Introduction to Differential Geometry has been planned for use, in an Honours Mathematics Course or as an introduction to the subject at postgraduate level.
This textbook gives all that is likely to be required at the undergraduate level and most of the material has in fact been taught to undergraduate. The book also gives a useful introduction to the methods of differential geometry or to tensor calculus for research students (e.g. in physics or engineering) who may wish to apply them.Part I is devoted to the classical theory of curves and surfaces, vector methods being used throughout. The last chapter dealing with global differential geometry of surface contains material which does not appear in any standard English text.Part II introduces the idea of a tensor, first in algebra and then in calculus. It gives the basic theory of absolute calculus and fundamentals of Riemannian geometry. The final chapter gives a brief account of the application of tensor methods to yield results previously obtained in Part I and some new results in addition.
1. Several worked examples and exercises have been interpreted in the text and, with the exception of Chapter IV, each chapter concludes with a set of exercises designed to test the understanding of the subject matter that has been learnt.2. A list of references is given at the end of each chapter and, at the end of the book, a collection of miscellaneous exercises.
PART ITHE THEORY OF CURVES AND SURFACES IN THREE DIMENTIONAL EUCLIDEAN SPACEThe Theory of Space CurvesThe Metric: Local Intrinsic Properties of a SurfaceThe Second Fundamental Form: Local Non-intrinsic Properties of a surface.Differential Geometry of the Surfaces in the LargePART IIDIFFERENTIAL GEOMETRY OF n- DIMENSIONAL SPACETensor Algebra.Tensor Calculus.Riemannian GeometryApplications for Tensor Methods to Surface Theory.EXERCISESSUGGESTIONS FOR FURTHER READINGINDEX