Differential Geometry: A First Course is an introduction to the classical theory of space curves and surfaces offered at the Graduate and Post- Graduate courses in Mathematics. Based on Serret-Frenet formulae, the theory of space curves is developed and concluded with a detailed discussion on fundamental existence theorem. The theory of surfaces includes the first fundamental form with local intrinsic properties, geodesics on surfaces, the second fundamental form with local non-intrinsic properties and the fundamental equations of the surface theory with several applications.

• Motivation of different concepts • Outline of the methods of proofs before giving details • Simple but interesting immediate illustrations in each section after the theory • Illuminating miscellaneous examples • Illustrations of the methods of proofs of the fundamental theorems of space curves and surfaces • Exercises with hints

Theory of Space Curves: Introduction / Representation of space curves / Unique parametric representation of a space curve / Arc – length / Tangent and Osculating Plane / Principal normal and binormal / Curvature and Torsion / Behaviour of a curve near one of its points / Curvature and torsion of the curve of intersection of two surfaces / Contact between curves and surfaces / Osculating circle and osculating sphere / Locus of centres of spherical curvature / Tangent surfaces, Involutes and evolutes / Betrand curves / Spherical Indicatrix / Intrinsic equations of space curves / Fundamental Existence Theorem for space curves / Helices / Examples 1 / Exercises 1 / The First Fundamental Form and Local Intrinsic Properties of a Surface: Introduction / Definition of a surface / Nature of points on a surface / Representation of a surface / Curves on surfaces / Tangent plane and surface normal / The general surface of revolution / Helicoids / Metric on a surface / Direction coefficients on a surface / Families of curves / Orthogonal Trajectories / Double Family of curves / Isometric correspondence / Intrinsic properties / Examples II / Exercises II / Geodesics on a Surface: Introduction / Geodesics and their differential equations / Canonical geodesic equations / Geodesics on surfaces of revolution / Normal property of geodesics / Differential equations of geodesics using normal property / Existence theorems / Geodesic parallels / Geodesic curvature / Gauss – Bonnet theorem / Gaussian Curvature / Surfaces of constant curvature / Conformal mapping / Geodesic mapping / Examples III / Exercises III / The Second Fundamental form and local Non - Intrinsic Properties of Surfaces: Introduction / The second fundamental form / The Classification of points on a surface / Principal curvatures / Lines of curvature / The Dupin indicatrix / Developable surfaces / Developables associated with space curves / Developables associated with curves on surfaces / Minimal surfaces / Ruled surfaces / Three fundamental forms / Examples IV / Exercises IV / The Fundamental Equations of Surface Theory: Introduction / Tensor notations / Gauss equations / Weingarten Equations / Mainardi – Codazzi equations / Parallel Surfaces / Fundamental existence theorem for surfaces / Examples V / Exercises V

Graduate and Postgraduate Students