Differential Geometry 1
                               Summer Term 2008
                                Michael Kunzinger
                             michael.kunzinger@univie.ac.at
                                  Universit¨at Wien
                                Fakult¨at fu¨r Mathematik
                                  Nordbergstraße 15
                                   A-1090 Wien
            Preface
            These lecture notes form the basis of an introductory course on differential geom-
            etry which I first held in the summer term of 2006. Several boundary conditions
            made the choice of material to be included quite delicate. On the one hand, in
            the mathematics curriculum of the Faculty of Mathematics in Vienna, the course
            ‘Differential Geometry 1’ is the only compulsory course on the subject for students
            not specializing in geometry and topology. On the other hand, the course duration
            is only three hours per week. Therefore, an approach which first focuses on clas-
            sical differential geometry and then gently moves on to the theory of differentiable
            manifolds is ruled out by time constraints.
            The course therefore puts its main emphasis on a concise introduction to modern
            differential geometry in order to provide the necessary tools for applications in
            other branches of mathematics or for a continued study of differential geometry.
            Nevertheless, an introduction to local curve theory in chapter 1 and applications
            to the theory of hypersurfaces in chapter 3 are intended to provide a link to more
            classical aspects of the subject.
            Throughout I have tried to motivate all basic concepts thoroughly. As a rule, all
            proofs are given in full detail, and comprehensibility is given prevalence over ele-
            gance whenever the need arises. I have also refrained from including more material
            than can be covered in one semester in order to make a clear statement on what I
            consider essential in an introductory course of this kind.
            I would like to thank Christoph Marx who typed a first (German) version of these
            notes and David Langer who supplied the beautiful pictures and diagrams included
            here. Also, I am grateful for many comments of students participating in the course
            which, I hope, have led to improvements in the text. Further comments and cor-
            rections are always welcome!
                               Michael Kunzinger, summer term 2008
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