I will not necessarily claim that one must know differential manifolds to study differential geometry but it is entirely possible that it is necessary in order to understand the subject properly. michael spivak's treament spends the first volume on a detailed introduction to differential manifolds and then in the second volume, explains in equal detail the fundamental controbutions to differential geometry of Gauss and Riemann, including translations of their original papers. Of course the theory in spivak's volume one did not exist when gauss and riemann did their work, so it cannot have been necessary to it, but it is very helpful to us in understadning what they did. here are spivaks own remarks from his preface:
"For many years I have wanted to write the Great American Differential Geometry book. Today a dilemma confronts anyone intent on penetrating the mysteries of differential geometry. On the one hand, one can consult numerous classical treatments of the subject in an attempt to form some idea how the concepts within it developed. Unfortunately, a modern mathematical education tends to make classical mathematical works inaccessible, particularly those in differential geometry. On the other hand, one can now find texts as modern in spirit, and as clean in exposition, as Bourbaki's Algebra. But a thorough study of these books usually leaves one unprepared to consult classical works, and entirely ignorant of the relationship between elegant modern constructions and their classical counterparts. ... no one denies that modern definitions are clear, elegant, and precise; it's just that it's impossible to comprehend how anyone ever thought of them. And even after one does master a modern treatment of differential geometry, other modern treatments often appear simply to be about totally different subjects.
There are two main premises on which these notes are based. The first premise is that it is absurdly inefficient to eschew the modern language of manifolds, bundles, forms, etc., which was developed precisely in order to rigorize the concepts of classical differential geometry.
The second premise for these notes is that in order for an introduction to differential geometry to expose the geometric aspect of the subject, an historical approach is necessary; there is no point in introducing the curvature tensor without explaining how it was invented and what it has to do with curvature. The second volume of these notes gives a detailed exposition of the fundamental papers of Gauss and Riemann."
i myself suggest getting both books and reading volume two for the exciting geometry and referring back to volume one as one finds oneself wanting to understand volume two better. i.e. use vol 2 as motiovation to plow through vol 1.
). Keep up the good work.
