Introduction to Differential Geometry with Tensor Applications

Dipankar De, PhD, received his BSc and MSc in mathematics from the University of Calcutta, India and his PhD in mathematics from Tripura University, India. He has over 40 years of teaching experience and is an associate professor and guest lecturer in India. He has published many research papers in various reputed journal in the field of fuzzy mathematics and differential geometry.

• Preface xvAbout the Book xviiIntroduction 1Part I: Tensor Theory 71 Preliminaries 91.1 Introduction 91.2 Systems of Different Orders 91.3 Summation Convention Certain Index 101.3.1 Dummy Index 111.3.2 Free Index 111.4 Kronecker Symbols 111.5 Linear Equations 141.6 Results on Matrices and Determinants of Systems 151.7 Differentiation of a Determinant 181.8 Examples 191.9 Exercises 232 Tensor Algebra 252.1 Introduction 252.2 Scope of Tensor Analysis 252.2.1 n-Dimensional Space 262.3 Transformation of Coordinates in S n 272.3.1 Properties of Admissible Transformation of Coordinates 302.4 Transformation by Invariance 312.5 Transformation by Covariant Tensor and Contravariant Tensor 322.6 The Tensor Concept: Contravariant and Covariant Tensors 342.6.1 Covariant Tensors 342.6.2 Contravariant Vectors 352.6.3 Tensor of Higher Order 402.6.3.1 Contravariant Tensors of Order Two 402.6.3.2 Covariant Tensor of Order Two 412.6.3.3 Mixed Tensors of Order Two 422.7 Algebra of Tensors 432.7.1 Equality of Two Tensors of Same Type 452.8 Symmetric and Skew-Symmetric Tensors 452.8.1 Symmetric Tensors 452.8.2 Skew-Symmetric Tensors 462.9 Outer Multiplication and Contraction 512.9.1 Outer Multiplication 512.9.2 Contraction of a Tensor 532.9.3 Inner Product of Two Tensors 542.10 Quotient Law of Tensors 562.11 Reciprocal Tensor of a Tensor 582.12 Relative Tensor, Cartesian Tensor, Affine Tensor, and Isotropic Tensors 602.12.1 Relative Tensors 602.12.2 Cartesian Tensors 632.12.3 Affine Tensors 632.12.4 Isotropic Tensor 642.12.5 Pseudo-Tensors 642.13 Examples 652.14 Exercises 713 Riemannian Metric 733.1 Introduction 733.2 The Metric Tensor 743.3 Conjugate Tensor 753.4 Associated Tensors 773.5 Length of a Vector 843.5.1 Length of Vector 843.5.2 Unit Vector 853.5.3 Null Vector 863.6 Angle Between Two Vectors 863.6.1 Orthogonality of Two Vectors 873.7 Hypersurface 883.8 Angle Between Two Coordinate Hypersurfaces 893.9 Exercises 954 Tensor Calculus 974.1 Introduction 974.2 Christoffel Symbols 974.2.1 Properties of Christoffel Symbols 984.3 Transformation of Christoffel Symbols 1104.3.1 Law of Transformation of Christoffel Symbols of 1st Kind 1104.3.2 Law of Transformation of Christoffel Symbols of 2nd Kind 1114.4 Covariant Differentiation of Tensor 1134.4.1 Covariant Derivative of Covariant Tensor 1144.4.2 Covariant Derivative of Contravariant Tensor 1154.4.3 Covariant Derivative of Tensors of Type (0,2) 1164.4.4 Covariant Derivative of Tensors of Type (2,0) 1184.4.5 Covariant Derivative of Mixed Tensor of Type (s, r) 1204.4.6 Covariant Derivatives of Fundamental Tensors and the Kronecker Delta 1204.4.7 Formulas for Covariant Differentiation 1224.4.8 Covariant Differentiation of Relative Tensors 1234.5 Gradient, Divergence, and Curl 1294.5.1 Gradient 1304.5.2 Divergence 1304.5.2.1 Divergence of a Mixed Tensor (1,1) 1324.5.3 Laplacian of an Invariant 1364.5.4 Curl of a Covariant Vector 1374.6 Exercises 1415 Riemannian Geometry 1435.1 Introduction 1435.2 Riemannian-Christoffel Tensor 1435.3 Properties of Riemann-Christoffel Tensors 1505.3.1 Space of Constant Curvature 1585.4 Ricci Tensor, Bianchi Identities, Einstein Tensors 1595.4.1 Ricci Tensor 1595.4.2 Bianchi Identity 1605.4.3 Einstein Tensor 1665.5 Einstein Space 1705.6 Riemannian and Euclidean Spaces 1715.6.1 Riemannian Spaces 1715.6.2 Euclidean Spaces 1745.7 Exercises 1756 The e-Systems and the Generalized Kronecker Deltas 1776.1 Introduction 1776.2 e-Systems 1776.3 Generalized Kronecker Delta 1816.4 Contraction of δijk αβγ 1836.5 Application of e-Systems to Determinants and Tensor Characters of Generalized Kronecker Deltas 1856.5.1 Curl of Covariant Vector 1896.5.2 Vector Product of Two Covariant Vectors 1906.6 Exercises 192Part II: Differential Geometry 1937 Curvilinear Coordinates in Space 1957.1 Introduction 1957.2 Length of Arc 1957.3 Curvilinear Coordinates in E 3 2007.3.1 Coordinate Surfaces 2017.3.2 Coordinate Curves 2027.3.3 Line Element 2057.3.4 Length of a Vector 2067.3.5 Angle Between Two Vectors 2077.4 Reciprocal Base Systems 2107.5 Partial Derivative 2167.6 Exercises 2198 Curves in Space 2218.1 Introduction 2218.2 Intrinsic Differentiation 2218.3 Parallel Vector Fields 2268.4 Geometry of Space Curves 2288.4.1 Plane 2318.5 Serret-Frenet Formula 2338.5.1 Bertrand Curves 2358.6 Equations of a Straight Line 2528.7 Helix 2548.7.1 Cylindrical Helix 2568.7.2 Circular Helix 2588.8 Exercises 2629 Intrinsic Geometry of Surfaces 2659.1 Introduction 2659.2 Curvilinear Coordinates on a Surface 2659.3 Intrinsic Geometry: First Fundamental Quadratic Form 2679.3.1 Contravariant Metric Tensor 2709.4 Angle Between Two Intersecting Curves on a Surface 2729.4.1 Pictorial Interpretation 2749.5 Geodesic in R n 2779.6 Geodesic Coordinates 2899.7 Parallel Vectors on a Surface 2919.8 Isometric Surface 2929.8.1 Developable 2939.9 The Riemannian–Christoffel Tensor and Gaussian Curvature 2949.9.1 Einstein Curvature 2969.10 The Geodesic Curvature 3089.11 Exercises 31910 Surfaces in Space 32110.1 Introduction 32110.2 The Tangent Vector 32110.3 The Normal Line to the Surface 32410.4 Tensor Derivatives 32910.5 Second Fundamental Form of a Surface 33210.5.1 Equivalence of Definition of Tensor b αβ 33310.6 The Integrability Condition 33410.7 Formulas of Weingarten 33710.7.1 Third Fundamental Form 33810.8 Equations of Gauss and Codazzi 33910.9 Mean and Total Curvatures of a Surface 34110.10 Exercises 34711 Curves on a Surface 34911.1 Introduction 34911.2 Curve on a Surface: Theorem of Meusnier 35011.2.1 Theorem of Meusnier 35311.3 The Principal Curvatures of a Surface 35811.3.1 Umbillic Point 36011.3.2 Lines of Curvature 36111.3.3 Asymptotic Lines 36211.4 Rodrigue’s Formula 37611.5 Exercises 37912 Curvature of Surface 38112.1 Introduction 38112.2 Surface of Positive and Negative Curvatures 38112.3 Parallel Surfaces 38312.3.1 Computation of aαβ and b αβ 38312.4 The Gauss-Bonnet Theorem 38712.5 The n-Dimensional Manifolds 39112.6 Hypersurfaces 39412.7 Exercises 395Part III: Analytical Mechanics 39713 Classical Mechanics 39913.1 Introduction 39913.2 Newtonian Laws of Motion 39913.3 Equations of Motion of Particles 40113.4 Conservative Force Field 40313.5 Lagrangean Equations of Motion 40513.6 Applications of Lagrangean Equations 41113.7 Himilton’s Principle 42313.8 Principle of Least Action 42713.9 Generalized Coordinates 43013.10 Lagrangean Equations in Generalized Coordinates 43213.11 Divergence Theorem, Green’s Theorem, Laplacian Operator, and Stoke’s Theorem in Tensor Notation 43813.12 Hamilton’s Canonical Equations 44213.12.1 Generalized Momenta 44313.13 Exercises 44414 Newtonian Law of Gravitations 44714.1 Introduction 44714.2 Newtonian Laws of Gravitation 44714.3 Theorem of Gauss 45114.4 Poisson’s Equation 45314.5 Solution of Poisson’s Equation 45414.6 The Problem of Two Bodies 45614.7 The Problem of Three Bodies 46214.8 Exercises 467Appendix A: Answers to Even-Numbered Exercises 469References 473Index 475