These notes were written by Camilla Horst on the basis of the lectures I gave during the week of June 22-26, 1981 at the DMV Seminar on Complex Differential Geometry in Dusseldorf. My aim was to make the contents of my survey lecture at the DMV annual meeting in 1980 (published in Jahresberichte, 1981) accessible to beginning research students by providing a little more details. I suggest therefore that the Jahresberichte paper be read as an introduction to these notes.

Whoever you are! How can I but offer you divine leaves . . . ? Walt Whitman The object of study in modern differential geometry is a manifold with a differ­ ential structure, and usually some additional structure as well. Thus, one is given a topological space M and a family of homeomorphisms, called coordinate sys­ tems, between open subsets of the space and open subsets of a real vector space V. It is supposed that where two domains overlap, the images are related by a diffeomorphism, called a coordinate transformation, between open subsets of V. M has associated with it a tangent bundle, which is a vector bundle with fiber V and group the general linear group GL(V). The additional structures that occur include Riemannian metrics, connections, complex structures, foliations, and many more. Frequently there is associated to the structure a reduction of the group of the tangent bundle to some subgroup G of GL(V). It is particularly pleasant if one can choose the coordinate systems so that the Jacobian matrices of the coordinate transformations belong to G. A reduction to G is called a G-structure, which is called integrable (or flat) if the condition on the Jacobians is satisfied. The strength of the integrability hypothesis is well-illustrated by the case of the orthogonal group On. An On-structure is given by the choice of a Riemannian metric, and therefore exists on every smooth manifold.

From the Preface: "This book was written for the active reader. The first part consists of problems, frequently preceded by definitions and motivation, and sometimes followed by corollaries and historical remarks… The second part, a very short one, consists of hints… The third part, the longest, consists of solutions: proofs, answers, or contructions, depending on the nature of the problem…

The question of existence of c10sed geodesics on a Riemannian manifold and the properties of the corresponding periodic orbits in the geodesic flow has been the object of intensive investigations since the beginning of global differential geo­ metry during the last century. The simplest case occurs for c10sed surfaces of negative curvature. Here, the fundamental group is very large and, as shown by Hadamard [Had] in 1898, every non-null homotopic c10sed curve can be deformed into a c10sed curve having minimallength in its free homotopy c1ass. This minimal curve is, up to the parameterization, uniquely determined and represents a c10sed geodesic. The question of existence of a c10sed geodesic on a simply connected c10sed surface is much more difficult. As pointed out by Poincare [po 1] in 1905, this problem has much in common with the problem ofthe existence of periodic orbits in the restricted three body problem. Poincare [l.c.] outlined a proof that on an analytic convex surface which does not differ too much from the standard sphere there always exists at least one c10sed geodesic of elliptic type, i. e., the corres­ ponding periodic orbit in the geodesic flow is infinitesimally stable.

This English edition could serve as a text for a first year graduate course on differential geometry, as did for a long time the Chicago Notes of Chern mentioned in the Preface to the German Edition. Suitable references for ordin­ ary differential equations are Hurewicz, W. Lectures on ordinary differential equations. MIT Press, Cambridge, Mass., 1958, and for the topology of surfaces: Massey, Algebraic Topology, Springer-Verlag, New York, 1977. Upon David Hoffman fell the difficult task of transforming the tightly constructed German text into one which would mesh well with the more relaxed format of the Graduate Texts in Mathematics series. There are some e1aborations and several new figures have been added. I trust that the merits of the German edition have survived whereas at the same time the efforts of David helped to elucidate the general conception of the Course where we tried to put Geometry before Formalism without giving up mathematical rigour. 1 wish to thank David for his work and his enthusiasm during the whole period of our collaboration. At the same time I would like to commend the editors of Springer-Verlag for their patience and good advice. Bonn Wilhelm Klingenberg June,1977 vii From the Preface to the German Edition This book has its origins in a one-semester course in differential geometry which 1 have given many times at Gottingen, Mainz, and Bonn.

A working knowledge of differential forms so strongly illuminates the calculus and its developments that it ought not be too long delayed in the curriculum. On the other hand, the systematic treatment of differential forms requires an apparatus of topology and algebra which is heavy for beginning undergraduates. Several texts on advanced calculus using differential forms have appeared in recent years. We may cite as representative of the variety of approaches the books of Fleming [2], (1) Nickerson-Spencer-Steenrod [3], and Spivak [6]. . Despite their accommodation to the innocence of their readers, these texts cannot lighten the burden of apparatus exactly because they offer a more or less full measure of the truth at some level of generality in a formally precise exposition. There. is consequently a gap between texts of this type and the traditional advanced calculus. Recently, on the occasion of offering a beginning course of advanced calculus, we undertook the expe- ment of attempting to present the technique of differential forms with minimal apparatus and very few prerequisites. These notes are the result of that experiment. Our exposition is intended to be heuristic and concrete. Roughly speaking, we take a differential form to be a multi-dimensional integrand, such a thing being subject to rules making change-of-variable calculations automatic. The domains of integration (manifolds) are explicitly given "surfaces" in Euclidean space. The differentiation of forms (exterior (1) Numbers in brackets refer to the Bibliography at the end.

This book is an introductory graduate-level textbook on the theory of smooth manifolds. Its goal is to familiarize students with the tools they will need in order to use manifolds in mathematical or scientific research–- smooth structures, tangent vectors and covectors, vector bundles, immersed and embedded submanifolds, tensors, differential forms, de Rham cohomology, vector fields, flows, foliations, Lie derivatives, Lie groups, Lie algebras, and more. The approach is as concrete as possible, with pictures and intuitive discussions of how one should think geometrically about the abstract concepts, while making full use of the powerful tools that modern mathematics has to offer.