A. C. 10. 11. CoiloBbeB, A. T. CBOPHI4K 110 JU4NEPEHLU4AJThHOfl ITEOMETPI4H H Touonor'1114 MocKoscKoro YHMBepcwreTa MocxBa A.5.Mishch.nko. Yu.P. and AT. Fominko Probisms in DifF.r.ntiat G.omstry and Topology Translated from the Russian by Oleg Efimov Mir Pubtishsrs Moscow First published 1985 Revised from the 1981 Russian edition TO THE READER Mir Publishers would be grateful for your comments on the content, translation and design of this book. We would also be pleased to receive any other suggestions you may wish to make. Our address is: Mir Publishers, 2 Pervy Pizhsky Pereulok, 1-110, GSP, Moscow, 129820, USSR. Ha au2JIuuCKoM © H3naTenbcTao MocKoBcKoro YHHBePCUTeTa, 1981 © English translation, Mir Publishers, 1985 Preface This book of problems is the result of a course in differential geometry and topology, given at the mechanics- and-mathematics department of Moscow State University. It contains problems practically for all sections of the seminar course. Although certain textbooks and books of problems indicated in the bibliography list were used in preparation of this volume, a considcrable number of the problems were prepared for this book expressly. The material is distributed over the sections as in text- book [3]. Some problems, however, touch upon topics outside the lectures. In these cases, the corresponding sec- tions are supplied with additional definitions and explanations. In conclusion, the authors express their sincere gratitude to all those who helped to publish this work. • Contents Preface 5 1. Application of Linear Algebra to Geometry 7 2. Systems of Coordinates 9 3. Riemannian Metric 14 4. Theory of Curves 16 5. Surfaces 34 6. Manifolds 53 7. Transformation Groups 60 8. Vector Fields 64 9. Tensor Analysis 70 10. Differential Forms, Integral Formulae, De Rham Cohomology 75 11. General Topology 81 12. t-Iomotopy Theory 87 13. Covering Maps, Fibre Spaces, Riemaun Surfaces 97 14. Degree of Mapping 105 15. Simplest Variational Problems 108 Answers and Hints 113 Bibliography 208 1 Application of Linear Algebra to Geometry 1.1. Prove that a vector set a1 ak in a Euclidean space is linearly independent if and only if det O(ai, 0. 1.2. Find the relation between a complex matrix A and the real matrix rA of the complex linear mapping. 1.3. Find the relations between det A and det rA, Tr A and Tr rA, det (A — XE) and det (rA XA). 1.4. Find the relation between the invariants of the matrices A, B and A B, A ® B. Consider the cases of det and Tr. 1.5. Prove the formula det e4 = 1.6. Prove that = + + C' [A, BIC" for a convenient choice of the matrices C' and C", where [A, B] = AB BA. 1.7. Prove that if A is a skewsymmetric matrix, then is an orthogonal matrix. 1.8. Prove that if A is a skewhermitian matrix, then is a unitary matrix. 1.9. Prove that if [A, A*] = 0, then the matrix A is similar to a diagonal one. 1.10. Prove that a unitary matrix is similar to a diagonal one with eigenvalues whose moduli equal unity. 1.11. Prove that a hermitian matrix is similar to a diagonal one with real eigenvalues. 1.12. Prove that a skewhermitian matrix is similar to a diagonal one with imaginary eigenvalues. 7 1.13. Let A ((aijO be a matrix of a quadratic form, and Dk = det Prove that A is positive definite if and only if for all k, I k the inequalities Dk > 0 are valid. 1.14. With the notation of the previous problem, prove that a matrix A is negative definite if and only if for all k, I k n, the inequality (_1)kDk > 0 holds. 1.15. Put ((A = > Prove the inequalities ((JI —i— B(( ( ( —F (I B ((A (( . ((B 1.16. Prove that if A2 = then the matrix A is similar to the matrix /'Ek 0 \ I,k4-I=n. \0 —E,/ 1.17. Prove that if A2 —E, then the order of the matrix A is (2n x 2n), and it is similar to a matrix of the form 0 ( 0 1.18. Prove that if A2 = A, then the matrix A is similar to a matrix (E 0 of the form \0 0 1.19. Prove that varying continuously a quadratic form from the class of non-singular quadratic forms does not alter the signature of the form. 1.20. Prove that varying continuously a quadratic form from the class of quadratic forms with constant rank does not alter its signature. 1.21. Prove that any motion of the Euclidean plane R2 can be resolved into a composition of a translation, reflection in a straight line, and rotation about a point. 1.22. Prove that any motion of the Euclidean space R3 can be resolved into a composition of a translation, reflection in a plane and rotation about a straight line. 1.23. Generalize Problems 1.21 and 1.22 for the case of the Euclidean space 8 2 Systems of Coordinates A set of numbers q', q2,..., qfl determining the position of a point in the space is called its curvilinear coordinates. The relation between the Cartesian coordinates X2 of this point and curvilinear coordinates is expressed by the equalities , qfl), X5 = Xs(q', q2, . . (1) or, in vector form, by qfl), r = r(q', q2 where r is a radius vector. Functions (1) are assumed to be continuous in their domain and to have continuous partial derivatives up to the third order inclusive. They must be uniquely solvable with respect to qfl; q2 this condition is equivalent to the requirement that the Jacobian (2) should not be equal to zero. The numeration of the coordinates is assumed to be chosen so that the Jacobian is positive. Transformation (I) determines n families of the coordinate hypersurfaces qr = The coordinate hypersurfaces of one and the same family do not intersect each other if condition (2) is fulfilled. Owing to condition (2), any n — I coordinate hyperplanes which belong to different families meet in a certain curve. They are called coor- dinate curves or coordinate lines. The vectors rk = are directed as the tangents to the coordinate lines. They determine the infinitesimal vector dr rkdq" , in a neighbourhood of the point M(q', q2, . .. q"). The square of its length, if expressed in terms of curvilinear coordinates, can be found from the equality ds2 = (dr, dr) r,dq', rkdci) = th15 dqk = k1 where (,) is the scalar product defined in 9 The quantities = = rk) define a metric in the adopted coor- dinate system. An orthogonal curvilinear coordinate system is one for which (0 s = k The quantities are called the Lamé coefficients. They are equal to the moduli of the vectors I /dxi'Y fax2\2 = + = + + ... The square of the linear element in orthogonal curvilinear coordinates is given by the expression ds2 = + + ... + 2.1. Calculate the Jacobian J = of transition from Cartesian coordinates (x1, . . . , to orthogonal curvilinear coordinates (q', q2 qfl) in the space 2.2. Calculate the gradient grad f of the function f: R3 R in an or- thogonal curvilinear coordinate system. 2.3. Calculate the divergence div a of a vector a E R3 in an orthogonal curvilinear coordinate system. 2.4. Find the expression for the Laplace operator Lxf of the function f: R3 R in an orthogonal curvilinear coordinate system. 2.5. Cylindrical coordinates in R3 q1=r, q3=z are related to Cartesian coordinates by the formulae x = r cosça, y = r Z = Z. (a) Find the coordinate surfaces of cylindrical coordinates. (b) Compute the Lamé coefficients. (c) Find expression for the Laplace operator in cylindrical coordinates. 2.6. Spherical coordinates in R3 q'=r, q2=O, q3 = are related to rectangular coordinates by the formulae z=rcosO. (a) Find the coordinate surfaces of spherical coordinates. (b) Compute the Lamé coefficients. (c) Find expression for the Laplace operator in spherical coordinates. 10