that

gS3 =1

4(d d+ d d+ d d + cos()d d + cos()d d) .

(v) Consider the differential one forms e1, e2, e3 defined by e1e2e3

=x4 x3 x2 x1x3 x4 x1 x2

x2 x1 x4 x3

dx1dx2dx3dx4

.Show that

gS3 =de1 de1+ de2 de2+ de3 de3 .(vi) Show that

dej =3

k,=1

jkek e

i.e. de1 = 2e2 e3, de2 = 2e3 e1, de3 = 2e1 e2.

Problem 41. LetV , Wbe two smooth vector fields defined on R3. Wewrite

V = V1(x)

x1+ V2(x)

x2+ V3(x)

x3

W= W1(x)

x1

+ W2(x)

x2

+ W3(x)

x3Let

= dx1wegdedx2 dx3be the volume form in R3. ThenLV = ((V)), where LV(.) denotesthe Lie derivative and denotes the diveregence of the vector field. Findthe divergence of the vector field given by the commutator [ V, W]. Apply itto the vector fields asscociated with the autonomous systems of first orderdifferential equations

dx1dt

=x2 x1, dx2dt

=x1x1x1x1x1, dx3

dt =x1x2 bx3

and

dx1dt

=x1x1, dx2dt

=x1x1x1x1, dx3dt

=x1x1x1x1 .


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The first system is theLorenz modeland the second system is Chens model.

Problem 42. Let A be a differential one-form in space-time with themetric tensor field

g= dx1 dx1+ dx2 dx2+ dx3 dx3 dx4 dx4withx4 = ct. LetF =dA. FindFF, where is the Hodge star operator.

Problem 43. Let f : R2 R2 be a two-dimensional analytic map.(i) Find the condition on f such that dx1 dx2 is invariant, i.e. f shouldbe area preserving.(ii) Find the condition on f such that x1dx1+ x2dx2 is invariant.(iii) Find the condition on f such that x1dx1 x2dx2 is invariant.(iv) Find the condition on f such that x1dx2+ x2dx1 is invariant.(v) Find the condition on f such that x1dx2 x2dx1 is invariant.

Problem 44. Consider the smooth one-form in R3

= f1(x)dx1+f2(x)dx2+f3(x)dx3, = g1(x)dx1+g2(x)dx2+g3(x)dx3 .

Find the differential equation from the condition

d( ) = 0and provide solution of it.

Problem 45. Letc >0. Consider the elliptical coordinates

x1(, ) = c cosh() cos(), x2(, ) = c sinh() sin().

Find the differential two-form = d1

dx2 in this coordinate system.

Problem 46. Let,, be the Euler angles and consider the differentialone-forms

1= cos d+ sin sin d

2= sin d+ cos sin d3= d+ cos d.

Find1 sigma2+ 2 3+ 3 1, 1 2 3.

Problem 47. Let B be a vector field in R3. Calculate

( B) B.


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Formulate the problem with differential forms.


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Chapter 5

Lie Derivative and

Applications

Problem 1. LetVbe a smooth vector field defined on Rn

V =n

i=1

Vi(x)

xi.

LetTbe a (1, 1) smooth tensor field defined on Rn

T=n

i=1n

j=1 aij(x)

xi dxj.

LetLVTbe the Lie derivative ofTwith respect to the vector field V. Showthat ifLVT= 0 then

LVtr(a(x)) = 0

wherea(x) is the n n matrix (aij(x)) and tr denotes the trace.

Problem 2. LetV, Wbe vector fields. Let f, g be C functions andbe a differential form. Assume that

LV= f , LW= g.

Show thatL[V,W]= (LVf LWg). (1)

50


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Lie Derivative and Applications 51

Problem 3. Letf andVbe smooth function and smooth vector field inRn. Find

Vdf.

Problem 4. LetVj(j = 1, . . . , n) be smooth vector fields anda smoothdifferential one-form. Assume that

LVj= dj , j = 1, 2, . . . , n

wherej are smooth functions.(i) Find

L[Vj,Vk].

(ii) Assume that the vector fields Vj (j = 1, . . . , n) form basis of a Liealgebra, i.e.

[Vj , Vk] =

n

=1cjkV

wherecjk are the structure constants. Find the conditions on the functionsj .

Problem 5. Find the first integrals of the autonomous system of ordinaryfirst order differential equations

dx1dt

= x1x2+ x1x3

dx2dt

= x2x3 x1x2dx3dt

= x1x3 x2x3.

Problem 6. (i) Consider the smoth vector fields

X=X1(x1, x2)

x1+ X2(x1, x2)

x2

and the two differential form

= dx1 dx2.Find the equation

d(X) = 0wheredenotes the contraction (inner product). One also writes

d((X)) = 0.


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Calculate the Lie derivative LX.(ii) Consider the smoth vector fields

X=

4

j=1

Xj(x)

xj

and the differential two form

= dx1 dx2+ dx3 dx4.Find the equation

d(X) = 0where denotes the contraction (inner product). One also writes d((X)) =0. CalculateLX.

Problem 7. Let be a smooth differential one-form andVbe a smooth

vector field. Assume that LV= f

wherefis a smooth function. Define the function F as

F :=Vwheredenotes the contraction. Show that

dF =f Vd.

Problem 8. Let V, Wbe two smooth vector fields defined on R3. Wewrite

V = V1(x) x1

+ V2(x) x2

+ V3(x) x3

W= W1(x)

x1+ W2(x)

x2+ W3(x)

x3.

Let= dx1 dx2 dx3

be the volume form in R3. ThenLV = (div(V)), where LV(.) denotesthe Lie derivative and divVdenotes the diveregence of the vector field V .Find the divergence of the vector field given by the commutator [ V, W].Apply it to the vector fields asscociated with the autonomous systems offirst order differential equations

dx1dt

=(x2 x1), dx2dt

=x1 x2 x1x3, dx3dt

=x3+ x1x2


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whereLV(.) denotes the Lie derivative. Find

LV(dx dy) .

Problem 13. LetV, Wbe two smooth vector fields

V =n

j=1

Vj(x)

xj, W =

nj=1

Wj(x)

xj

defined on Rn. Assume that

[V, W] = f(x)W .

Let =dx1 dxn

be the volume form and := W. Find the Lie derivativeLV .

Discuss.

Problem 14. LetM= R2 and letx, y denote the Euclidean coordinateson R2. Consider the differential one-form

=1

2(xdy ydx) .

Consider the vector field defined on R2 \ {0}

V = 1x2 + y2 x x y y .Find

Vdand the Lie derivative LV.

Problem 15. Consider the two smooth vector fields in R2

V =V1(x)

x1+ V2(x)

x2, W=W1(x)

x1+ W2(x)

x2.

Assume that [W, V] = 0. Find the Lie derivatives

LV(V1W2 V2W1), LW(V1W2 V2W1) .


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Discuss.

Problem 16. Consider the smooth manifold M = R3 with coordintes(x,p,z) and the differential one form

= dz pdx .(i) Show that d= 0. Consider the vector fields

V =

p, W =

x+ p

z.

FindV, W .

(ii) Consider the smooth manifold M= R5 with coordinates (x1, x2, p1, p2, z)and the differential one-form

= dz 2

j=1

pjdzj.

Show that d = 0. Consider the vector fields

V1 =

p1, V2 =

p2, W1 =

x1+p1

z, W2 =

x2+p2

z.

FindV1, V2, W1, W2 .

Problem 17. LetVbe a smooth vector field in R3. Find the conditiononV such that

LV(x1dx2+ x2dx3+ x3dx1) = 0 .

Problem 18. LetM= R2. Consider

V =x1

x2 x2

x1, = dx1 dx2 .

Calculate the Lie derivative LV.

Problem 19. Let du1/dt = V1(u) , . . . , dun/dt = Vn(u) be an au-tonomous system of ordinary differential equations, whereVj(u)C(Rn)for all j = 1, . . . , n. A function C(Rn) is called conformal invariantwith respect to the vector field

V =V1(u) u1+ + Vn(u) un


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if

LV=

where

C(Rn). Letn = 2 and consider the vector fields

V =u1

u2 u2

u1, W =u1

u1+ u2

u2.

Show that(u) = u21 + u22 is conformal invariant underV andW. Find the

commutator [V, W].

Problem 20. Consider the mainfoldR2 and the smooth vector field

V =V1(x1, x2)

x1+ V2(x1, x2)

x2.

FindV1, V2 such that

LV(dx1 dx1+ dx2 dx2) = 0

LV

x1 dx1+

x2 dx2

= 0

LV

x1

x1+

x2

x2

= 0 .

Problem 21. Let Mbe differentiable manifold and : M R be asmooth function. Let be a smooth diferential one form defined on M.Show that ifVis a vector field defined on M such that d = Vd, then

LV= d(V + ) .

Problem 22. Consider the manifoldM= Rn and the volume form

= dx1 dxn .

Consider the analytic vector field

V =n

j=1

Vj(x)

xj.

(i) Find = V

.(ii) Find LV.


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Problem 23. Consider the autonomous system of first order ordinarydifferential equations

duj

dt =Vj(u), j = 1, 2, . . . , n

where theVj s are polynomials. The corresponding vector field is

V =n

j=1

Vj

xj.

Letfbe an analytic function. The Lie derivative off is

LVf=n

j=1

Vjf

xj.

A Darboux polynomialis a polynomial g such that there is another polyno-mial p satisfying

LVg= pg .

The couple is called a Darboux element. If m is the greatest of degVj(j = 1, . . . , n), then degpm 1. All the irreducible factors of a Darbouxpolynomial are Darboux. The search for Darboux polynomials can be re-stricted to irreducibleg. If the autonomous system of first order differentialequations is homogeneous of degreem, i.e. allVj are homgeneous of degreem, thenp is homogeneous of degreem 1 and all homgeneous componentsofg are Darboux. The search can be restricted to homgeneous g .(i) Show that the product of two Darboux polynomials is a Darboux poly-nomial.(ii) Consider the Lotka-Volterra model for three species

du1dt

= u1(c3u2+ u3)

du2dt

= u2(c1u3+ u1)

du3dt

= u3(c2u1+ u2)

wherec1, c2, c3 are real parameters. Find the determining equation for theDarboux element.

Problem 24. Consider a smooth vector field inR3

V =V1(x) x1+ V2(x) x2

+ V3(x) x3


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and the differential two-form

= dx1 dx2+ dx2 dx3+ dx3 dx1 .

Find V and d(V). Thus find LV. Find solutions of the partialdifferential equations given by LV= 0.

Problem 25. Consider the unit ball

x2 + y2 + z2 = 1 .

and the vector field

V = (a0+ a1x + a2y+ a3z+ x(e1x + e2y+ e3z))

x

+(b0+ b1x + b2y+ b3z+ y(e1x + e2y+ e3z))

y

+(c0+ c1x + c2y+ c3z+ z(e1x + e2y+ e3z))

z .

Find the coefficients from the conditions

LV(x2 + y2 + z2) = 0, x2 + y2 + z2 = 1 .

Problem 26. Some quantities in physics owing to the transformationlaws have to be considered as currents instead of differential forms. LetMbe an orientablen-dimensional differentiable manifold of class C. Wedenote by k(M) the set of all differential forms of degree k with compactsupport. Letk(M) and let be an exterior differential form of degreen k with locally integrable coefficients. Then, as an example of a current,we have

T()() :=M

.

Define the Lie derivative for this current.

Problem 27. LetH: R2n Rbe a smooth Hamilton function with thecorresponding vector field

VH=n

j=1

H

pj

qj H

qj

pj

.

Let

W =

n

j=1

fj(p, q) qj + gj(p, q) pj


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be another smooth vector field. Assume that

[VH, W] = W (1)

where is a smooth function ofp and q. Let

= dq1 dqn dp1 dpnbe the standard volume differential form. Let

= W .

Show that (1) can be written as

LVH= .

Problem 28. Consider the vector field V associated with the Lorenzmodel

du1dt

= (u2 u1)du2dt

= u1u3+ ru1 u2du3dt

= u1u2 bu3 .

Let= u1du2+ u2du3+ u3du1 .

Calculate the Lie derivative

LV

Discuss.


g=cdt cdt + dr dr+ r2d d+ r2 sin2 d d

and the vector field

V = 1

1 2r2 sin2 /c2

t+

wherecis the speed of light and a fixed frequency. Find the Lie derivativeLVg.


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Problem 30. Consider the 2n + 1 dimensional anti-de Sitter spaceAdS2n+1. This is a hypersurface in the vector space R

2n+2 defined bythe equation R(x) =1, where

R(x) =(x0)2 (x1)2 + (x2)2 + + (x2n+1)2 .One introduces the even coordinates p and odd coordinates q. Then wecan write

R(p, q) =p21 q21+ p22+ q22 + +p2n+1+ q2n+1 .We consider R2n+2 as a symplectic manifold with the canonical symplecticdifferential form

=n+1k=1

dpk dqk.

Let

=

1

2

n+1

k=1

(pkdqk qkdpk) .Consider the vector field V in R2n+2 given by

V =1

2

n+1k=1

pk

pk+ qk

qk

.

Find the Lie derivative LVR and V.

Problem 31. Consider the Lotka-Volterra equation

du1dt

= (a bu2)u1, du2dt

= (cu1 d)u2

where a, b, c, d are constants and u1 > 0 and u2 > 0. The correspondingvector field V is

V = (a bu2)u1 u1

+ (cu1 d)u2 u2

.

Let= f(u1, u2)du1 du2

wherefis a smooth nonzero function. Find a smooth functionH(Hamiltonfunction) such that

V =dH .Note that from this condition sinceddH= 0 we obtain

d(V) = 0 .


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Problem 32. Let I, f be analytic functions of u1, u2. Consider theautonomous system of differential equations

du1/dtdu2/dt = 0 f(u)f(u) 0 I/u1I/u2 .Show that I is a first integral of this autonomous system of differentialequations.

Problem 33. Consider the smooth vector field

V =V1(u)

u1+ V2(u)

u2

in R2. Let f1(u1),f2(u2) be smooth functions.(i) Calculate the Lie derivative

LV f1(u1)du1 u1 + f2(u2)du2 u2 .Find the condition arising from setting the Lie derivative equal to 0.(ii) Calculate the Lie derivative

LV (f1(u1)du1 du1+ f2(u2)du2 du2).

Find the conditions arising from setting the Lie derivative equal to 0. Com-pare the conditions to the conditions from (i).

Problem 34. Let V, W be smooth vector fields defined in Rn. Letf, g : Rn R be smooth functions. Consider now the pairs (V, f), (W, g).One defines a commutator of such pairs as

[(V, f), (W, g)] := ([V, W], LVg LWf).

Let

V =u2

u1 u1

u2, W=u1

u1+ u2

u1

andf(u1, u2) = g(u1, u2) = u21+ u22. Calculate the commutator.

Problem 35. Consider the two differential form in R3

= x3dx1 dx2+ x1dx2 dx3+ x2dx3 dx1 .

Find d. Find the Lie dervivative LV

. Find the condition on the vectorfieldV such that LV= 0.


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Problem 36. LetVbe the vector field for the Lorenz model

V =(u1+ u2) u1

+ (u1u3+ ru1 u2) u2

+ (u1u2 bu3) u3

.

Find the Lie derivative LV(du1 du2), LV(du2 du3), LV(du3 du1).Discuss.

Problem 37. (i) Consider the tensor fields inR2

T1 =

2j,k=1

tjk(x)dxjdxk, T2 =2

j,k=1

tjk(x)dxj xk

, T3 =

2j,k=1

tjk(x)

xj

xk.

Find the condition on the vector field

V =

2=1

V(x)

x

such thatLVT1 = 0, LVT2 = 0, LVT3 = 0 .

(ii) Simplify for the case tjk(x) = 1 for allj, k= 1, 2.

Problem 38. Letn2. Consider the smooth vector field in Rn

V =

nj=1

Vj(x)

xj.

Find the Lie derivative of the tensor fields

xj

xk, dxj

xk, dxj dxk

with j, k = 1, . . . , n. Set the Lie derivative to 0 and study the partialdifferential equations ofVj.

Problem 39. V, Wbe smooth vector fields in R3. Let

LV(dx1dx2dx3) = (div(V))dx1dx2dx3, LW(dx1dx2dx3) = (div(W))dx1dx2dx3 .Calculate

L[V,W]dx1 dx2 dx3 .

Problem 40. LetVbe a smooth vector field in R2. Assume that

LV(dx1 dx1+ dx2 dx2) = 0, LV x1 x1 + x2 x2 = 0 .


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Can we conclude that

LV

dx1

x1+ dx2

x2

= 0 ?

Problem 41. The Heisenberg groupH is a non-commutative Lie groupwhich is diffeomorphic to R3 and the group operation is defined by

(x,y,z) (x, y, z) := (x + x, y+ y, z+ z xy+ xy).(i) Find the idenity element. Find the inverse element.(ii) Consider the metric tensor

g=dx dx + dy dy+ x2dy dy+ xdy dz+ xdz dy+ dz dzand the vector fields

V1 = z , V2 = y xz , V3 = x.

Show that the vector fields form a basis of a Lie algebra. Classify the Liealgebra. Calculate the Lie derivatives

LV1g, LV2g, LV3g .

Discuss.

Problem 42. Consider the 2n + 1 smooth vector fields

Xj =

xj yj

2

t, Yj =

yj+

xj2

t, T =

t

(j = 1, . . . , n) and the differential one form

= dt +1

2

nj=1

(yjdxj xjdyj) .

(i) Find the commutators

[Xj , Yj ], [Xj , T], [Yj , T] .

(ii) Findexp(Xj)xj , exp(Yj)yj , exp(T)t .

(iii) Find the Lie derivatives

LXj, LYj, LT .


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Problem 43. Let V, W be vector fields and be a differential form.Find the Lie derivative

LV(W ) .


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Chapter 6

Killing Vector Fields

Problem 1. Consider the two-dimensional Euclidean space with themetric tensor field

g= dx1 dx1+ dx2 dx2 .Find the Killing vector fields, i.e. the analytic vector fields V such that

LVg= 0

where LV denotes the Lie derivative. Show that the set of Killing vectorfields form a Lie algebra under the commutator.


g= 1

y2(dx dx + dy dy), < x


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where

gj,k=

zj

zkln(1 |z|2) .

Find the Killing vector fields ofg .


g= d d+ sin2 d d .Show thatg admits the Killing vector fields

V1= sin

+ cos cot

V2= cos

sin cot

V3 =

.

Is the Lie algebra given by the vector fields semisimple?

Problem 5. Ade Sitter universemay be represented by the hypersurface

x21+ x22+ x

23+ x

24 x20 = R2

whereR is a real constant. This hypersurface is embedded in a five dimen-sional flat space whose metric tensor field is

g = dx0 dx0 dx1 dx1 dx2 dx2 dx3 dx3 dx4 dx4 .Find the Killing vector fields V ofg , i.e. the solutions ofLVg= 0.

Problem 6. For the Poincare upper half plane

H={z= x1+ ix2 : y >0 }the metric tensor field is given by

g= 1

y(dx1 dx1+ dx2 dx2) .

Find the Killing vector fields for g , i.e.

V =V1(x1, x2)

x1+ V2(x1, x2)

x2

whereLVg= 0.


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Killing Vector Fields 67


g= dt dt eP1(t)dx dx eP2(t)dy dy eP3(t)dz dz

where Pj (j = 1, 2, 3) are smooth functions of t. Find the Killing vectorfields.


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Lie-Algebra Valued Differential Forms 69

for all xG, f(x) denoting the left translation gxg on G. LetX1, . . . ,Xn be a basis ofL and 1, . . . n the one-forms on G determined by

i( Xj) = ij (2)whereXi are the corresponding left invariant vector fields on G andij isthe Kronecker delta. Show that

di=12

nj,k=1

cijkj k, i= 1, 2, . . . , n (3)

where the structural constantscijk are given by

[Xj, Xk] =n

i=1

cijkXi. (4)

System (3) is known as the Maurer-Cartan equations.

Problem 4. LetG be a Lie group whose Lie algebra is L. L is identifiedwith the left invariant vector fields on G. Now suppose thatX1, . . ., Xn isa basis ofL and that1, . . . n is a dual basis of left invariant one-forms.There is a natural Lie algebra valued one-formon Gwhich can be writtenas := n

i=1

i Xi (1)

where(Xi, j) = ij . (2)

Show that

d+12 [, ] = 0 (3)where

[, ] := ni=1

nj=1

(i j) [Xi, Xj ]. (4)

Obviously, (3) are the Maurer-Cartan equations.

Problem 5. Consider the Lie algebra

G:=

e 0 1

:R , R

. (1)

Let

X := e 0 1 (2)


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and := X1dX. (3)

Show that

d + = 0. (4)

Problem 6. Let

X=

x11 x12x21 x22

(1)

wherex11x22 x12x21 = 1

so that X is a general element of the Lie group SL(2,R). Then X1dX,considered as a matrix of one-forms, takes its value in the Lie algebrasl(2,R), the Lie algebra ofS L(2,R). If

X1

dX= 1 23 1 (2)then{ j }are the left-invariant forms ofS L(2,R).(i) Show that there is a (local) SL(2,R)-valued functionA on R2 such that

A1dA=

1 2

3 1

= . (3)

Write for this sl(2,R)-valued one-form on R2.(ii) Show that then dG = G and that each row (r, s) of the matrix Gsatisfies

dr= r1+ s3, ds= r2 s1. (4)Note thatMaurer-Cartan equationsfor the forms

{1

, 2

, 3}

may be written

d + = 0. (5)(iii) Show that any element ofSL(2,R) can be expressed uniquely as theproduct of an upper triangular matrix and a rotation matrix (the Iwasawadecomposition). Define an upper-triangular-matrix-valued functionT anda rotation-matrix-valued functionR on R2 byA = T R1. Thus show that

T1dT =R1dR+ R1R.

Problem 7. Let

SL(2,R) := X= a bc d ad bc= 1 (1)


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be the group of all (22)-real unimodular matrices. Its right-invariantMaurer-Cartan form is

= dX X1 = 11 1221 22 (2)where

11+ 22 = 0. (3)

Show that satisfies the structure equationofSL(2,R), (also called theMaurer-Cartan equation)

d= or, written explicitly,

d11 = 12 21, d12 = 211 12, d21 = 221 11.

Problem 8. Let

SL(2,R) :=

X=

a bc d

ad bc= 1 (1)be the group of all (22)-real unimodular matrices. Its right-invariantMaurer-Cartan form is

= dX X1 =

11 1221 22

(2)

where11+ 22 = 0. (3)

Then satisfies (see previous problem) the structure equationofS L(2,R),

(also called the Maurer-Cartan equation)

d= or, written explicitly,

d11 = 12 21, d12 = 211 12, d21 = 221 11. (4)(ii) LetUbe a neighbourhood in the (x, t)-plane and consider the smoothmapping

f :USL(2,R). (5)The pull-backs of the Maurer-Cartan forms can be written

11

= (x, t)dx+A(x, t)dt, 12

= q(x, t)dx+B(x, t)dt, 21

= r(x, t)dx+C(x, t)dt(6)


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where the coefficients are functions ofx, t. Show that

t

+ A

x qC+ rB = 0 (7a)

qt

+ B

x 2B+ 2qA = 0 (7b)

rt

+ C

x 2rA + 2C= 0. (7c)

(ii) Consider the special case that r = +1 and is a real parameter inde-pendent ofx, t. Writing

q= u(x, t), (8)

show that

A(x, t) = C(x, t)+1

2

C

x, B(x, t) = u(x, t)C(x, t)(x, t) C

x 1

2

2C

x2.

(9)

Show that substitution into the second equation of (7) gives

u

t =

u

xC+ 2u

C

x = 22

C

x 1

2

3C

x3. (10)

(iii) Let

C= 2 12

u (11)

Show that (10) becomes

u

t =

1

4

3u

x3 3

2u

u

x, (12)

which is the well-known Korteweg-de Vries equation.

Problem 9. We consider the case whereM = R2 and L = sl(2,R). Inlocal coordinates (x, t) a Lie algebra-valued one-differential-form is givenby

= 3i=1

i Ti (1)

wherei := ai(x, t)dx + Ai(x, t)dt (2)

and{T1, T2, T3} is a basis of the semi-simple Lie algebra s(2,R). A con-venient choice is

T1 = 1 00 1 , T2 = 0 10 0 , T3 = 0 01 0 . (3)


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and for c = 1

df1dt

=4c1f3, df2dt

= 4c1f3, d2f3

dt2 =16c21f3 (9e)

wherec1 =c5.

Forc =1df1dt

= 4c1f3, df2

dt =4c1f3, d

2f3dt2

= 16c21f3 (9f)

wherec1 = c5.

(iv) Show that the solutions to these differential equations lead to the non-linear wave equations

2u

t22u

x2 =C1cosh u + C2sinh u (10)

and2u

t2

2u

x2 =C1sin u + C2cos u (11)

(C1, C2 R) can be written as the covariant exterior derivative of a Liealgebra-valued one-form, where the underlying Lie algebra is sl(2,R).

Problem 10. Let (M, g) be a Riemann manifold with dim(M) = m. Lets be an orthonormal local frame on Uwith dual coframe and let bethe Levi-Civita covariant derivative. Then we have

(1) g|U=mi=1

i i

(2) s= s., ij =ji , so 1(U,so(m))

(3) d+ = 0, di +m

k=1

ik k = 0

(4) Rs= s., = d + 2(U,so(m)), ij =dij+m

k=1

ik kj

(5) = 0,m

k=1ik k = 0, first Bianchi identity

(6) d + d + [, ]= 0, second Bianchi identity


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If (M, g) is a pseudo Riemann manifold,

ij =g(si, sj) = diag(1, . . . , 1, 1, . . . , 1)

the standard inner product matrix of the same signature (p, q) (p + q= m),then we have instead

(1) g=mi=1

iii i

(2) jjji =iiij thus = (ji )1(U,so(p, q))

(3) jjji =iiij thus = (ji )2(U,so(p, q)).

Consider the manifold S2 R3. Calculate the quantities given above.Consider the parametrization (leaving out one longitude)

f : (0, 2) (, ) R3, f(, ) = cos() cos()sin() cos()sin()

.

Problem 11. Show that the Korteweg-de Vries and nonlinear Schrodingerequations are reductions of the self-dual Yang-Mills equations. We work onR4 with coordinates xa = (x,y,u,t) and metric tensor field

g= dx dx dy dy+ du dt dt du

of signature (2,2) and a totally skew orientation tensor abcd = [abcd]. Weconsider a Yang-Mills connection Da := a Aa where the Aa where theAa are, for the moment, elements of the Lie algebra ofSL(2,C). The Aaare defined up to gauge transformations

AahAah1 (ah)h1

whereh(xa)SL(2,C). The connection is said to be self-dual when (sum-mation convention)

1

2cdab[Dc, Dd] = [Da, Db]. (3)

Problem 12. With the notation given above the self-dual Yang-Millsequationsare given by

De= D (1)Find the components of the self-dual Yang Mills equation.


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Problem 13. Consider the non-compact Lie group SU(1, 1) and thecompact Lie group U(1). Let z C and|z|< 1. Consider the coset spaceSU(1, 1)/U(1) with the element (R)

U(z, ) = 11 |z|2

1 zz 1

ei 00 ei

.Consequently the coset space SU(1, 1)/U(1) can be viewed as an open unitdisc in the complex plane. Consider the Cartan differential one-forms forms

= izdz zdz

1 |z|2 , + = idz

1 |z|2 , = idz

1 |z|2 .

Show that (Cartan-Mauer equations)

d= 2i +, d+ = i +, d =i .

Show that + = 1(1 |z|2)2 dz dz.


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Chapter 8

Lie Symmetries and

Differential Equations

Problem 1. Show that the second order ordinary linear differential equa-tion

d2u

dt2 = 0

admits the eight Lie symmetries

t,

u, t

t, t

u

u u

, ut

, utt

+ u2 u

, ut u

+ t2 t

.

Find the commutators. Classify the Lie algebra.

Problem 2. Show that the third order ordinary linear differential equa-tion

d3u

dt3 = 0

admits the seven Lie symmetries

t,

u, t

t, t

u

t2 u

, u u

, ut u

+ 12

t2 t

.

77


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Find the commutators.

Problem 3. Consider the nonlinear partial differential equation

3ux3

+ uut

+ c ux = 0

where c is a constant. Show that the partial differential equation admitsthe Lie symmetry vector fields

V1 =

t, V2 =

x,

V3 = 3t

t+ (x + 2ct)

x, V4 = t

t+ ct

x+ u

u.

Problem 4. Consider the stationary incompressible Prandtl boundarylayer equation

3u

3 =

u

2u

u

2u

.

Using the classical Lie method we obtain the similarity reduction

u(, ) = y(x), x= 1 + f()

where f is an arbitrary differentiable function of . Find the ordinarydifferential equation for y .

Problem 5. Show that theChazy equation

d3ydx3

= 2y d2y

dx2 3dy

dx2

admits the vector fields

x, x

x y

y, x2

x (2xy+ 6)

y

as symmetry vector fields. Show that the first two symmetry vector fieldscan be used to reduce the Chazy equation to a first order equation.

Problem 6. Show that the Laplace equation

2x2 + 2y2 + 2z2u= 0


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Problem 8. The Harry Dym equationis given by

u

t u3

3u

x3 = 0.

Show that it admits the Lie symmetry vector fields

V1 =

x, V2 =

t

V3 = x

x+ u

u, V4 =3t

t+ u

u, V5= x

2

x+ 2xu

u.

Is the Lie algebra spanned by these generators semi-simple?

Problem 9. Given the partial differential equation

2u

xt=f(u)

wheref : R Ris a smooth function. Find the condition that

V =a(x,t,u)

x+ b(x,t,u)

t+ c(x,t,u)

u

is a symmetry vector field of the partial differential equation. Start withthe corresponding vertical vector field

Vv = (a(x,t,u)ux b(x,t,u)ut+ c(x,t,u)) u

and calculate first the prolongation. Utilize the differential consequencieswhich follow from the partial differential equations

uxt f(u) = 0, uxxt dfdu

ux= 0, uxtt dfdu

ut = 0.

Problem 10. Consider then-dimensional smooth manifoldM= Rn withcoordinates (x1, . . . , xn) and an arbitrary smooth first order differentialequaion on M

F(x1, . . . , xn,u/x1, . . . , u/xn, u) = 0.

Find the symmetry vector fields (sometimes called the infinitesimal sym-metries) of this first order partial differential equation. Consider the cotan-genet bundle T(M) over the manifold Mwith coordinates

(x1, . . . , xn, p1, . . . , pn)


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Lie Symmetries and Differential Equations 81

and construct the product manifoldT(M)R. ThenT(M) has a canon-ical differential one-form

n

j=1pjdxj

which provides the contact differential one-form

= du n

j=1

pjdxj

onT(M)R. The solutions of the partial differential equation are surfacesinT(M) R

F(x1, . . . , xn, p1, . . . , pn, u) = 0

which annul the differential one-form . We construct the closed ideal Idefined by

F(x1, . . . , xn, p1, . . . , pn, u)

= du n

j=1

pjdxj

dF =n

j=1

F

xjdxj+

F

pjdpj

+

F

udu

d=n

j=1

dxj dpj .

The surfaces in T(M) R which annul Iwill be the solutions of the firstorder partial differential equation. Let

V(x1, . . . , xn, p1, . . . , pn, u) =

nj=1

Vxj

xj +

nj=1

Vpj

pj + Vu

u

be a smooth vector field. Let LV denote the Lie derivative. Then theconditions for Vto be a symmetry vector field are

LVF= gF

LV = + dF+

nj=1

(Ajdxj+ Bjdpj)

F.Here , , Aj, Bj are smooth functions ofx1, . . . , xn, p1, . . . , pn and u onT(Rn) R, where g, Aj , Bj must be nonsingular in a neighbourhood ofF= 0. Find V .


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Chapter 9

Integration

Problem 1. Let (t) = (x(t), y(t)) be a positive oriented simple closedcurve, i.e. x(b) = x(a),y (b) = y(a). Show that

A= ba

y(t)x(t)dt=

ba

x(t)y(t)dt=1

2

ba

(x(t)y(t) y(t)x(t))dt .

Problem 2. AnyS U(2) matrixA can be written as (x0, x1, x2, x3 R)

A=

x0 ix3 ix1 x2x2

ix1 x0+ ix3

, x20+ x21+ x

22+ x

23 = 1 (1)

i.e., det A= 1. UsingEuler angles, , the matrix can also be written as

A=

cos(/2)ei(+)/2 sin(/2)ei()/2sin(/2)ei()/2 cos(/2)ei(+)/2

. (2)

(i) Show that the invariant measure dg ofS U(2) can be written as

dg = 1

2(x20+ x

21+ x

22+ x

23 1)dx0dx1dx2dx3

where is the Dirac delta function.(ii) Show that dg is normalized, i.e.

dg = 1 .82


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Integration 83

(iii) Using (1) and (2) find x1(, , ), x2(, , ), x3(, , ). Find theJacobian determinant.(iv) Using the results from (iii) show that the invariant measure can bewritten as

1162

sin ddd .

Problem 3. LetMbe a smooth, compact, and oriented n-manifold. Letf : M Rn+1 \ { 0 } be a smooth map. TheKronecker characteristic isgiven by the following integral

K(f) := (volSn)1M

f(x)(n+1) det

f(x), f

x1(x), . . . ,

f

xn(x)

dx

where (x= (x1, x2, . . . , xn)) are local coordinates ofMand dx= dx1dx2 dxn.Express this integral in terms of differential forms.

Problem 4. Let Cbe the unit circle centered at the orign (0, 0). Calculate

1

2

C

P dQ QdPP2 + Q2

whereP(x, y) =y, Q(x, y) = x.

Problem 5. LetSn Rn+1 be given bySn :={ (x1, . . . , xn+1) : x21+ + x2n+1 = 1 } .

Show that the invariant normalized n-differential form on Sn is given by

= 12

n/2n2 dx1 dxn

|xn+1|where denotes the gamma function.

Problem 6. A volume differential form on a manifoldMof dimension nis an n-form such that (p)= 0 at each point pM. ConsiderM= R3(or an open set here) with coordiante system (x1, x,x3) with respect to theusual right-handed orthonormal frame. Then the volume differential formis defined as

= dx1 dx2 dx3and hence any differential three-form can be written as

= f(x1, x2, x3)dx1 dx2 dx3


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for some functionf. The integral of is (if it exists)

R3=

R3f(x1, x2, x3)dx1dx2dx3 .

(i) Express in terms of spherical coordinates (r,,) withr0, 0 0)

x1(, , ) = a sinh sin cos

x2(, , ) = a sinh sin sin

x3(, , ) = a cosh cos .

Problem 7. Consider the differential 1-form

=x2dx1 x1dx2

x21+ x22

defined onU= R2 \ {(0, 0)} .

(i) Calculate d.(ii) Calculate

using polar coordinates.


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Chapter 10

Lie Groups and Lie

Algebras

Problem 1. Let Rij denote the generators of an SO(n) rotation in thexi xj plane of then-dimensional Euclidean space. Give ann-dimensionalmatrix representation of these generators and use it to derive the Lie algebraso(n) of the compact Lie groupS O(n).

Problem 2. The Lie groupSL(2,C) consists all 2 2 matrices over Cwith determinant equal to 1. The group is not compact. The maximalcompact subgroup ofS L(2,C) isS U(2). Give a 2 2 matrixA which is anelement ofS L(2,C), but not an element ofS U(2).

Problem 3. Consider the Lie group G = O(2, 1) and its Lie algebrao(2, 1) ={K1, K2, L3}, where K1, K2 are Lorentz boosts and L3 and in-finitesimal rotation. The maximal subalgebras ofo(2, 1) are represented by{K1, K2 + L3}and{L3}, nonmaximal subalgebras by{K1}and{K2 + L3}.The two-dimensional subalgebra corresponds to the projective group of areal line. The one-dimensional subalgebras correspond to the groups O(2),O(1, 1) and the translationsT(1), respectively. Find theo(2, 1) infinitesimalgenerators.

85


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Problem 4. The group generator of the compact Lie group SU(2) canbe written as

J1 =1

2 z1

z2+ z2

z1 , J2 = i

2 z2

z1 z1

z2 , J3 = 12 z1

z1 z2

z2 .(i) Find

J+ = J1+ iJ2, J= J1 iJ2 .(ii) Letj = 0, 1, 2, . . .and m =j, j+ 1, . . . , 0, . . . , j. We define

ejm(z1, z2) = 1

(j+ m)!(j m)! zj+m1 z

jm2 .

FindJ+e

jm(z1, z2), Je

jm(z1, z2), J3e

jm(z1, z2)

(iii) Let

J2

=J2

1 + J2

2 + J2

31

2 (J+J+ JJ+) + J2

3.

FindJ2ejm(z1, z2) .

Problem 5. Show that the operators

L+ = zz, L= z

z

L3 =12

z

z+ z

z+ 1

, L0 =1

2

z

z z

z+ 1

.

form a basis for the Lie algebra su(1, 1) under the commutator.

Problem 6. Consider the semi-simple Lie algebras(3,R). The dimen-sion ofs(3,R) is 8. Show that the 8 differential operators

J13 =y2

y+ xy

x ny, J 12 =x2

x+ xy

y nx,

J23 =y

x, J21 =

x, J31 =

y, J32 =x

y,

Jd= y

y+ 2x

x n, Jd= 2y

y+ x

x n

wherex, y R and n is a real number. Find all the Lie subalgebras.


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Show thatU2t

V2x

+ [U2, V2] = 0. (3)

Problem 3. Consider the nonlinear Schrodinger equation in one spacedimension

i

t +

2

x2 + 2||2= 0 (1)

and the Heisenberg ferromagnet equation in one space dimension

S

t =S

2S

x2, S2 = 1 (2)

where S = (S1, S2, S3)T. Both equations are integrable by the inverse

scattering method. Both arise as the consistency condition of a system oflinear differential equations

t

=U(x,t,), x

=V(x,t,) (3)

where is a complex parameter. The consistency conditions have the form

U

t V

x + [U, V] = 0 (4)

(i) Show

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