W. Klingenberg

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.

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.

Das vorliegende Buch ist aus einer 1-semestrigen Vorlesung iiber DifTerentialgeometrie entstanden, die ich wiederholt in Gottingen, Mainz und Bonn gehalten habe. Mit dieser Vorlesung verfolgte ich das Ziel, den Studenten mittlerer Semester, die die Anfangervorlesungen gemeistert haben, eine Einfiihrung in die klassische Differentialgeometrie der Kurven und FIachen anzubieten und damit eine Alternative zu offerieren zu anderen Vorlesungen fiir mittlere Semester, wie etwa Funktionen theorie, Hohere Algebra oder Algebraische Topologie. Fiir den groBten Teil der Vorlesung wird nichts weiter als eine griindliche Kenntnis der Vorlesung iiber Analysis sowie die Kennt nis der reellen linearen Algebra und der euklidischen Geometrie vorausgesetzt. Nur in spiiteren Kapiteln, wo ich auch globale Fragen behandle, ist eine gewisse Vertrautheit mit der Topologie kompakter Fliichen von Nutzen; dabei wird aber nichts benutzt, was sich nicht in dem klassischen Topologie-Lehrbuch von Seifert und Threlfall findet. Fiir eine Obersicht iiber den Inhalt der Vorlesung verweise ich auf das Verzeichnis. Natiirlich muBte ich eine Auswahl treffen aus der Fiille des Materials, das fiir eine solche Vorlesung zur Verfiigung steht. Fiir mich ergab es sich ganz von seIber, daB da bei solche Gegenstiinde bevorzugt wurden, die der 2-dimensionalen riemannschen Geometrie zuzurechnen sind. Dennoch denke ich, daB meine Vorlesung eine brauchbare Grundlage fiir das Verstiindnis aller Teilgebiete der Differentialgeometrie liefert."