Di®erential Geometry Michael E. Taylor 1 2 Contents The Basic Object 1. Surfaces, Riemannian metrics, and geodesics Vector Fields and Di®erential Forms 2. Flows and vector ¯elds 3. Lie brackets 4. Integration on Riemannian manifolds 5. Di®erential forms 6. Products and exterior derivatives of forms 7. The general Stokes formula 8. The classical Gauss, Green, and Stokes formulas 8B. Symbols, and a more general Green-Stokes formula 9. Topological applications of di®erential forms 10. Critical points and index of a vector ¯eld Geodesics, Covariant Derivatives, and Curvature 11. Geodesics on Riemannian manifolds 12. The covariant derivative and divergence of tensor ¯elds 13. Covariant derivatives and curvature on general vector bundles 14. Second covariant derivatives and covariant-exterior derivatives 15. The curvature tensor of a Riemannian manifold 16. Geometry of submanifolds and subbundles The Gauss-Bonnet Theorem and Characteristic Classes 17. The Gauss-Bonnet theorem for surfaces 18. The principal bundle picture 19. The Chern-Weil construction 20. The Chern-Gauss-Bonnet theorem Hodge Theory 21. The Hodge Laplacian on k-forms 22. The Hodge decomposition and harmonic forms 23. The Hodge decomposition on manifolds with boundary 24. The Mayer-Vietoris sequence for deRham cohomology 3 Spinors 25. Operators of Dirac type 26. Cli®ord algebras 27. Spinors 28. Weitzenbock formulas Minimal Surfaces 29. Minimal surfaces 30. Second variation of area 31. The Minimal surface equation Appendices A. Metric spaces, compactness, and all that B. Topological spaces C. The derivative D. Inverse function and implicit function theorem E. Manifolds F. Vector bundles G. Fundamental existence theorem for ODE H. Lie groups I. Frobenius' theorem J. Exercises on determinants and cross products K. Exercises on the Frenet-Serret formulas L. Exercises on exponential and trigonometric functions M. Exponentiation of matrices N. Isothermal coordinates O. Sard's theorem P. Variational Property of the Einstein tensor Q. A generalized Gauss map R. Moser's area preservation result S. The Poincar¶edisc and Ahlfors' inequality T. Rigid body motion in Rn and geodesics on SO(n) U. Adiabatic limit and parallel transport V. Grassmannians (symmetric spaces, and KÄahlermanifolds) W. The Hopf invariant X. Jacobi ¯elds and conjugate points Y. Isometric imbedding of Riemannian manifolds Z. DeRham cohomology of compact symmetric spaces 4 Introduction These are notes for a course in di®erential geometry, for students who had a course on manifold theory the previous semester, which in turn followed a course on the elementary di®erential geometry of curves and surfaces. Section 1 recalls some basic concepts of elementary geometry, and extends them from surfaces in R3 to hypersurfaces in Rn; and then to manifolds with Riemannian metrics, de¯ning arc length and deriving the ODE for a geodesic. The ODE in general has a somewhat messy form; a more elegant form will be produced in x11, following some material on vector ¯elds and related topics in xx2{10. These sections also contain a review of material from the previous course on manifold theory, such as di®erential forms and deRham cohomology. The material in xx11{16, on geodesics, covariant derivatives, and curvature, is the heart of the course. We take the intrinsic de¯nitions of these objects as fun- damental, though the very important relations between curvature and the second fundamental form are studied in x16, in the general context of one Riemannian manifold imbedded in another (not necessarily Euclidean space). In xx17{20 we cover the famous Gauss-Bonnet Theorem, and its higher dimen- sional extension, which involves a study of characteristic classes, certain deRham cohomology classes derived from curvature. An essential tool is the Chern-Weil theory of characteristic classes, developed in x19. This in turn is treated most con- veniently in terms of a \principal bundle," a structure which is in a sense more fundamental than that of a vector bundle. Section 18 develops the basic theory of principal bundles. Sections 21{24 study the Hodge theory, representing elements of deRham co- homology by harmonic forms. This leads to simple proofs of some fundamental results, such as Poincare duality and the Kunneth formula, for products of man- ifolds. Parts of this study make use of the theory of elliptic partial di®erential equations, not presented here, but contained in the author's book [T1]. Sections 25{28 study spinors and some applications. This is a situation in which it is particularly helpful to use concepts involving principal bundles, developed in x18. Sections 29{31 are devoted to minimal surfaces, which to some extent are higher dimensional analogues of geodesics. At the end are a number of appendices, on various background or auxiliary ma- terial useful for understanding the main body of the text. This includes de¯nitions of metric and topological spaces, manifolds, vector bundles, and Lie groups, and proofs of some basic results, such as the inverse function theorem and the local existence of solutions to ODE. There are also sets of exercises, on determinants, the cross product, and trigonometric functions, intended to give the reader a fresh perspective of these elementary topics, which appear frequently in the study of 5 di®erential geometry. In addition, the ¯nal handful of appendices deal with some special topics in di®erential geometry, which complement material in the main text but did not ¯nd space there. We close this introduction with some comments on where this material comes from and where it is going. Most of the main body of the text (xx2{31) was adapted from material in di®erential geometry scattered through the three-volume text [T1]. This was augmented by the introductory material in x1, developed in an advanced calculus course. The material in xx1{10 has since been rewritten and appears, in more polished form, in [T4]. The text [T3] treats linear algebra, and contains further material on exterior algebra and Cli®ord algebra, as well as more detailed presentations of other linear algebra topics, such as given here in Appendices J and M. The theory of Lie groups, sketched here in Appendix H and used in a number of other sections, receives a fairly thorough treatment in [T2]. 6 1. Surfaces, Riemannian metrics, and geodesics ¡ ¢ n Suppose S is a smooth hypersurface in R : If γ(t) = x1(t); : : : ; xn(t) ; a · t · b; is a smooth curve in S, its length is Z b (1.1) L = kγ0(t)k dt: a where Xn 0 2 0 2 (1.2) kγ (t)k = xj(t) : j=1 A curve γ is said to be a geodesic if, for jt1 ¡ t2j su±ciently small, tj 2 [a; b]; the curve γ(t); t1 · t · t2 has the shortest length of all smooth curves in ­ from γ(t1) to γ(t2): Our ¯rst goal is to derive an equation for geodesics. So let γ0(t) be a smooth curve in S (a · t · b), joining p and q: Suppose γs(t) is a smooth family of such curves. We look for a condition guaranteeing that γ0(t) has minimum length. Since the length of a curve is independent of its parametrization, we may as well suppose 0 (1.3) kγ0(t)k = c0; constant, for a · t · b: Let N denote a ¯eld of normal vectors to S: Note that, with @sγs(t) = (@=@s)γs(t), (1.4) V = @sγs(t) ? N: Also, any vector ¯eld V ? N over the image of γ0 can be obtained by some variation γs of γ0; provided V = 0 at p and q: Recall we are assuming γs(a) = p; γs(b) = q: If L(s) denotes the length of γs; we have Z b 0 (1.5) L(s) = kγs(t)k dt; a and hence Z 1 b ¡ ¢ L0(s) = kγ0 (t)k¡1@ γ0 (t); γ0 (t) dt 2 s s s s (1.6) a Z b 1 ¡ 0 0 ¢ = @sγs(t); γs(t) dt; at s = 0: c0 a 7 Using the identity d ¡ ¢ ¡ ¢ ¡ ¢ (1.7) @ γ (t); γ0 (t) = @ γ0 (t); γ0 (t) + @ γ (t); γ00(t) ; dt s s s s s s s s s together with the fundamental theorem of calculus, in view of the fact that (1.8) @sγs(t) = 0 at t = a and b; we have Z b 0 1 ¡ 00 ¢ (1.9) L (s) = ¡ V (t); γs (t) dt; at s = 0: c0 a Now, if γ0 were a geodesic, we would have (1.10) L0(0) = 0; 00 for all such variations. In other words, we must have γ0 (t) ? V for all vector ¯elds V tangent to S (and vanishing at p and q), and hence 00 (1.11) γ0 (t)kN: This vanishing of the tangential curvature of γ0 is the geodesic equation for a hypersurface in Rn: We proceed to derive from (1.11) an ODE in standard form. Suppose S is de¯ned locally by u(x) = C; ru 6= 0: Then (1.11) is equivalent to 00 (1.12) γ0 (t) = Kru(γ0(t)) for a scalar K which remains to be determined. But the condition that u(γ0(t)) = C implies 0 γ0(t) ¢ ru(γ0(t)) = 0; and di®erentiating this gives 00 0 2 0 (1.13) γ0 (t) ¢ ru(γ0(t)) = ¡γ0(t) ¢ D u(γ0(t)) ¢ γ0(t) where D2u is the matrix of second order partial derivatives of u: Comparing (1.12) and (1.13) gives K; and we obtain the ODE ¯ ¯¡2h i 00 ¯ ¡ ¢¯ 0 2 ¡ ¢ 0 ¡ ¢ (1.14) γ0 (t) = ¡¯ru γ0(t) ¯ γ0(t) ¢ D u γ0(t) ¢ γ0(t) ru γ0(t) for a geodesic γ0 lying in S: A smooth m-dimensional surface M ½ Rn is characterized by the following property.