AM 303 : DIFFERENTIAL GEOMETRY (Common with paper MA 304 of Branch I(A) Mathematics) UNIT : –I The Theory Space Curves: Introductory remarks about space curves –Definitions – Arc length-Tangent, normal, and binormal and torsion of a curve given as –Curvature the intersection of two surfaces between curves and surfaces-Tangent –Contact surface, involutes and evolutes.( Sections 1 to 7 of Chapter 1). UNIT II: – The Metric: Local Intrinsic Properties of a Surface: Definitions of a Surface- Curves on a surface-Surfaces of revolution Metric-Direction –Helicoids- Families of Curves correspondence properties. ( Sections 1 to 9 –Isometric –Intrinsic of Chapter 11). UNIT III: – Geodesics-Canonical Geodesic Equations-Normal Property of geodesics –Existence theorems-Geodesic parallels-Geodesic curvature-Gauss-Bonnet theorems –Gaussian curvature surfaces of constant curvature mapping-Geodesic mapping –conformal (Sections 10 to 20 of Chapter 11). UNIT-IV: The second Fundamental Form: Local non intrinsic properties of a surface: The – second fundamental form-principal curvatures of curvature -Developables –Lines associated with space curves-Developables associated with curves on surfaces – Minimal surfaces-Ruled surfaces-The fundamental equations of surface theory – Parallel surfaces. ( Sections 1 to 10 Chapter III). Scope and Standard as in Sections and chapters as specified above of the book An “ Introduction to Differential Geometry" of T.J Willmore, Oxford University Press, Thirteenth Impression, 1997. References : 1. A first course in Differential Geometry- D. Soma sundaram Narosa – Publications
AM 303 : DIFFERENTIAL GEOMETRY (Common with paper MA 304 of Branch I(A) Mathematics) UNIT –I : The Theory Space Curves: Introductory remarks about space curves –Definitions – Arc length-Tangent, normal, and binormal –Curvature and torsion of a curve given as the intersection of two surfaces –Contact between curves and surfaces-Tangent surface, involutes and evolutes.( Sections 1 to 7 of Chapter 1). UNIT – II: The Metric: Local Intrinsic Properties of a Surface: Definitions of a Surface- Curves on a surface-Surfaces of revolution –Helicoids- Metric-Direction of Curves –Isometric correspondence –Intrinsic properties. ( Sections 1 to 9 of Chapter 11). UNIT – III: Geodesics-Canonical Geodesic Equations-Normal Property of geodesics –Existence theorems-Geodesic parallels-Geodesic curvature-Gauss-Bonnet theorems –Gaussian curvature surfaces of constant curvature –conformal mapping-Geodesic mapping (Sections 10 to 20 of Chapter 11). UNIT-IV: The second Fundamental Form: Local non – intrinsic properties of a surface: The second fundamental form-principal curvatures –Lines of curvature -Developables associated with space curves-Developables associated with curves on surfaces – Minimal surfaces-Ruled surfaces-The fundamental equations of surface theory – Parallel surfaces. ( Sections 1 to 10 Chapter III). Scope and Standard as in Sections and chapters as specified above of the book “ An Introduction to Differential Geometry" of T.J Willmore, Oxford University Press, Thirteenth Impression, 1997. References : 1. A first course in Differential Geometry- D. Soma sundaram – Narosa Publications Families